In this third lecture, we shall explore the paradox of giants, showcasing Galileo’s argument that the traditional giants of folklore—taking human form but at much larger scale—are physically impossible. He argued on the basis of an understanding of how size scales differently in different dimensions. Similar ideas lead to the paradox of Gabriel’s horn, the painter’s paradox, paradoxical fractals, and to many further paradoxes of dimension. By the end of the lecture, we shall glimpse some genuinely troubling conundrums in high dimension.

I shall be sharing the individual infinity lectures here on Infinitely More in the coming weeks and months.

Please enjoy!

• Find the lectures here on Infinitely More in the lectures-on-infinity tag.

• The lectures will appear on YouTube.

• The whole lecture course is hosted at Ergo: Lectures on Infinity.

• Find other philosophy lecture courses at Ergo.org.

• This lecture is based on my essay The Paradox of Giants.

• The essay also appears in my new book, The Book of Infinity.

The Paradox of Giants—Strange Consequences in High Dimension

A lightly edited transcript. Timestamps link to the video on the Ergo website.

0:02

Giants of Legend and Literature

Welcome to these lectures on infinity. Today, I want to tell you about the paradox of giants. According to legend, giants once roamed the Earth. Everyone knows that Odysseus met the Cyclops, who lived in a great cave and grabbed sheep and men with his hands and ate them whole. And in the time of King Arthur, there was the young boy who earned the title Jack the Giant Killer because he was able to use his sharp wit to outsmart and slay the various giants that plagued the land.

It is the same Jack, I think, as the Jack of Jack and the Beanstalk, who planted the seeds that grew into the beanstalk, climbed to the castle in the sky, and tricked that giant as well. But there is also Jonathan Swift’s character Gulliver, who travels to distant lands and finds the Lilliputians, those tiny human beings to whom Gulliver himself seemed the giant. And yet in those same travels he also encountered the Brobdingnagians, who were giants to whom Gulliver seemed Lilliputian, even though Gulliver had never changed his size at all.

In all of these legends, the giants tend to have an ordinary human form and they do ordinary human things. They walk around, they stomp, they dance, they drink wine from goblets, they carry heavy stones, they climb ladders. They move in a human manner, but simply at scale.

1:47

Why Physics Makes Giants Impossible

Galileo wrote in his Dialogues Concerning Two New Sciences a wonderful criticism of this whole manner of thinking about giants, arguing that giants are actually impossible. Physics cannot work like that. The very idea of a giant is, on his account, contradictory. Let me explain his argument.

He asks us to imagine a great oak beam, sturdy enough to hold up a heavy stone or a load of bricks. Now imagine making it ten times bigger, keeping exactly the same dimensions and proportions but scaling everything up: ten times longer, ten times thicker, ten times wider, made of the same material. This larger beam might be the kind of beam you would find in a giant’s house, as opposed to the ordinary beam in our own.

Of course, we expect the bigger beam to be stronger and able to hold a greater load. But how much stronger, exactly? Galileo argued that the strength of a beam is related to its cross-sectional area, because the fibers of the wood run lengthwise, and when the beam breaks, it breaks along the cross-section. It is the strength of the fibers passing through that cross-section that determines the overall strength of the beam. If the beam is ten times bigger in every direction, the cross-sectional area is multiplied by one hundred, since it scales as ten times ten. So the larger beam is one hundred times stronger, which seems quite promising.

But now consider the stone the beam was holding up. If we make the stone ten times bigger in every direction, its volume increases by a factor of one thousand, since it scales as ten times ten times ten across all three dimensions. A stone ten times larger in every direction therefore weighs one thousand times as much, assuming it is made of the same material. Suddenly the situation looks far less favorable: the beam is one hundred times stronger, but the load it must bear is one thousand times heavier.

If the original beam was just barely holding up the original stone, the scaled-up beam will not be strong enough to hold the scaled-up stone. Galileo went further still, arguing that the beam would not even be able to support its own weight, because the mass of the enlarged beam itself grows by that same factor of one thousand, while its strength grows by only one hundred. The structure collapses under itself. This is why, on Galileo’s reasoning, giants are not merely unlikely but physically impossible.

5:31

The Square-Cube Law Destroys Giants

Galileo argued that because of this difference in dimension, strength increases with the square of the scaling, but mass increases with the cube of the scaling, which is significantly greater. This mismatch between the two quantities means that the whole concept of a giant becomes incoherent. The beams of the giant’s house would not be able to hold up the roof. The goblet that is ten times bigger would not be able to hold the wine it contains. The giant would not be able to climb a ladder, because the ladder would not even be able to support itself, let alone a giant.

The bones of a giant are essentially beams, and if we take a human being and scale up to make the giant ten times bigger in every direction, ten times taller, ten times thicker, and so on, then the bones become one hundred times stronger, but the mass of the giant increases by one thousand. Therefore the giant will not be able to stand up or walk around. The whole concept of a giant is incoherent.

Galileo writes: “Clearly then, if one wishes to maintain in a great giant the same proportion of limb as that found in an ordinary man, he must either find a harder and stronger material for making the bones, or he must admit a diminution of strength in comparison with men of medium stature, for if his height be increased inordinately, he will fall and be crushed under his own weight.”

7:22

Why Elephants Are Stocky and Bugs Are Thin

We can see this as almost obvious if we think about the nature of large animals versus small animals. Consider the typical large animals: elephants, rhinoceroses, and so on. They are characteristically stocky, with very stocky limbs. The reason is precisely the dimensional issue that Galileo pointed out: in order to support greater weight, the limbs need to be not only bigger, but proportionally bigger, and that is what produces a stocky animal.

The same principle works in the other direction. Tiny animals and insects typically have very slender, thin limbs, and yet they support their weight just fine. But if you were to take a housefly and make it ten times bigger, it could no longer crawl along a wall, because the electrostatic forces would simply not be strong enough. Making it ten times bigger makes it one thousand times heavier, and the electrostatic forces do not scale to match that increase.

Similarly, a bug that walks on the surface of water relies on surface tension, and surface tension does not scale in the right way either. The paradox of giants, then, is that you cannot simply take a functioning animal with a given architecture, scale it up, and expect it to work in the same way. It simply will not work that way.

8:53

The Paradox of Miniature Humans

A similar issue arises not just with giants but with what we might call the paradox of the miniature human. You may have encountered this theme in Hollywood films such as Downsizing, Ant-Man, or Honey, I Shrunk the Kids, or indeed in the Lilliputians of Gulliver’s Travels. The idea of a miniature human is a recurring cultural fascination, but it runs into the same scaling problems we have been discussing.

If you take an ordinary human and make them ten times smaller, they will be proportionally stronger for their height, for exactly the same kind of reason we considered with giants. This is precisely why grasshoppers can jump many times their own height: a tiny human would likewise be able to jump very high in comparison with their height, not in absolute terms, but relative to its new, smaller stature. Such a creature would not move through the world the way ordinary humans do.

Interacting with water, for instance, would become very complicated, because at that scale water would behave as far stickier than it does at our scale. The nature of physical existence simply does not scale in that way, and this is the core of Galileo’s argument.

10:09

Evolution and Body Size Genes

This is related to evolution and body size. It seems to be the case that the body-size architecture for many different kinds of animals must be controlled by relatively few genes, because when you look at the evolutionary history of certain animals, their size varies quite a lot. In prehistoric times, for instance, there were enormous dragonflies, far larger than the ones we see today. Horses, too, were much smaller when they first evolved, and their size went up and down repeatedly over their evolutionary history.

We can see some residual evidence of this in miniature horse breeds, those very tiny horses that still exist, which carry what we might call smallness genes still present in the horse population. One can imagine natural selection acting on those genes to produce changes in the body-size architecture of a species. There might be some advantage to becoming larger, even though greater size makes an animal heavier and proportionately less strong, if that size helps the animal compete more effectively within its ecological niche. So it is easy to imagine evolution acting on those genes to shift body size in response to environmental pressures.

There are closely related effects that arise from differences in dimension, particularly the difference between surface area and volume. I want to turn now to some interesting mathematical examples that illustrate this distinction.

11:56

Gabriel’s Horn Has Finite Volume

One of my favorite examples is the Paradox of Gabriel’s Horn. We begin with the function y = 1/x, looking at the portion of the curve starting at x = 1 and extending out to infinity. To form Gabriel’s Horn, we take that curve and revolve it around the x-axis, producing a kind of symmetric shape. It is, in a sense, like a horn from heaven, perhaps sounding some sonorous, multi-toned note, and that is why it carries the name Gabriel’s Horn.

Now, the paradox centers on a simple question: what is the volume of Gabriel’s Horn? It is an infinite object, since it extends forever, but the function is 1/x, so the horn becomes very thin as we move far out along the x-axis. We can compute the volume using a standard technique from calculus for finding the volume of a solid of revolution. The idea is to slice the solid into thin disks perpendicular to the x-axis.

At a given point x, the radius of such a disk is 1/x, since that is the value of the function being rotated, and the thickness of the disk is dx. The volume of one disk is therefore the area of the disk times its thickness, and since the area is π r squared, the volume of a single disk is π times (1/x)² dx. To find the total volume, we integrate this expression from 1 to infinity, adding up the contributions of all the disks. The total volume is thus the integral from 1 to infinity of π over x squared dx.

This is an elementary calculus integral. The antiderivative of 1/x squared is minus 1/x, so we evaluate minus π/x from 1 to infinity. As x goes to infinity, minus π/x approaches zero, and by the Fundamental Theorem of Calculus we subtract the value at x = 1, giving zero minus (minus π/1), which equals π. The volume of Gabriel’s Horn is precisely π, a finite number. That is the first paradoxical observation: Gabriel’s Horn is an infinite object, yet it encloses a perfectly finite volume.

16:10

Gabriel’s Horn Has Infinite Surface Area

The second part of the paradox is to ask: what is the surface area of Gabriel’s horn? Rather than computing the volume of each disk, we now concentrate on the band around the outside, which is what is called the frustum of a cone. It is slightly angled, and the infinitesimal length along that angled piece is commonly written as ds, equal to the square root of dx squared plus dy squared. Factoring out a dx, this becomes the square root of 1 plus (dy/dx) squared, times dx.

The total surface area is therefore the integral from one to infinity of the circumference of each frustum times that infinitesimal slant length. The circumference is π times the diameter, which gives 2π over x, so the surface area integral becomes the integral from one to infinity of (2π over x) times the square root of 1 plus (dy/dx) squared, dx. Since y equals 1 over x, which is x to the minus one, we get dy/dx equals minus 1 over x squared, so (dy/dx) squared equals 1 over x to the fourth.

The resulting integral is more complicated, but we can sidestep the difficulty with a simple observation. The square root of 1 plus 1 over x to the fourth is always at least 1, so the surface area is greater than or equal to the integral from one to infinity of 2π over x, dx. That integral equals 2π times the natural log of x, evaluated from one to infinity, which diverges to infinity. Therefore the surface area of Gabriel’s horn is infinite.

18:20

Can You Paint an Infinite Surface?

The surface area of Gabriel’s Horn is infinite, but its volume is finite. How can that be? Consider this: if we point the horn downward and fill it with paint, we use only a finite amount of paint, and that paint would be touching every part of the interior surface. It seems, then, that with a finite amount of paint we have painted Gabriel’s Horn. This is the heart of the paradox. Gabriel’s Horn is a geometrical object we can understand in a deep way, and yet it has finite volume and infinite surface area.

But does the filling argument really work? Should filling a container with paint count as painting its surface? I would say we are cheating a little, because Gabriel’s Horn grows thinner and thinner as it extends outward. The paint inside the horn is spread more and more thinly the farther out you go. If we require a uniform thickness of paint on the surface, say one millimeter, then eventually that condition is violated, because the horn itself becomes less than one millimeter across. So even though the horn is full of paint, it does not follow that we have painted the surface to any uniform thickness.

This reveals why filling Gabriel’s Horn with paint should not count as painting its surface: the paint is spread so thin in the region far out along the tail. And it is precisely that tail region which accounts for the infinite surface area. If we chop the tail off, what remains has obviously only a finite area. So the infinite area lives out in the part where the paint has been spread vanishingly thin. Filling the volume with paint is therefore entirely the wrong way to think about painting the surface, and the apparent paradox dissolves once we see how the argument was cheating all along.

20:55

Extended Real Numbers and Infinity

There is one thing I want to mention. I have been writing the infinity symbol ∞ on the board, and since this whole lecture series is about the infinite, I want to discuss this particular use of infinity, which is often the first instance of infinity that many students encounter in a mathematics class, in a calculus class. So what does this mean? What is that number? Is it a number? How should we think about it?

We begin with the real number system, the set of all real numbers. It is an ordered field: we can add and multiply its elements, compare their order, and identify them with points on the number line. From there, we have what is called the extended real numbers. This is a number system obtained by starting with the real numbers and simply adding infinity and minus infinity as idealized objects. We adjoin these two extra elements to the set and then define how arithmetic works with them.

For example, in the extended real numbers, infinity plus two equals infinity; indeed, adding any finite number to infinity leaves it infinite. Infinity plus infinity is infinity. Similarly, minus infinity plus any finite number remains minus infinity. As for multiplication, infinity times a equals infinity if a is positive, but equals minus infinity if a is negative, so infinity times minus five is minus infinity, and so on.

One has to keep in mind, however, that certain combinations are simply not defined in the extended real numbers. Infinity minus infinity has no meaning, and neither does infinity times zero. Within those constraints, you can work quite intuitively with these symbols according to these rules, and it is remarkable how far this way of treating infinity actually goes.

I think of it, philosophically, as ontologically very light, even deflationary in a way. It says: we do not need to give a robust or heavy meaning to infinity; we can simply add it as a symbol, define how to calculate with it, and things work out beautifully. It is rather remarkable that such a light attitude toward an apparently heavy concept can be so productive. For many mathematicians, this is precisely the use of infinity they encounter most often, and it is the one we already relied on when examining the nature of Gabriel’s horn.

25:05

Testing the Paint-Based Theory of Area

We discussed the idea of a paint-based theory of surface area. The proposal is this: a surface has finite area if and only if one can coat it to a uniform finite thickness using a finite volume of paint. If every point on the surface is covered to some fixed depth, say one millimeter, and the total volume of paint required is finite, then perhaps that is a reasonable criterion for saying the surface has finite area.

But let me criticize this proposal, because it does not quite work. Suppose we have an infinite line, such as the x-axis. A line has zero area, and yet one cannot cover it to a uniform thickness with a finite volume of paint, because any uniform coating around an infinite line would form an infinite cylinder, which has infinite volume. So here we have something with finite area, namely zero area, that nevertheless cannot be painted to uniform thickness with a finite volume of paint. This is a counterexample to the paint-based account.

One might object that a line is not a surface at all. It is a one-dimensional object, not a surface, and so it should not count as a test case for a theory about surfaces. Fair enough. So let me offer a different kind of counterexample, a variant of Gabriel’s horn.

27:34

A Modified Horn with Finite Area

In this modified version of Gabriel’s horn, I am using the function one over x squared instead of one over x. The two functions look roughly similar, but one over x squared decreases to zero far more rapidly. When x is 100, one over x squared equals one ten-thousandth, which is 100 times smaller than one one-hundredth. When x is a million, one over x squared is a million times smaller than one over x, since it equals one over a million times a million, and so on.

Because one over x squared goes to zero faster, the resulting horn tapers toward the x-axis much more quickly, though it never actually touches it. It is extremely thin in the tail, but we can construct a Gabriel’s horn-type surface from it in exactly the same way. The key difference is that for this version, both the surface area and the volume are finite. Recall that the paradox of the original Gabriel’s horn was the contrast between a finite volume and an infinite surface area; this modified horn has neither of those infinities.

Now consider what happens when we apply the paint-based criterion for finite surface area. The proposal was that a surface has finite area if and only if it can be painted to a uniform thickness using a finite volume of paint. If we try to apply a uniform coat of paint to this tighter, more rapidly tapering horn, we still run into trouble. Even though the horn is extremely close to the x-axis out in the tail, there remains a thin cylindrical shell of paint of, say, one millimeter radius running along that entire infinite tail, and covering it to a uniform thickness requires an infinite volume of paint.

So this modified Gabriel’s horn is a surface with finite surface area that nevertheless cannot be painted to uniform thickness with a finite volume of paint. Together with the original Gabriel’s horn, we now have two examples on the same side of the ledger: surfaces of finite area that fail the painting criterion. What I want to do next is produce a counterexample on the other side, namely a surface that satisfies the painting criterion but does not have finite area in the standard sense.

30:32

Koch Snowflake Breaks the Paint Rule

The paint-based criterion for finite surface area turns out to be wrong in both directions: it is neither necessary nor sufficient. To see why, I want to introduce an example we will return to more fully in a later lecture on the infinite coastline paradox and the concept of fractals. The example is the Koch snowflake curve.

The construction works as follows. You start with a line segment of a certain length, chop it into thirds, and replace the middle third with two sides of an equilateral triangle, producing a shape with a small outward kink. Where you had one segment, you now have four segments, each of length one-third. You then repeat the process: each of those four segments gets its own kink in the middle. You do this again, and again, adding smaller and smaller triangular bumps at every scale, producing a curve that is ever more wiggly at ever finer scales. If you carry this process all the way around a triangle rather than along a single segment, the resulting shape looks like a snowflake, which is why it bears that name.

The Koch snowflake curve has infinite length, and you can see why directly from the construction. Each iteration of the process replaces three segments of length one-third with four segments of length one-third, so the total length is multiplied by four-thirds at every step. Since this is done infinitely many times, and since these curves converge in a way that makes the infinite iteration well-defined, the length cannot be any finite value. A finite length would have to equal four-thirds times itself in order to satisfy the generation rule, which is impossible. So the length of the curve is infinite.

Now I want to build a surface out of this curve by extending it into a third dimension, producing something like a corrugated roof whose cross-section is exactly the snowflake curve. The surface is extremely wiggly in one direction but consists of straight lines in the other. I then enclose this corrugated lid in a rectangular box to make a solid. Because the snowflake curve has infinite length, the lid of this box has infinite area: there are so many nooks and crannies, at such fine scales, that the cross-sectional length is infinite, and therefore the area of the roof is infinite, larger than any finite quantity.

And yet the whole object is bounded. If you dunk it in a vat of paint, a finite amount of paint will cover every part of the surface to within one millimeter, thereby satisfying the paint-based criterion, even though the surface area is genuinely infinite. This is the opposite situation from Gabriel’s Horn. In the Koch box, one small region of paint simultaneously covers many different parts of the surface, because the surface folds back on itself so tightly that a one-millimeter thickening of the surface produces enormous overlaps. In the Gabriel’s Horn case, the geometry runs the other way: to cover even a tiny patch of surface area, you need a large volume of paint wrapping all the way around an extremely thin tube. One bit of paint covers very little surface there, whereas here one bit of paint covers a great deal.

This pair of examples together constitutes what I call the painter’s paradox. The paint-based account of finite surface area simply does not work, and these two constructions show exactly why: one gives infinite surface area that can be painted with finite paint, and the other gives finite surface area that cannot be painted with finitely much paint. With that, we can move on to some other paradoxes of higher dimension, beginning with curves in the plane.

37:47

Beautiful Spirals in the Plane

There are some beautiful curves that can be drawn in the plane. If you are familiar with polar coordinates, where a point is specified not by its x and y coordinates but by its radial and angular coordinates, then consider the curve r equals e to the minus theta, where theta is the angle and r is the radius. This specifies the radius as a function of the angle, and the resulting curve spirals inward, because as theta increases the value of e to the minus theta becomes very small, so the curve spirals very rapidly into the origin. This is called a logarithmic spiral, and one can prove that even though the curve winds around the origin infinitely many times, it still has finite length.

There is another spiral given by r equals theta, called the Archimedean spiral. A characteristic feature of the Archimedean spiral is that the spacing between successive arms is quite regular: each time you go around, the distance between turns is the same. If we traverse it inward toward the origin, we go around only finitely many times, and the curve has finite length.

Another example is the hyperbolic spiral, given by r equals one over theta. This curve also winds around the origin infinitely many times, but it approaches the origin more slowly, and it has infinite length. So we have a contrast: the logarithmic spiral winds around infinitely many times and has finite length, the hyperbolic spiral winds around infinitely many times and has infinite length, and the Archimedean spiral, traversed inward, winds around only finitely many times and has finite length. These examples illustrate some of the range of possible behavior for these one-dimensional curves in the plane.

With that, let us move to higher dimensions and ask: what is the volume of a sphere in higher dimensions?

40:50

Hypersphere Volume Peaks at Dimension 5

Let me begin with something familiar. The unit circle in dimension two has radius one, and its area is π r-squared, which at r equals one gives us simply π. Moving up to dimension three, the unit sphere has volume four-thirds π r-cubed, and again with r equal to one, that is four-thirds π. So from dimension two to dimension three, the hyper volume has increased by a factor of one-third. The natural question is what happens as we continue into higher dimensions.

Before going up, it is worth asking what happens when we go down. What is the one-dimensional sphere? A circle is the set of all points at distance one from a given center, and we can apply exactly that definition in one dimension. The result is just two points, one on each side of the center, forming a line segment of length two between them. The relevant notion of size in dimension one is length, in dimension two it is area, in dimension three it is volume, and in higher dimensions we call it hyper volume. All of these are instances of the same concept, and we can use the term hyper volume to cover all cases uniformly.

So the sequence begins: in dimension one, the hyper volume is two; in dimension two, it is π; in dimension three, it is four-thirds π. The question is whether this keeps increasing forever. It turns out there is a recurrence relation one can derive, which I will state without proof. If v sub n denotes the hyper volume of the unit sphere in dimension n, then v sub n equals two π over n, times v sub n minus two. In other words, if you know the hyper volume of the unit hypersphere two dimensions below, you multiply by two π over n to obtain the hyper volume in dimension n.

Applying this formula, we can build a table. v₄ equals two π over 4, times v₂, which is two π over 4 times π, giving π-squared over 2. v₅ works out to eight π-squared over 15. v₆ then comes to π-cubed over 6. In approximate decimal terms, π-squared over 2 is about 4.9, eight π-squared over 15 is approximately 5.264, and π-cubed over 6 is approximately 5.168. So the hyper volume increases up through dimension five and then begins to fall in dimension six.

We can see directly from the recurrence why this must happen. The factor two π over n is less than one whenever n is greater than two π, and two π is approximately 6.28. So for n equal to seven and beyond, each step multiplies the previous hyper volume by something less than one, and the values decrease. In fact, comparing dimension six with dimension four already shows a decrease relative to dimension five, which is why the maximum is achieved at dimension five rather than at six or seven. In all dimensions greater than five, the hyper volume of the unit hypersphere is strictly smaller, and it continues to shrink toward zero as the dimension grows. The upshot is that the hyper volume of the unit hypersphere is maximized in dimension five, which is, on reflection, a rather surprising fact.

47:27

Why Hypercubes Are All Corners

I want to talk about this in connection with the paradox of giants, because Galileo’s argument was fundamentally about understanding the nature of giants by understanding how scaling works in different dimensions. That is exactly what we are doing here. He was mainly concerned with dimensions up to three, but I see no reason to be limited to three dimensions only.

I want to understand hyperspheres and how they sit inside the cubes that naturally bound them. We have the unit circle sitting inside a square, the unit sphere sitting inside its bounding cube, and similarly, in higher dimensions, we have a hypercube bounding the corresponding hypersphere. In the one-dimensional case, the unit sphere and the unit cube are the same object. As the dimension increases, the sphere begins to fill less and less of the cube.

The question is: what fraction of the cube’s volume does the sphere fill? In two dimensions, we are asking what fraction of the area of the square lies inside the circle. The unit circle has radius one, so its area is π times one squared, which is π. The bounding square is two by two, since the diameter is two, so the fraction of the area inside the circle is π fourths, a little more than three quarters.

In three dimensions, the volume of the unit sphere is four thirds π, and the bounding cube is two by two by two, giving a volume of eight. The fraction is therefore π over six, which is already noticeably smaller. This makes intuitive sense: in the square there are only four small extra corner regions not covered by the circle, whereas in the cube there are eight corners, accommodating more of the volume outside the sphere.

What happens in higher dimensions? Recall the formula where the volume of the hypersphere is multiplied by two π over n when passing from dimension n to the next. The volume of the bounding hypercube is two to the n, since it is a product of n factors of two. So the ratio we care about is v sub n divided by two to the n. Each time the dimension increases, the denominator doubles, while the numerator is multiplied by two π over n. That fraction becomes tinier and tinier as n grows large.

The picture that emerges is striking. As the dimension increases, more and more of the points in the hypercube lie outside the sphere. The points near the center are precisely those inside the sphere, but the proportion of such points, compared to all points in the hypercube, goes to zero. Almost all the points in a high-dimensional hypercube are not near the center. Instead, they are concentrated in the corners. This is the phrase people use: the hypercube is very “endy.” Almost all the hypervolume comes from the corners, and very little of it comes from the center.

This represents a fundamentally different geometric character from the dimensions we are familiar with. Our ordinary intuition is built on dimensions one, two, and three, perhaps with dimension four imagined as time. But in dimensions five, six, and beyond, while visualization becomes difficult, we can still calculate, and what we observe is that existence inside the hypercube has the property that almost all points are far from the center. If you are running a numerical simulation that involves picking points at random from a high-dimensional hypercube, almost all of those points will be stuck in some corner. Points near the origin are not typical; they are, in fact, extremely rare as a proportion of all points in high dimension.

53:18

The Blue Sphere That Escapes Its Box

Let me show you some more examples of this kind of phenomenon. Take four unit spheres and stack them inside a square. Since each sphere has diameter two, the containing square is four by four. Now place a small blue ball in the middle of the four spheres, and ask: how big is that ball? We can calculate this exactly. If we place the origin at the center of the square, the centers of the four unit circles sit at coordinates (1, 1), (1, −1), (−1, 1), and (−1, −1). The distance from the origin to any one of those centers is the square root of one squared plus one squared, which is the square root of two. Since that distance equals the radius of the blue circle plus the radius of one of the unit circles, we get r plus one equals the square root of two, and therefore r equals the square root of two minus one, which is approximately 0.414.

Now let us do the same thing in three dimensions. Take eight unit spheres, like billiard balls, arranged in a perfectly orthogonal stack inside a cube, and fit a blue sphere in the center. The containing cube is four by four by four, and the centers of the eight unit spheres sit at coordinates such as (1, 1, 1), (1, 1, −1), and so on. The distance from the origin to any one of those centers is the square root of one squared plus one squared plus one squared, which is the square root of three. By the same reasoning as before, r plus one equals the square root of three, so the radius of the blue sphere in three dimensions is the square root of three minus one.

Exactly the same analysis applies in any number of dimensions, and in general the radius of the blue hypersphere that fits snugly in the center of the arrangement of unit hyperspheres in dimension n is the square root of n minus one. Let us think about what this means as n grows. When n equals four, the square root of four is two, so the radius of the blue hypersphere is two minus one, which equals one. In dimension four, the blue hypersphere in the middle is exactly the same size as each of the surrounding unit hyperspheres.

When n equals nine, the square root of nine is three, so the radius of the blue hypersphere is three minus one, which equals two. In dimension nine, the blue hypersphere is twice as large as each of the surrounding unit hyperspheres. More strikingly, its radius of two carries it all the way from the center of the hypercube to the wall, so in dimension nine the blue hypersphere is actually touching the walls of the hypercube that contains all the others.

For dimensions greater than nine, the square root of n minus one exceeds two, and the blue hypersphere is so large that it actually protrudes outside the hypercube that bounds the surrounding hyperspheres. This is very hard to imagine if you think only in two or three dimensions, but it is exactly what the mathematics tells us. Nine is not even a particularly large number, yet the geometry has already become radically different from anything our low-dimensional intuition would suggest. In a million dimensions, the blue hypersphere is so enormous that it is difficult to grasp, and yet that is precisely what follows from the formula.

I hope you enjoyed this account of the paradox of giants, which led us from Galileo’s analysis of how volume and structural strength scale with dimension into these further paradoxes of higher-dimensional geometry. I hope to see you next time.

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The Lectures on Infinity will appear in the lectures-on-infinity tag. The full collection of essays is available on Infinitely More at The Book of Infinity. And the book is also now available in printed form:

In this second lecture, we explore the concept of supertask—a task involving infinitely many separate actions or steps. We shall play with Thomson’s lamp, turning it on and off infinitely in a finite duration of time, before encountering the dangerous Deal with the Devil. And then infinitely many billiard balls in a sack! Will the Glutton win in the Chocolatier’s game? Let’s find out.

I shall be gradually sharing the individual lectures here on Infinitely More in the coming weeks and months.

Please enjoy!

• Find the lectures here on Infinitely More in the lectures-on-infinity tag.

• The lectures will appear on YouTube.

• The whole lecture course is hosted at Ergo: Lectures on Infinity.

• Find other philosophy lecture courses at Ergo.org.

• This lecture is based on my essay Supertasks.

• The essay also appears in my new book, The Book of Infinity.

Supertasks: Doing Infinitely Many Things

A lightly edited transcript. Timestamps link to the video on the Ergo website.

0:00

What Are Supertasks?

Let’s talk about supertasks, which are tasks involving infinitely many steps.

0:10

Thompson’s Lamp: On or Off After Infinity?

There is a famous supertask puzzle known as Thomson’s lamp, where you turn a lamp on and off infinitely many times in a finite period. It is hard to believe, but there are hundreds of papers in the academic literature devoted to it. The setup is domestic and familiar: you are at home reading in your study as the twilight fades, so you turn on the light. But it is a little too bright, so you switch it off again. Then it is a little too dark, so you switch it back on again, and so on.

Thompson’s Lamp is a supertask. You turn the lamp on for half a minute, then off for a quarter of a minute, then on for an eighth of a minute, then off for a sixteenth of a minute, and so on, following the geometric series discussed in another lecture. After exactly one minute, you have turned the light on and off infinitely many times, cycling on, off, on, off, on, off, faster and faster. The question Thompson asked about this scenario concerns the intelligibility of the supertask itself: what is the state of the lamp after one minute? Is it on, or is it off?

To be precise about the timing: we run from time zero to time one, representing one minute. The lamp is on for the first half-minute, then off for the quarter-minute that follows, then on, then off, alternating infinitely many times within that single minute. The question is whether, at T equal one, the lamp is on or off, and indeed whether that question is even well-posed or determined.

There are, of course, all kinds of physics-based objections one can raise against the thought experiment. It is not physically possible to flick a light switch so rapidly toward the end of the sequence. Electric current flows at a finite rate and could not switch the lamp on and off in those vanishingly narrow intervals. And every interval in which the lamp is on requires the emission of photons, yet there may be only finitely many photons available, so the lamp could not genuinely be on during infinitely many distinct intervals. These objections are quite strong.

But they are really beside the point, because what is at stake is not whether we can actually perform a supertask in physical reality. The real question is whether it is even logically coherent, whether it is intelligible to speak of a task involving infinitely many steps. Mathematically, for instance, we can easily construct a step function that behaves exactly as described: on between zero and one-half, then off, then on, then off, alternating in the familiar pattern. The question then becomes what the function is doing at the limit point T equal one. And mathematically, of course, a function can behave in precisely this way and take any value you like at T equal one. There is no mathematical objection to stipulating that it is on at that point, or off, or anything else.

4:00

Zeno’s Paradox Makes Supertasks Real

Let us return to the Zeno situation. Zeno argued that it is not possible to walk from here to there, and I want to consider not the first version of that paradox but the second, in which to go from here to there you first go halfway, then halfway of what remains, then halfway of what remains again, and so on. When you walk from here to there, you have done infinitely many things along the way: you first reached the halfway point, then the halfway point of what remained, then the halfway point of what remained after that, and so on.

The picture looks just like this. In order to walk from here to there, you first had to reach this point, then this point, then this point, then this point, and so on. It seems, then, that we can perform a supertask, and that the idea is perfectly intelligible. If you believe that you can walk from here to there, you should also believe that you can do infinitely many things, provided we view each of those intermediate accomplishments as a separate thing you have done in the course of that walk.

5:03

Trivial Supertasks Hide in Plain Sight

This suggests a different way of thinking about Thompson’s Lamp. Suppose we simply leave the lamp on for the first half-minute, then continue to leave it on for the next quarter-minute, then for half of what remains after that, and so on. You have left the lamp on for one minute altogether, but in doing so you have, in a sense, done infinitely many things: you left it on for the first half of that period, then again for half of what remained, then for half of what remained after that, and so on indefinitely.

That doesn’t seem problematic in any way. You simply left the lamp on for a minute, and we can view that as an ordinary task which can equally well be thought of as a supertask, if we divide the action into these infinitely many successive intervals. Equally, we could have left the lamp off the entire time, which would constitute another trivial supertask of the same kind. So it seems that in at least some cases, we can perform a supertask without any difficulty at all.

6:22

The Deal with the Devil

Let me tell you about a particular supertask that I find especially compelling. It is called the Deal with the Devil. Suppose you have made some shrewd investments and now carry with you infinitely many dollar bills, numbered with all the odd numbers: one, three, five, and so on. You walk into an underground bar where the Devil is sitting at a table piled high with money. He holds all the even-numbered bills: two, four, six, and so on.

The Devil takes a particular liking to your dollar bills and offers to pay a premium for them. Specifically, he will give you two dollar bills for each one of yours. That sounds like it might not matter much: two for one still leaves you with infinitely much money, so you seem no worse off. You think, “What’s the harm?” and agree to the deal.

The Devil draws up a contract specifying exactly how the exchange will be carried out, and of course it will be carried out as a supertask. In the first half hour, he gives you two dollar bills and takes one from you. In the next quarter hour, he gives you two more and takes one more. In the next eighth of an hour, the same again, and so on through the geometric series. After one hour, all infinitely many trades are complete.

But the contract is very fussy about the order of the exchanges. The Devil always buys from you your currently lowest-numbered bill, and he always pays you with higher-numbered bills. So at the start you hold bills one, three, five, seven, nine, and so on. In the first round, he gives you bills two and four and takes bill number one. In the next round, he gives you six and eight and takes bill number two, which he had just paid you. In the round after that, he gives you ten and twelve and takes bill number three, and so on.

You can see what is happening. Because the Devil always buys your currently lowest-numbered bill and always pays you with higher-numbered bills, your lowest-held bill keeps growing without bound as the process continues. Every single bill is eventually purchased from you at some stage of the supertask. When all the trades are complete, the Devil holds every bill, and you have nothing at all.

I like this example because it shows how supertask transactions do not always behave the way finite transactions do. Even though each individual trade looked favorable, or at worst neutral, the details of how the process unfolds in this infinitary manner produce a result that is genuinely surprising. We have to pay close attention to the order and structure of the process, not just its local character at each step.

12:03

Balls in a Sack: Empty After Infinity?

Here is another puzzle in the same spirit. Suppose you have lost a bet with your friends, and the agreed penalty is that you must stand in humiliation holding a large, empty wool sack. Nearby is a pile of infinitely many billiard balls. The procedure you must carry out is this: at each step, you take two billiard balls from the pile and put them into the sack, and then you take one billiard ball out of the sack and discard it permanently. Moreover, you must perform each step faster and faster. If the first step takes half a minute, the second a quarter of a minute, the third an eighth, and so on, then after exactly one minute you will have completed infinitely many steps, finishing the supertask in finite time.

You will be redeemed in the eyes of your friends if, at the end of this procedure, the sack is empty. At first this seems impossible. The sack is empty at the start, yet at every step you are putting two balls in and taking only one out, so after n steps there are n balls in the sack. The sack grows heavier and heavier as you go. How could it possibly be empty after infinitely many steps?

And yet redemption is achievable, provided you arrange the procedure in the right way. Think of the billiard balls as numbered by the natural numbers: ball zero, ball one, ball two, and so on. At every stage, you place the next two balls into the sack, but you always remove the lowest-numbered ball currently in the sack. So on the first step you put balls zero and one into the sack and remove ball zero, the lowest. On the second step you put balls two and three in and remove ball one. On the third step you put balls four and five in and remove ball two. And so on.

The key observation is that ball n is removed from the sack on the nth step and is never returned. So when you ask what balls remain in the sack after the supertask is complete, the answer must be: none. Any particular ball you name, say ball n, was removed at step n and discarded. There is no ball that could be sitting in the sack at the end, because every ball has been accounted for and removed at some finite stage.

This is another instance of the strange character of supertasks. If you performed only finitely many steps, the count of balls in the sack after n steps would simply be n, growing without bound. Yet when the infinitely many steps are arranged in precisely this way, the sack is empty at the conclusion. The details of how you carry out the infinite procedure turn out to matter enormously.

16:04

Controlling What Remains in the Sack

We can also arrange for the sack to be completely full, with infinitely many balls remaining at the end. Suppose we always remove the highest-numbered ball in each step. We put in balls zero and one and take out ball one, then we put in balls two and three and take out ball three, then we put in balls four and five and take out ball five. In this way, all the even-numbered balls end up in the sack, and all the odd-numbered balls are the ones we remove.

This naturally raises further questions. How could one arrange the process so that exactly the prime numbers remain in the sack at the end? Or could one design a process so that exactly the multiples of 17 remain? The answer turns out to be quite robust: one can actually design a process so that any given target set is exactly what remains at the end. One can even find necessary and sufficient criteria on the target set that make this possible.

17:01

Random Removal and Vanishing Probability

There is perhaps a stochastic way of thinking about the balls-in-a-sack puzzle, meaning a way of treating it as a random process. Suppose that whenever you put two balls into the sack, the ball you then remove is chosen randomly from all the balls currently in the sack. You start with an empty sack, put two balls in, and pick one of them at random. Then you put two more balls in, giving you three, and again remove one at random, leaving two. Two more go in, one comes out at random, and so on. The question is: what should we expect at the end? Will there be any balls left in the sack, and if so, how many?

Let us focus on one particular ball from the very first step and ask how likely it is to survive. On the first step, two balls are in the sack, so our chosen ball has a one-half chance of not being the one removed. Given that it survives that first step, two more balls are added, making three in total, and our ball survives if either of the other two is chosen, giving it a two-thirds chance of surviving the second step. On the third removal, there are four balls in the sack, so the probability of surviving that step, given survival through the first two, is three-quarters. The pattern is now clear: at step n, the probability of surviving that step, conditional on having survived all previous steps, is n over n plus one.

The probability that our ball survives all n steps is therefore the product of these conditional probabilities: one-half times two-thirds times three-quarters, and so on up to n over n plus one. Because of the telescoping cancellation, the twos cancel, the threes cancel, the fours cancel, and every intermediate factor cancels, leaving simply one over n plus one. So the probability that a given ball has not been discarded after n steps is one over n plus one, which approaches zero as n grows large.

Since the probability of surviving n steps is one over n plus one for every n, the probability of surviving infinitely many steps must be less than this quantity for every n. That forces the probability to be zero. In other words, the chance that any particular ball is never chosen for removal is exactly zero.

21:18

Probability Zero Does Not Mean Impossible

The probability that a given ball is never picked is zero. The same argument applies to the balls added at later stages. If I consider the balls remaining in the sack at stage k, there is a k over k+1 chance of surviving the next round, since that is the number of balls in the sack, then k+1 over k+2, and so on, multiplying k+n over k+n+1. We get the same cancellation phenomenon, leaving us with k over k+n+1. For any fixed k, as n grows large, this quantity goes to zero.

Therefore, for any particular ball at any particular stage, the probability that it survives infinitely many steps is zero. For every ball, almost surely it will be chosen at some stage. I should note that “almost surely” is a technical term as probabilists use it, with a very particular meaning: it means that the probability of that event is 100%. That is different from being logically certain, and this is the philosophical point I want to make.

In this kind of stochastic reasoning, the difference between a probability-zero event and an impossible event is not trivial. Just because something has probability zero does not mean it is logically impossible. It is logically possible, for example, that on the first round one of the balls is red and that red ball is simply never chosen. That is logically possible, and yet it is probability zero, as we calculated: the probability of the red ball never being chosen is less than 1 over n+1 for every n, and therefore the probability of that event is zero.

One has to keep in mind that probability zero and impossible are not the same thing.

23:40

Every Ball vs. All Balls: A Subtle Gap

We can still reason probabilistically here. For any particular ball, it is very likely to be chosen at some stage, and so what we expect at the end is that the sack should be empty. With probability one, we should expect the sack to be empty, because with probability one any particular ball is almost surely chosen at some stage.

But notice what happened in that reasoning. What we argued first is that for every ball, almost surely it is chosen at some stage. What I said afterwards, however, was that almost surely every ball is chosen. Those are not quite the same thing. For any particular ball, it is very likely to be chosen at some stage and removed, but that is different from the sack being empty. For the sack to be empty, I want to say it is very likely that every ball is chosen at some stage.

So the question is: how do we move from a probability calculation about every individual ball to the claim that almost surely something is true of every ball? Perhaps it is helpful here to think about the situation of a dartboard.

25:00

Dartboards and Countable Additivity

Suppose we are playing darts and I am throwing darts at a dartboard with a uniform probability distribution over where the dart lands. The probability that the dart lands in any particular region is simply the ratio of that region’s area to the total area of the dartboard. So, for example, the probability that the dart lands on the left side is one-half, since the left and right sides have equal area. Similarly, the probability of landing in any given quadrant is one-quarter, and so on.

Now, the probability of hitting any particular point is zero, because a point has zero area. If we think of the dartboard as a continuum of points, the probability of hitting any exact point is zero. But here is the tension: I want to say, for every point, almost surely the dart does not hit that point. Am I then willing to say that almost surely the dart does not hit any point at all? No, because the dart is going to hit some point. I will throw it, it will land, and it will land exactly somewhere. That means the dart will hit a point that was itself a probability-zero event.

There is a useful way of putting this: it is very likely that rare things happen. I throw the dart and it hits whichever point it hits. For any particular point, that outcome was a very rare event, yet it is virtually certain that some such rare event will occur. The space of possible outcomes is so enormous that, even though each individual outcome is vanishingly unlikely, one of them is guaranteed to happen.

We can see the same phenomenon with a coin. Suppose I flip a coin ten times and obtain some particular sequence of heads and tails. Almost surely I will get some sequence, but any particular sequence has probability 1 over 2 to the 10, which is a very small number. So it is very likely that a rare event occurs. It is a little paradoxical, but it should not be too paradoxical, because in a sense it is obvious: the space of things that could occur is enormous, and it is very likely that one of them will happen.

Returning to the stochastic process with balls in a sack, we seem to be reasoning in exactly the same way in both cases. For any particular ball, it is very likely that it gets removed, and I want to conclude from that that it is very likely all the balls are removed. But I do not want to make the same move with the dartboard: for any particular point, it is very likely the dart will not land there, yet I cannot conclude that it is very likely the dart will not land anywhere, because it certainly will land somewhere. So how can the inference be justified in the balls case but not in the dartboard case?

The answer lies in the philosophy of probability and in a fundamental asymmetry: the number of balls is only a countably infinite collection, whereas the number of points on the dartboard is an uncountably infinite collection. Probability theory is countably additive. When we have a countable list of probability-zero events, we can conclude that the probability of any one of them occurring is still zero. But we cannot make that move in an uncountable setting. To examine the subtle differences between these two situations is to enter deeply into the philosophy of probability, and in particular into the question of why we want our probability measures, and Lebesgue measure, to be countably additive but not required to be more than that.

30:12

The Chocolatier’s Game: Finite Servings

Let us consider another supertask, which I call the Chocolatier’s Game. It is a game played between two players: the chocolatier, who serves exquisite chocolate creations on a kind of serving platter, and the glutton, who eats them as the game proceeds. The game has infinitely many rounds, and on every round the chocolatier serves finitely many new chocolates, while the glutton is allowed to eat only one. The uneaten chocolates accumulate on the platter.

So perhaps the chocolatier serves 17 chocolates, and the glutton picks one and eats it. Then 37 more arrive on the next round, and the glutton again picks one. Then perhaps just two more are added, and the glutton picks one from among the accumulating chocolates on the platter. We can think of this entire infinite process as completing in a finite amount of time, using the geometric series reasoning we have already discussed, but in fact nothing is at stake in that decision. The logic of the game is simply that there are infinitely many steps, and then we ask who won afterwards. It is irrelevant whether the process took finite or infinite time.

The glutton wins if he eats every single chocolate that was ever served. At first glance this might seem impossible, because at every stage of the game the number of chocolates on the platter is increasing and growing without bound. It would appear absurd for the glutton to eat all of them. But if we apply the ideas from the deal-with-the-devil scenario or the balls-in-a-sack puzzle, we can see that the glutton can in fact win.

Here is one strategy. The glutton will be systematic, mentally organizing the chocolates on the platter into a queue. New chocolates, however many arrive, are always added to the back of the queue, and the glutton always eats from the front. This is precisely the stock-rotation method a restaurant uses: new supplies go to the back of the cupboard, and the oldest items at the front are always used first, ensuring that everything is turned over in time.

This queue strategy is a winning strategy for the glutton. For any particular chocolate that is ever served, it occupies a specific place in the queue at the moment it arrives, and there are only finitely many chocolates in front of it. Therefore we know exactly which turn that chocolate will be eaten on, and at that turn it will indeed be eaten. Since this reasoning applies to every chocolate ever served, every chocolate will be eaten at some stage. After infinitely many steps, the glutton will have eaten every single chocolate. The glutton has a winning strategy in the Chocolatier’s Game.

35:31

Infinite Servings and the Zigzag Strategy

There is another version of the chocolatier’s game, a slightly harder version. I like it a lot, because it begins with the easy case we just discussed, which is quite clear, and then we can make it progressively harder. In the end, the chocolatier’s game becomes quite sophisticated mathematically, and I will be hinting at those deeper developments. But we can go at least one step further.

Consider the version of the chocolatier’s game in which the chocolatier is allowed to serve infinitely many chocolates on a single turn. Each turn, the chocolatier puts down infinitely many chocolates, but the glutton can still eat only one. The chocolatier serves infinitely many, the glutton eats one; then infinitely many more, and the glutton eats one; and so on. I claim that the glutton can still win.

The queuing strategy does not work here at all. If we think of the chocolates from the first round as forming a queue, with the second-round chocolates placed behind them, the glutton will never reach the round-two chocolates. At every one of the countably many stages, the glutton will still be eating through the chocolates served in round one, and the round-two and round-three chocolates will never be reached. We have to think a little more imaginatively.

What the glutton does instead is imagine the chocolates on the serving platter as filling up an infinite matrix. The first-round chocolates are placed in the first row, the second-round chocolates in the second row, the third-round chocolates in the third row, and so on. Each entry in this matrix is a single chocolate, all of them distinct, with their exquisite glazing and cherries and whatever else. The matrix simply organizes, in the glutton’s mind, all the chocolates that will ever be served.

The glutton then eats the chocolates according to a winding zigzag path through this matrix. On the first turn the glutton eats the chocolate in position one, then moves along the path to the next position, then the next, and so on, traversing the matrix in this diagonal winding order. It is clear that every chocolate that will ever be served appears somewhere on this winding path. Moreover, every chocolate on the path has only finitely many chocolates preceding it, corresponding to the triangular region above and to the left of it in the matrix. Therefore, each chocolate will be eaten on precisely the turn whose number equals the count of chocolates preceding it on the path.

So even though the chocolatier is serving infinitely many chocolates on each turn, and the glutton is eating only one, the glutton can nonetheless proceed systematically so that every chocolate is eventually eaten at some finite stage. The glutton wins.

39:32

Can the Glutton Win Without Memory?

The strategy for the glutton we have been considering requires him to pay attention to the order in which the chocolates are served. But one might ask whether the glutton really needs to pay so much attention. Perhaps there is a strategy that tells him which chocolates to eat based only on the set of chocolates currently on offer, regardless of the order in which they have been served. This is called a memory-free strategy, and it corresponds to the distinction in game theory between a tactic and a strategy: a tactic depends only on the current situation, whereas a strategy depends on the entire history of play up to that point.

The situation becomes quite interesting once we ask what is on the menu for the chocolatier. Specifically, can the chocolatier serve the same chocolate more than once? If a memory-free strategy existed, the chocolatier could exploit it in the following way: place two chocolates before the glutton, observe which one the glutton’s tactic selects, and then simply replace that chosen chocolate with an identical one. Since the tactic depends only on what is currently on offer, the glutton would be forced to make the same choice again, and again, and again. The other chocolate would therefore never be chosen.

This gives us a straightforward way to see that if the chocolatier is permitted to serve identical chocolates repeatedly, there can be no memory-free strategy for the glutton. So let us rule that out, and stipulate that the chocolatier must not repeat identical chocolates.

41:56

Countable vs. Uncountable Creativity

The chocolatier is not allowed to repeat a chocolate ever. That constraint breaks the previous argument, and we still want to know: is there a winning tactic for the glutton? The answer turns out to depend on how creative the chocolatier is.

Suppose the list of possible chocolates the chocolatier could make is infinite in the manner of the natural numbers. There is menu item number zero, menu item number one, menu item number two, menu item number three, and so on. The chocolatier need not serve all those chocolates, nor serve them in that order; those are simply the possible chocolates available to serve. In that case, the glutton has a winning tactic, which is simply to always eat the chocolate with the lowest menu item number available.

At any stage there may be finitely many or even infinitely many chocolates on the serving platter, but one of them will always be the lowest menu item. If the glutton always eats the one with the lowest menu item number, he will eventually eat every chocolate, because any particular chocolate has only finitely many items ahead of it on the menu. Therefore no chocolate could remain uneaten at infinity, since it would have been the lowest one at some point and would already have been eaten.

In other words, if the chocolatier is merely countably creative, in the sense that the space of possible chocolates they could serve is only countable, then the glutton has a winning tactic. But what about the case where the chocolatier is uncountably creative? Perhaps there is a distinct chocolate for every real number, with some parameter such as the width of the glazing or the density of the liqueur coming from a real number, so that each real number corresponds to a strictly different chocolate type.

In that case, one can prove that it is not possible for the glutton to have a winning tactic. The argument is mathematically sophisticated, so I will not give it here, but this is one way in which the problem becomes quite deep. There is, however, something very close to a winning tactic even in the uncountably creative case.

Specifically, there is a winning tactic for the glutton in the uncountably creative case provided the glutton is also allowed to use the information of the most recently eaten chocolate. At any stage, the glutton looks at the chocolates on offer, which may be infinite, and also has the taste of the previously eaten chocolate on his tongue, and is allowed to use that information to determine which chocolate to pick. It turns out that in the case where the chocolatier is allowed to serve only finitely many chocolates at each stage, if the axiom of choice is true, then the glutton has a winning tactic that uses this memory of the previous chocolate, and the argument involves the axiom of choice and well-orders, which makes it all quite fascinating.

45:35

Wrapping Up Supertasks

That is all for this lecture. I hope you enjoyed the discussion of supertasks and the chocolatier’s game. See you next time.

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The lectures will all appear in the lectures-on-infinity tag. The full collection of essays is available on Infinitely More at The Book of Infinity. And the book is also now available in printed form:

In this first lecture, we introduce the ancient puzzle of Zeno’s paradox and follow the thread where it leads—to the very meaning of our number expressions, to infinite summations, to convergent and divergent series. We will discuss what I call the most contested equation in middle school, and by the end we shall reach the remarkable Riemann rearrangement theorem, concerning the malleable nature of infinite sums.

I shall be gradually sharing the individual lectures here on Infinitely More in the coming weeks and months.

Please enjoy!

• Find the lectures here on Infinitely More in the lectures-on-infinity tag.

• The lectures will appear on YouTube.

• The whole lecture course is hosted at Ergo: Lectures on Infinity.

• Find other philosophy lecture courses at Ergo.org.

• This lecture is based on my essay Zeno’s Paradox.

• The essay also appears in my new book, The Book of Infinity.

Zeno’s Paradox and Infinite Sums

A lightly edited transcript. Timestamps link to the video on the Ergo website.

0:00

Welcome to Infinity

I’m Joel David Hamkins, and in this series of lectures I want to tell you about infinity, that most fascinating of topics. I want to start with a classical thinker.

0:13

Zeno’s Paradox: Why All Motion Is Impossible

Zeno of Elea made a very interesting argument, around 450 BC, to the effect that all motion is impossible. You cannot go from here to there. Now, of course we know the conclusion is false, because we can simply walk from one place to another. But it is no good to simply say, “Well, we know Zeno’s conclusion is false because we can go from here to there.” Rather, we need to understand the argument itself and find the flaw in his reasoning.

Zeno argued as follows. Suppose you want to go from point A to point B. Before you can go from A to B, you must first get halfway to B. But before you can get halfway to B, you must first get halfway to the halfway point. And before you can get halfway to the halfway point, you must first get halfway to that point, and so on. Before you move at all, it seems you would already have to have gone halfway as far, which generates an infinite regress of tasks that must be completed before any motion can begin. Zeno concluded from this that all motion is impossible.

I find this quite interesting. The conclusion is of course absurd, but one may still struggle with the reasoning. The core problem Zeno is identifying seems to be this: is it possible to do infinitely many things?

2:07

Achilles vs the Tortoise

Let me tell you about another of Zeno’s paradoxes: Achilles and the tortoise. Achilles and the tortoise are going to have a race around the stadium, both starting from the same point and running all the way around. The tortoise challenges Achilles, claiming to be able to win, on the condition that it is given a small head start.

The tortoise’s argument runs as follows. Suppose the tortoise begins one-quarter of the way around the track, at a point we will call A, while Achilles starts from the beginning. They set off, and Achilles very rapidly reaches point A. But by the time Achilles arrives at A, the tortoise has moved on to a new point, B. So the tortoise is still ahead.

Now the same reasoning applies again. By the time Achilles reaches B, the tortoise has moved on to a further point C, and is still ahead. Then by the time Achilles reaches C, the tortoise has moved on to a point D, and so on. At every stage, whenever the tortoise occupies some point and Achilles has not yet arrived there, by the time Achilles does arrive, the tortoise will have moved on.

The tortoise therefore argues that it is impossible for Achilles ever to catch up, because doing so would require completing infinitely many such steps.

3:57

What Are Supertasks?

These puzzles are perhaps related to the concept of supertasks: a task involving infinitely many steps or infinitely many actions. We will have another lecture devoted to supertasks itself, but Zeno’s paradox may well be the origin of the concept.

4:16

Turning Zeno into an Infinite Sum

I want to discuss a slightly different way of understanding Zeno. Zeno had argued that all motion is impossible because, before you get from here to there, you need to get halfway, and before you do that you need to get halfway to that halfway point, and so on. But we can turn the argument around and put it this way: we cannot go from here to there because, before going from here to there, we must have already gone halfway. Then after reaching the halfway point, we must get halfway of what remains, and then halfway of what remains again, and so on. Before arriving, we must have done infinitely many things, which Zeno argues is impossible.

There is a way of understanding this argument in a more contemporary manner by thinking about the line segment from zero to one. If we want to go from zero to one, we must first get halfway there. Now standing at the halfway point, before we go from there to the end, we must get halfway to there, which brings us to the three-quarter point. Standing at the three-quarter point, we go halfway again, covering one-eighth of the total segment, then one-sixteenth, and so on.

Consider the numbers we have written down. The total length traversed is one-half, the initial one-half, plus one-quarter, plus one-eighth, plus one-sixteenth, and so on. We are adding up infinitely many numbers, and what does it really mean to add up infinitely many numbers? It is quite clear that the total length of all of these segments will exhaust the original interval, and so this infinite series adds up to one, the total original length. This is a contemporary way of understanding what is going on in Zeno’s paradox: one-half plus one-quarter plus one-eighth plus one-sixteenth, and so on, adds up in total to one.

There is another way of seeing this infinite summation. Consider the unit square, a one-by-one square with total area one. If I take half of it, that portion has area one-half, and then half of what remains is one-quarter. Taking half of that gives one-eighth, half again gives one-sixteenth, and so on, each time chopping what remains in half. The total area, which is one, can therefore be thought of as one-half plus one-quarter plus one-eighth plus one-sixteenth, and so on. This is another way of seeing that an infinite sum of numbers can nevertheless add up to a finite number, namely one.

7:48

The most contested equation in middle school

Next I would like to tell you about the most contested equation in middle school. Perhaps some of you have heard of it, and perhaps you have had arguments about it, which is precisely what I mean by calling it the most contested equation in middle school. The equation concerns the number 0.9999 repeating, where the nines never stop, and it says that this number is in fact equal to one. That is, 0.999 repeating is the same as 1.000 forever. Let me give a couple of arguments for why this is true.

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Sometimes people feel that 0.999 repeating should be a little bit less than one, since there seem to be all these nines with something extra left over. I want to argue that this intuition is mistaken and that the equation is correct. Let x be 0.999 repeating. If I multiply x by 10, I obtain 10x, and multiplying a decimal number by 10 simply moves the decimal point one place to the right, giving 9.999 repeating, with infinitely many nines after the decimal point.

Now consider the subtraction 10x minus x. On the left side this is simply 9x. On the right side, 9.999 repeating minus 0.999 repeating causes all the nines after the decimal point to cancel, leaving exactly 9. So 9x equals 9, and therefore x equals 1, just as claimed: 0.999 repeating is equal to one.

Mathematicians sometimes criticize this argument on the grounds that it presupposes 0.999 repeating is meaningful. When I said “let x be that number,” the step only makes sense if the expression actually denotes something. In fact it does denote something, and an argument can be given for that, but we have not given such an argument here.

10:32

What Decimal Notation Actually Means

Let me give another argument for this contested equation. Perhaps it is less controversial to say that 0.3 repeating is equal to one-third. Many people know this, and if you divide one by three using the long division algorithm, you arrive at exactly that expression. Now, 0.9 repeating is simply three times 0.3 repeating, and therefore its value must be three times one-third, which is one. So that is another way of seeing that 0.9 repeating must equal one.

Let us get a little more basic and fundamental about what these expressions actually mean. When you write a number in decimal notation, say 8,547, this is a positional number system, and the notation means that we have eight thousands, five hundreds, four tens, and seven ones. We can write this out as 8 times 10 to the 3, plus 5 times 10 squared, plus 4 times 10 to the 1, plus 7 times 10 to the 0. The meaning of the notation is precisely a sum with one term for each digit.

The same principle applies when we write digits after the decimal point. Take a number like 3.14159265 and so on, the beginning of pi. What it means is 3, plus 1 times one-tenth, plus 4 times one-hundredth, plus 1 times one-thousandth, plus 5 times one ten-thousandth, and so forth. Each digit corresponds to one term in the sum, multiplied by a power of ten, except that now those powers of ten carry negative exponents.

The meaning of a number with infinitely many digits is therefore precisely an infinite sum, with one term for each digit. When we write 0.999 repeating, what it means is nine-tenths, plus nine one-hundredths, plus nine one-thousandths, plus nine ten-thousandths, and so on. The very meaning of that expression is already an infinite sum of the kind we were examining earlier, and this particular sum has the property that each term is one-tenth as large as the previous one. That is what is called a geometric series.

14:26

Deriving the Geometric Series Formula

A geometric series is an infinite sum in which each term is a constant multiple of the previous term. In our earlier example, drawn from Zeno, we had one-half plus one-fourth plus one-eighth plus one-sixteenth, and so on. Each term is half as large as the one before it, so we are multiplying by one-half each time. In general, a geometric series begins with a term a and proceeds through ar, ar-squared, ar-cubed, and so on, where each term is r times the previous one.

To find the value of such a series in full generality, it is easiest to take a equal to 1, so we are considering 1 plus r plus r-squared plus r-cubed, and so on. When we try to understand the value of an infinite sum, what we should do is think about what happens to the value when we take only finitely many of those terms. This approach has a distinctly potentialist character. The distinction between potential infinity and actual infinity is relevant here: potential infinity is the idea that one never completes the infinite task but instead takes more and more, finitely much at a time. The very meaning of an infinite sum carries this potentialist character, because we look at what happens to the values of the finite partial sums as we include more and more terms.

So let us call x the value obtained by summing the geometric series up to the nth power: that is, x equals 1 plus r plus r-squared plus r-cubed, all the way up to r to the n. Now consider what happens when we add the very next term, r to the n plus 1. We get x plus r to the n plus 1, which equals 1 plus r plus r-squared, and so on up to r to the n, and then one further term, r to the n plus 1. After the leading 1, every remaining term is a multiple of r, so we can factor r out of those terms, reducing each exponent by one. The result is that x plus r to the n plus 1 equals 1 plus r times the quantity 1 plus r plus r-squared, and so on up to r to the n, which is simply 1 plus r times x.

We have therefore derived the equation x plus r to the n plus 1 equals 1 plus rx, and this is an equation we can solve for x. Moving the rx term to the left-hand side gives x minus rx equals 1 minus r to the n plus 1. Factoring the left-hand side yields x times 1 minus r equals 1 minus r to the n plus 1, and so altogether we find that x equals 1 minus r to the n plus 1, divided by 1 minus r. This gives us the exact value of the finite partial sum out to that point.

With this formula in hand, we can now understand what happens as n becomes larger and larger. Perhaps n is 100, or 1,000, or 1 million. The formula tells us precisely the value of the finite partial sum at each such stage, and by letting n grow without bound we can grasp the meaning of the infinite sum itself, in keeping with that potentialist conception of infinity.

19:51

When Do Infinite Sums Converge?

Let us summarize what we have. We are interested in the infinite sum 1 plus R plus R squared plus R cubed and so on. We have observed that if we take N terms, we obtain a specific finite value. If R is greater than or equal to 1, then this infinite sum will become infinite, because the finite partial sums will simply grow without bound. All the terms will be at least as large as 1, and adding more and more such numbers causes the sum to grow indefinitely.

But if the absolute value of R is less than 1, including the possibility that R is negative, then we can reason carefully about what happens. This is precisely the situation in Zeno’s geometric series, where R was one-half and each term was half as large as the previous one. The puzzle of Zeno is exactly this: how can infinitely many numbers add up to a finite sum? That is what we are trying to explain.

When the absolute value of R is less than 1, the finite partial sum is given by a certain expression, and the key point is that as N becomes very large, we are multiplying a number less than 1 by itself many, many times. A number less than 1 raised to higher and higher powers gets smaller and smaller, and we can make it as small as we like. Therefore, as N grows large, the value of the finite partial sum approaches 1 over 1 minus R, which is all that remains in the expression.

By taking enough terms, we can make the finite partial sum as close to 1 over 1 minus R as we like. This means that the total value of the geometric series 1 plus R plus R squared plus R cubed and so on is equal to 1 over 1 minus R. More generally, if the series begins with a leading coefficient A, so that we have A plus AR plus AR squared plus AR cubed and so on, the same reasoning gives us A over 1 minus R as the value of the geometric series.

What this means precisely is that no matter how close you wish to be to this limiting value, if you take enough terms, every finite partial sum with at least that many terms will fall within your chosen tolerance of the limit. That is the proper understanding of what it means for an infinite series to have a value.

23:13

Applying the Formula to Zeno and 0.999...

Let us apply this formula to the Zeno case, where we had one-half plus one-quarter plus one-eighth plus one-sixteenth and so on. This is the case where a is one-half, since that is the term we started with, and r is also one-half, because each term is obtained by multiplying the previous one by r. The total sum is therefore a over 1 minus one-half, and one-half over one-half is simply 1. That is exactly what we said: this sum is exactly 1 in the case of Zeno.

Now consider what is perhaps the most contested equation in middle school: 0.9 repeating. As we noted, this equals nine-tenths plus nine-hundredths plus nine-thousandths and so on. This is a case where a is nine-tenths, since that is the first term, and r is one-tenth, since each term is one-tenth as large as the previous one. The total sum is therefore a over 1 minus r, which is nine-tenths over 1 minus one-tenth, and that is nine-tenths over nine-tenths, which equals 1.

This gives us another way of seeing that 0.9 repeating must equal 1. Not only must it equal 1 if it has any meaning at all, but this argument shows that it does have a meaning: it converges to 1, because by taking more and more terms, more digits, we can make the value as close to 1 as we like. That is perhaps the most contested equation of middle school.

25:11

The Harmonic Series Blows Up

Let us consider some other interesting series. The confusing thing about infinite series is that it can seem impossible, at first, that one could add up infinitely many numbers and arrive at a finite answer. Nevertheless, we argued that there are cases where a finite answer does result: one-half plus one-quarter plus one-eighth plus one-sixteenth, and so on. We add up all of those infinitely many numbers and still obtain a finite answer, equal to one. That is an instance of the geometric series.

Obviously, there are also cases where adding up infinitely many numbers does not yield a finite answer. If I add one plus two plus three plus four and so on, this cannot converge to any finite number. But that is a case where the individual terms are getting larger, so of course their sum will not be finite. Similarly, if I add up infinitely many ones, one plus one plus one plus one forever, I will never reach a finite answer; I can make that sum exceed any given bound simply by taking enough terms.

One might therefore think: perhaps what it takes for an infinite sum to be finite is that the individual terms must be getting smaller, as they do in the geometric series, where the terms one-half, one-quarter, one-eighth, and so on decrease toward zero. Let us consider another famous series of exactly that kind. It is called the harmonic series, and it goes: one plus one-half plus one-third plus one-quarter plus one-fifth plus one-sixth, and so on, taking the reciprocal of each successive integer. The individual terms are indeed getting smaller, which we said should be a necessary condition for the sum to be finite. But let us think more carefully about whether the series actually converges.

Suppose we have added up many terms, say the first n terms: one plus one-half plus one-third plus one-quarter, all the way out to one over n. Perhaps n is a billion. Now consider doubling the number of terms. The additional terms are one over n plus one, plus one over n plus two, and so on, up to one over 2n. That is exactly n new terms. Each of these new terms is at least as large as the last one, which is one over 2n. So the total contribution of this additional block is at least n times one over 2n, which equals one-half, since the n’s cancel.

What this shows is that no matter how many terms you have already summed, you can always add at least one-half more to the total simply by doubling the number of terms. If you want to add another one-half on top of that, double again. If you want to add yet another one-half, double again. You can add an extra 17 to the sum, if you like, just by doubling 34 times. There is no ceiling.

Therefore, the harmonic series cannot converge to a finite value. For convergence to a finite value would require that once you have taken enough terms, you are very close to that value and remain close no matter how many additional terms you take. But that is precisely what fails here: we can always add one-half more, or two more, or five more, simply by doubling the number of terms enough times. The harmonic series is thus famous for being divergent. It does not converge to a finite value, even though its individual terms go to zero. This is an entirely different situation from the Zeno case and the geometric series.

31:26

The Alternating Harmonic Series

Let me twist things around a little. Perhaps it seems surprising that I have been talking about all this mathematics, about series and sums, when we started with the philosophical idea of Zeno’s paradox. But the point I want to make is that, when you are looking at the philosophy of infinity, it blends into mathematics so gradually and naturally that one is simply pushed toward these mathematical ways of thinking. Our mathematical knowledge is a quite natural tool for understanding the nature of infinity.

So let us look at what is called the alternating harmonic series. This is the series 1 minus 1/2 plus 1/3 minus 1/4 plus 1/5, and so on. It is called alternating because the signs vary: positive, negative, positive, negative, and so on. We saw earlier that if all the terms are positive, the harmonic series does not converge to a finite number; it adds up to infinity. But what about this one, where we are sometimes subtracting instead of adding?

Consider a graph with the number of terms along one axis and the running total along the other. We start at 1. Then we subtract 1/2, dropping to 1/2. Then we add 1/3, rising to 1/2 plus 1/3, which is less than 1. Then we subtract 1/4, going down, but by less than we just went up. Then up by 1/5, down by 1/6, up by 1/7, and so on. The series has this zigzag character, but with a crucial feature: every time we go up, the subsequent downward step is smaller than the upward step that preceded it, and every time we go down, the subsequent upward step is smaller than the downward step that preceded it.

This means that the upper envelope of the zigzag, the values recorded just after each upward step, is descending, while the lower envelope, the values recorded just after each downward step, is ascending. Moreover, the distance between these two envelopes equals the magnitude of the steps we are zigzagging by, and since those steps are shrinking, the two envelopes are drawing closer and closer together. One can prove that they converge to exactly the natural log of 2, which is approximately 0.69. The alternating harmonic series therefore converges; it has a finite value, and that value is the natural log of 2.

So here we have a striking contrast. When all the terms are positive, the harmonic series adds up to infinity. But when the terms alternate in sign, the series adds up to log 2. The mere introduction of subtraction transforms a divergent sum into a convergent one.

35:28

Rearranging Terms Changes the Answer

There is something remarkable about this kind of series. It is called a conditionally convergent series: a series that converges to a finite answer, but if you take the absolute value of each term, making them all positive, it no longer converges to a finite value. In particular, you cannot view such a series as first summing the positive terms and then summing the negative terms and subtracting the two. You might think that 1 minus 1/2 plus 1/3 minus 1/4 and so on should be the same as taking all the positive terms, 1 plus 1/3 plus 1/5 plus 1/7 and so on, and subtracting all the negative terms, 1/2 plus 1/4 plus 1/6 and so on. But you cannot group them that way, because the positive terms add up to infinity and the negative terms also add up to infinity, and so the subtraction does not make sense at all.

There is a profound theorem that addresses exactly this kind of situation: the Riemann Rearrangement Theorem. It states that if you have a conditionally convergent series, you can rearrange its terms to make the sum equal to whatever target value you like. For any conditionally convergent series, not just this one, and for any target value, there is a rearrangement of the terms that converges to that target.

Let me show you how the proof works, using the alternating harmonic series as our example. Suppose we want the new sum to equal 1.4. Since log 2 is approximately 0.69, that target is comfortably above the original value. The strategy is straightforward: keep adding positive terms until you exceed the target. For instance, 1 plus 1/3 is about 1.33, which is not yet above 1.4, but 1 plus 1/3 plus 1/5 is already above 1.4, so we stop there. Once we have exceeded the target, we begin adding negative terms until we fall below it. Subtracting 1/2 already brings us below 1.4, so with just one negative term we are back under the target. Then we resume adding positive terms, starting with 1/7, and continue the process.

The key point is that this zigzagging procedure always works. Because the positive terms alone sum to infinity, you can always take enough of them to climb back above the target. Because the negative terms alone also sum to infinity in magnitude, you can always take enough of them to fall back below the target. By repeating this process, you zero in on the target value, and the rearranged series converges to exactly that value.

I find this genuinely profound. What it means is that when you are adding up infinitely many numbers, you do not necessarily get the same answer when you rearrange them. If the terms come from a conditionally convergent series, the order in which you add them can affect the numerical result. That is surprising and, I think, deeply illuminating. And all of these observations shed light, in my view, on what is really going on with Zeno’s paradox.

40:38

Recap and What’s Next: Supertasks

Zeno is troubled by the possibility of doing infinitely many things in a finite space of time. This concern connects directly with the concept of supertasks, which we will take up in the next lecture. To understand specifically the idea of moving from one place to another, we were led to the concept of the geometric series: going halfway, then a quarter, then an eighth, and so on. Adding up all those lengths and arriving at a finite answer gives us a way of understanding how an infinite sum can nevertheless have a finite result.

Using the geometric series, we found that in cases where we can describe the series exactly, we can determine precisely what the numerical answer is. Then came the complication introduced by alternating series, which have sometimes negative and sometimes positive terms. Such a series can converge to an answer, and yet the order in which you take those terms can affect the final result, as the Riemann Rearrangement Theorem shows.

I hope you enjoyed the story of Zeno’s paradox and the path it opened up into geometric series, into the possibility of adding up infinitely many numbers to obtain a finite answer, and then into the twist provided by the alternating harmonic series. Ultimately, we arrived at Riemann’s Rearrangement Theorem, a profound idea: when you are adding up infinitely many numbers, the order in which you take those numbers can affect the result.

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The other lectures will appear in the lectures-on-infinity tag. The full collection of essays is available at The Book of Infinity. And the book is now available:

These lectures are an exploration of all my favorite paradoxes and conundrums. We’ll get into Zeno’s paradox, supertasks, the paradox of giants, the paradox of the largest tweetable number, Galileo’s paradox, the dispute between potentialist and actualist conceptions of infinity, and much more. Eventually, we’ll dive into the details of Cantor’s discovery of uncountable infinity, the problem of the continuum hypothesis, and much more. Please enjoy!

The lectures are based on my new book, The Book of Infinity, MIT Press 2026.

I have been serializing the chapters here on Infinitely More as I wrote them—find them in the section The Book of Infinity.

The full lecture series is hosted at Ergo.org in the form of a lecture course Ergo: Lectures on Infinity, part of the broad collection of philosophy lecture courses they have assembled there. Highly recommended, and I am proud to be part of it.

The lectures will also appear on the Ergo YouTube channel, in collaboration with my YouTube channel.

I shall be releasing the videos here on Infinitely More over the coming weeks and months, with full transcripts—subscribe now for access to these and all my other Infinitely More content. I’ll be sending out the first lecture very soon: Zeno’s paradox and infinite sums.

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Which numbers arise in this field? Can we represent √2 or √ω this way? Which surreal numbers are algebraic or transcendental over the ordinals? Do the ordinals reveal new algebraic relations concerning e or π? How can we know?

We shall discuss all this and more in today’s installment, part of my series of essays on the ordinal numbers—find them in the ordinals tag.

Let’s get into it!

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when the exponent n is larger than 2. For example, there is no nontrivial solution in the integers of a³ + b³ = c³ and no nontrivial solution of a⁴ + b⁴ = c⁴. By nontrivial, we just mean that the numbers a, b, c are all nonzero, since we don’t want to count. 2³ + 0³ = 2³ as a counterexample instance.

The theorem is named for Pierre de Fermat, who in 1637 scribbled a note claiming the result in the margin of book, saying that he had found a truly wonderful proof, but alas, the margin was too small to contain it.

Mathematicians spent centuries since that time struggling to find the missing proof, or indeed any proof at all, always failing, until finally Andrew Wiles proved the theorem in 1994. Wiles’s argument uses sophisticated contemporary ideas—almost surely not what Fermat had in mind. Indeed, many mathematicians believe that Fermat was probably mistaken about having a proof of the general result in the first place.

I propose that we should consider the question of Fermat’s last theorem in the natural ring of ordinals. Namely, are there nonzero numbers a, b, and c in the natural ring of ordinals ⟨Ord⟩ that solve

for an integer exponent n > 2? What do you think?

Think about it...

Let’s get into it!

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We discussed a sweeping selection of topics—the nature of mathematical truth, proof, platonism, fictionalism, the roles of set theory in mathematics, pluralism in the foundations of mathematics, the multiverse view, the continuum hypothesis, the junk-theorem phenomenon, the dispute between classical logic and constructive mathematics, and much more.

I should confess that one of my hosts and I are known to disagree on a few core philosophical matters—there are some Twitter exchanges to prove it—and our conversation here did not shy away from these points of contention. In a friendly spirit of constructive philosophical exchange, there were probing questions, challenges, and witty counterpoints. What an enjoyable and fruitful meeting of the minds it was. On which side of these questions do you find yourself? Post your views in the comments.

Please enjoy!

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Subscribe for full access to Infinitely More, with regular posts on infinity, philosophy, and mathematics.

Learn more at these links:

• Deniz and Thorston’s About Logic YouTube channel, where you’ll find numerous interviews with prominent logicians, mathematicians, and philosophers. Highly recommended.

• Joel David Hamkins YouTube channel

• My book, Lectures on the Philosophy of Mathematics

• My new book, The Book of Infinity

• Joel David Hamkins, “How the continuum hypothesis could have been a fundamental axiom,” Journal for the Philosophy of Mathematics (2024), DOI:10.36253/jpm-2936, arxiv:2407.02463.

In the previous essay, The Natural Ring of Ordinals Has Prime Factorization, we proved that this ring is an integral domain and indeed it is a unique factorization domain—every number factors uniquely as a finite product of primes. Thus, the ordinal analogue of the fundamental theorem of arithmetic holds in the natural ring of ordinals. Find the whole essay series in the ordinals tag.

Today, I should like to discuss several features in this ring that regrettably do not work out as one might have hoped or expected.

• The concept of even goes awry. Not every odd number has the form 2a＋1.

• The alternating even/odd pattern fails in the natural ring of ordinals. There are consecutive odd numbers!

• Indeed, there are arbitrarily long chains of consecutive odd numbers.

• The Collatz conjecture fails badly in the natural ring of ordinals.

• There are consecutive infinite prime numbers.

• There are arbitrarily long intervals of consecutive prime numbers.

• In this sense, the prime pair conjecture holds in the natural ring of ordinals in a very strong formulation.

• Greatest common divisors exist, but they do not always fulfill Bézout’s identity, by which the GCD is represented as a linear combination:

• The natural ring of ordinals ⟨Ord⟩ is not a Euclidean domain—we cannot always divide with remainder.

• We therefore cannot reliably implement the Euclidean algorithm.

• Indeed, the natural ring of ordinals is not a principal ideal domain, nor is it a Noetherian ring (these concepts will be explained).

• The natural ring of ordinals is not an integer part of the surreal field. There are arbitrarily large intervals in the surreal field containing no member of the natural ring of ordinals.

Let’s get into it!

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A ring is a certain kind of algebraic structure generalizing the familiar arithmetic of the integers ⟨ℤ,+,·⟩ to a more general or abstract realm of number objects. In ring theory, we aim to leverage our understanding of the integers to these more abstract realms. And indeed the natural ring of ordinals generalizes many features of the integers to a context including the transfinite ordinals. Today we shall how prime factorization is manifested in the natural ring of ordinals, with an ordinal version of the fundamental theorem of arithmetic—every number factors uniquely as a finite product of primes.

We introduced the basic construction of ⟨Ord⟩ in my previous essay, The Natural Ring of Ordinals, where we saw how the numbers of this ring can be represented as formal ordinal differences α ─ β, taken with respect to the same-difference equivalence relation. We outlined several attractive algebraic features to which this leads. The natural ring of ordinals, for example, is a discretely ordered cancellative commutative ring with identity. We provided several normal forms for representing the numbers in this ring, such as the signed Cantor normal form

where the coefficients k_i are taken from the integers, including negative integers. We saw the closely related normal form based on finite-support formal polynomial expressions ∑_β ω^β · k_β and another normal form based on finite signed sums of distinct powers of 2. The natural ring of ordinals, it turns out, is isomorphic to the subring of the surreal numbers generated by the ordinals, and so one may legitimately imagine these numbers, if desired, as surreal numbers. Meanwhile, the direct construction is simple and proceeds independently of any need for the surreal field.

We ended the previous essay on a cliff-hanger with several tantalizing questions left open, which I shall presently begin to take up in this essay:

• Does ⟨Ord⟩ admit a robust concept of prime numbers?

• Does ⟨Ord⟩ admit a robust concept of even and odd?

• Does ⟨Ord⟩ admit greatest common divisors?

• Is √2 irrational with respect to ⟨Ord⟩?

• What about √ω?

I shall aim to provide a general algebraic analysis of the natural ring of ordinals, showing that it is an integral domain and indeed, a unique factorization domain, which will be the key to several of the questions above. Next time, however, we will see that the natural ring of ordinals is not a Euclidean domain, nor a principal ideal domain, nor a Noetherian ring.

Our analysis here will be based on the extremely fruitful structural observation that the natural ring of ordinals is isomorphic to a vast discretely ordered polynomial ring over the integers, arising in a presentation as a transfinite tower of rings extending endlessly upward with newly created indefinite variables x_α, one for each ordinal α, with each new generator x_α added on top, larger in the order than every polynomial using only earlier variables.

Once we do this, it follows on general ring-theoretic grounds that ⟨Ord⟩ is a unique factorization domain and therefore supports a robust theory of irreducibility and prime numbers, by which every number factors uniquely as a finite product of primes. This observation in turn is the key to several of the questions asked above.

Let’s get into it!

You can find my whole essay series on the ordinals in the ordinals tag.

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A ring is a certain kind of algebraic structure generalizing the familiar arithmetic of the integers ⟨ℤ,+,·⟩ to a more general or abstract realm of number objects, such as polynomials or matrices. In the natural ring of ordinals, for example, we shall find the ordinals as well as formal ordinal differences α ─ β. Ring theory is about leveraging our deep understanding of the integers to these more abstract objects and contexts, which can often retain many of the attractive features of integer arithmetic in a manner that fruitfully aids our thinking about them, often leading to insight. We can add and multiply polynomials, for example, and factor them; in certain contexts we can perform long division of polynomials with remainders in a manner that is exactly analogous to what we do in the integers with the Euclidean algorithm. In the natural ring of ordinals, we shall find a robust concept of prime number and indeed every number in this ring will factor uniquely as a product of primes—thus we shall discover an ordinal analogue of the fundamental theorem of arithmetic. This feature is completely lacking, in contrast, in the ring of omnific integers in the surreal numbers, which exhibits only weaker ring-theoretic features.

To be more specific about the definition, a ring consists of a realm of objects with accompanying concepts of addition and multiplication for them exhibiting certain regular features, namely, both addition and multiplication are associative; addition is commutative; there is an additive identity 0; every number has an additive inverse; and lastly, multiplication distributes over addition.

Examples of rings would include the ring of integers ⟨ℤ,+,·⟩, of course, with the usual arithmetic; but also modular arithmetic ℤ/nℤ with addition and multiplication modulo n; the ring ℤ[x] of polynomials in the indefinite variable x with integer coefficients; the polynomial rings ℤ[x, y, z,…] allowing additional variables; the various field structures, such as the rational field ℚ; the real field ℝ; the complex field ℂ; and the associated polynomial rings over these fields, such as ℚ[x, y, z,…]. These are all commutative rings, meaning that multiplication as well as addition is commutative, but there are also many noncommutative rings, such as the various matrix rings ℝ^2×2 or ℂ^n×n, and many others.

Meanwhile, the natural numbers ℕ with the usual addition and multiplication do not form a ring, since they lack additive inverses, but the natural numbers do form what is called a semiring, which drops that requirement. Similarly, the ordinals are not a ring, since they also lack additive inverses, although with the natural sum and product they do form a semiring. We shall ultimately expand the class of ordinals with ideal number objects that will represent the various possible ordinal differences α ─ β, thereby providing a presentation of the natural ring of ordinals, which has many attractive features.

I denote the natural ring of ordinals by ⟨Ord⟩ in order to suggest that it is generated by but not identical to the class of ordinals. The natural ring of ordinals, it turns out, is a discretely ordered commutative ring, just like the integers themselves, and it is moreover an integral domain, which means nonzero numbers never multiply to zero. Further, it will turn out that the natural ring of ordinals is a unique factorization domain, which means that we shall find in it a robust concept of prime number and every number will factor uniquely as a finite product of primes.

Perhaps the reader will be surprised to learn that the natural ring of ordinals can be realized isomorphically as a vast discretely ordered polynomial ring over the integers, arising in a presentation as a transfinite tower of rings extending endlessly upward with newly created indefinite variables x_α, one for each ordinal α, with each new generator x_α added on top, larger in the order than every polynomial using only earlier variables.

We shall also find a variety of useful normal forms for the elements of the natural ring of ordinals, such as the signed analogue of the Cantor normal form,

with the difference that we now allow the coefficients k_i to be arbitrary integers, including negative integers, in order to accommodate the ordinal differences. Numbers in the natural ring of ordinals can also be represented as a signed sum of distinct powers of 2:

with α_n > ··· > α₀, although this representation is not unique, in light of such examples as 8 - 1 = 4 + 2 + 1.

The natural ring of ordinals, it turns out, is the same as the ring generated by the ordinal numbers inside the surreal number field—but it is strictly contained within the omnific integers.

To mention a few intriguing features, we shall eventually prove that √2 is irrational in the natural ring of ordinals, just as it is with the classical Pythagorean result in the integers. Of course, this would be the expected result, at least until one recalls that this wasn’t true in the omnific integers, since we saw earlier that √2 is rational with respect to the omnific integers and indeed every surreal number is rational with respect to the omnific integers. The number 2 is prime in the natural ring of ordinals and every number factors uniquely into primes, which might seem initially to give a solid basis for the concepts of even and odd in the natural ring of ordinals. However, these concepts are regrettably a little less successful there than in the integers, for matters go somewhat awry in regard to the order—we lose the uniformly regular even/odd pattern and indeed, there will be arbitrary long intervals in the natural ring of ordinals having no even numbers at all. This phenomenon can be seen as an instance of the failure of the Euclidean algorithm in the natural ring of ordinals—⟨Ord⟩ is not a Euclidean domain, nor even a principal ideal domain, nor is it Noetherian. Although we shall have a concept of greatest common divisor in the natural ring of ordinals, nevertheless the ordinal analogue of Bézout’s identity will fail—the greatest common divisor of two numbers will not always arise as a linear combination of them.

Eventually we shall also perform the quotient field construction and thereby construct the natural field of ordinals, as well as its real algebraic closure. Let us aim to explore all these ordinal number rings and fields together.

Let’s get into it!

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At one point, we turned to John Conway and the surreal numbers, and so please enjoy this excerpt from the conversation.

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Lex Fridman(02:46:47) So speaking of the land of nonsense, I have to ask you about surreal numbers, …there’s this aforementioned wonderful blog post on the surreal numbers and that there’s quite a simple surreal number generation process that can basically construct all numbers. So maybe this is a good spot to ask what are surreal numbers and what is the way we can generate all numbers?

Joel David Hamkins(02:47:20) So the surreal number system is an amazing, an amazingly beautiful mathematical system that was introduced by John Conway.

Lex Fridman(02:47:30) Rest in peace, one of the great mathematicians ever on this earth.

Joel David Hamkins(02:47:33) Yes, absolutely. And I really admire his style of mathematical thinking and working in mathematics and the surreal number system is a good instance of this. So the way I think about the surreal numbers system is what it’s doing is providing us a number system that unifies all the other number systems. So it extends the real numbers. Well, not only does it extend the integers, the natural numbers, the rational numbers, and the real numbers, but also the ordinals and the infinitesimals. So they’re all sitting there inside the surreal numbers, and it’s this colossal system of numbers. It’s not a set even. It’s a proper class, it turns out, because it contains all the ordinal numbers.

Joel David Hamkins(02:48:19) But it’s generated from nothing by a single rule, and the rule is, so we’re going to generate the numbers in stages, in a transfinite sequence of stages. And at every stage, we take the numbers that we have so far and in all possible ways, we divide them into two sets, a lower set and an upper set, or a left set and a right set. So we divide them into these two sets so that everything in the left set is less than everything in the right set, and then at that moment, we create a new number that fits in the gap between L and R. Okay? That’s it. That’s all we do. So let me say it again.

Joel David Hamkins(02:49:05) The rule is we proceed in stages, and at any stage, in all possible ways, we divide the numbers we have into two collections, the left set and the right set, so that everything in the left set is less than everything in the right set. And we create a new number, a new surreal number that will fit in that gap. Okay. So for example, we could start… Well, at the beginning, we don’t have any numbers. We haven’t created anything yet, and so, we could take nothing and we could divide it into two sets, the empty lower set and the empty upper set. I mean, the two empty sets. And everything in the empty set is less than everything in the empty set because that’s a vacuous statement.

Joel David Hamkins(02:49:48) So we’re, we satisfy the conditions and we apply the number generation rule, which says we should create a new number. And this is what I call the big bang of numbers, the surreal genesis when the number zero is born. Zero is the firstborn number that is bigger than everything in the empty set and less than everything in the empty set. Okay, but now we have this number zero, and so therefore, we now can define new gaps. Because if we put zero into the left set and have an empty right set, then we should create a new number that’s bigger than zero and less than everything in the empty set, and that number is called the number one.

Joel David Hamkins(02:50:30) And similarly, at that same stage, we could have put zero into the right set, and so that would be the firstborn number that’s less than zero, which is called minus one. So now we have three numbers, minus one, zero, and one, and they have four gaps because there could be a number below minus one or between minus one and zero or between zero and one or above one, and so we create those four new numbers. The first number above one is called two. The first number between zero and one is called 1/2, and then on the negative side, we have minus 1/2 and minus two and so on. So now we have, what is that, seven numbers. So there’s eight gaps between them.

Joel David Hamkins(02:51:10) So at the next birthday, they call them, the next stage will be born all the numbers between those gaps, and then between those and between those and so on. And as the days progress, we get more and more numbers. But those are just the finite birthdays, because as I said, it’s a transfinite process. So at day omega, that’s the first infinite day, we’re going to create a lot of new surreal numbers. So every real number will be born at that stage, because every real number fills a gap in the previously born rational numbers that we had just talked about. It’s not all the rationals, because actually the rational numbers that are born at the finite stages are just the rationals whose denominator is a power of two, it turns out. Those are called the dyadic rationals.

Joel David Hamkins(02:51:57) So the real numbers are all born on day omega, but also some other numbers are born on day omega. Namely, the ordinal omega itself is the firstborn number that’s bigger than all those finite numbers, and minus omega is the firstborn number that’s less than all those finite numbers. But also, we have the number epsilon, which is the firstborn number that’s strictly bigger than zero and strictly less than all the positive rational numbers. So that’s going to be an infinitesimal number in that gap, and so on. On day omega plus one, we get more numbers, and then omega plus two and so on. And the numbers just keep coming forever. So, this is how you build the surreal number system.

Joel David Hamkins(02:52:39) And then it turns out you can define the arithmetic operations of addition and multiplication in a natural way that is engaging with this recursive definition. So we have sort of recursive definitions of plus and times for the surreal numbers. And it turns out you can prove that they make the surreal numbers into what’s called an ordered field. So they satisfy the field axioms, which means that you have distributivity and commutativity of addition and multiplication, and also you have reciprocals for every non-zero number. You can divide by the number. So you can add and multiply and divide and subtract. And furthermore, you can take square roots.

Joel David Hamkins(02:53:21) And furthermore, every odd degree polynomial has a root, which is true in the real numbers, because if you think about, say, a cubic or a fifth degree polynomial, then you know it’s going to cross the axis, because it has opposite behaviors on the two infinities, because it’s an odd degree polynomial. So on the positive side, it’s going to the positive infinity. On the negative side, it would be going to minus infinity. So it has to cross. So we know in the real numbers, every odd degree polynomial has a root. And that’s also true in the surreal numbers. So that makes it what’s called a real closed field which is a very nice mathematical theory. So it’s really quite interesting how we can find copies of all these other number systems inside the surreal numbers.

Lex Fridman(02:54:09) But the surreal numbers are fundamentally discontinuous as you’re worried about. What are the consequences of this?

Joel David Hamkins(02:54:14) Right. So the surreal numbers have a property that they form a non-standard model of the real field, which means that they provide a notion of infinitesimality that one can use to develop calculus on the grounds of Robinson’s non-standard theory that I had mentioned earlier. But they don’t have the least upper bound property for subcollections. There’s no set of surreal numbers, no non-trivial set of surreal numbers has at least upper bound, and there are no convergent sequences in the surreal numbers. And so for the sort of ordinary use in calculus based on limits and convergence, that method does not work in the surreal numbers at all. So that’s what I mean when I say the surreal numbers are fundamentally discontinuous. They have a fundamental discontinuity going on.

Joel David Hamkins(02:55:07) But you can still do calculus with them, because you have infinitesimals if you use these non-standard methods, the infinitesimal based methods to calculus. And people do that. I once organized a conference in New York, and we had John Conway as a speaker at that conference. And there was a question session, and someone asked him, I mean, it’s a bit of a rude question, I think, but they asked it and the question was, “What is your greatest disappointment in life?” I mean, I would never ask a question like that at a conference in a very public setting.

Joel David Hamkins(02:55:41) But Conway was extremely graceful and he answered by saying that, “The surreal numbers…” Not the numbers themselves, but the reception of the surreal numbers, because he had ambition that the surreal numbers would become a fundamental number system used throughout mathematics and science, because it was able to do nonstandard -set analysis, it was able to do calculus, it unified the ordinals and so on. And it’s such a unifying, amazing structure, beautiful structure with elegant proofs and sophisticated ideas all around it. And he was disappointed that it never really achieved that unifying status that he had the ambition for. And this, he mentioned as his greatest disappointment.

Lex Fridman(02:56:32) Yeah, Donald Knuth tried to celebrate it, but it never quite took hold.

Joel David Hamkins(02:56:36) So I don’t want to give the impression, though, that the surreal numbers are not widely studied, because there are thousands of people who are…

Lex Fridman(02:56:41) Sure

Joel David Hamkins(02:56:42) …studying it. In fact, Philip Ehrlich, who is one of the world experts on the surreal numbers, mentioned to me once that Conway was his own worst enemy with regard to that very issue because in the Conway style, everything is a game. And he treated the surreal numbers as a kind of plaything, a toy, and maybe that makes people not take it seriously. Although my view is that it is extremely serious and useful and profound, and I’ve been writing a whole series of essays on the surreal numbers for my Substack at Infinitely More. And I just find the whole subject so fascinating and beautiful. I mean, it’s true. I’m not applying it in engineering, which maybe was part of this Conway ambition.

Lex Fridman(02:57:30) And I just wanted to, before I forget, mention Conway turning everything into a game. It is a fascinating point that I didn’t quite think about, which I think the Game of Life is just an example of exploration of cellular automata. I think cellular automata is one of the most incredible, complicated, fascinating… It feels like an open door into a world we have not quite yet explored. And it’s such a beautiful illustration of that world, the Game of Life, but calling it a game… Maybe life balances it, because that’s your powerful word, but it’s not quite a game. It’s a fascinating invitation to an incredibly complicated and fascinating mathematical world.

Lex Fridman(02:58:09) I think every time I see cellular automata and the fact that we don’t quite have mathematical tools to make sense of that world, it fills me with awe. Speaking of a thousand years from now, it feels like that is a world we might make some progress on.

Joel David Hamkins(02:58:23) The Game of Life is a sort of playground for computably undecidable questions because, in fact, you can prove that the question of whether a given cell will ever become alive is computably undecidable. In other words…

Lex Fridman(02:58:39) Yeah

Joel David Hamkins(02:58:39) …given a configuration, and you ask, “Will this particular cell ever, you know, be alive—” …in the evolution?” And you can prove that that question is equivalent to the halting problem. It’s computably undecidable. It’s semi-decidable in the sense that if it will become alive, then you will know it at a finite stage because you could just run the Game of Life algorithm and let it run. And if it ever did come alive, you could say, “Yeah, it was alive.” But if you’ve run it for a thousand years and it hasn’t come alive yet, then you don’t necessarily seem to have any basis for saying, “No, it won’t ever come alive,” if the behavior was very complicated.

Joel David Hamkins(02:59:18) Maybe if you have a complete understanding of the evolution of the behavior, then you can say no, but you can prove you won’t always have that understanding— …precisely because the problem is equivalent to the halting problem.

Lex Fridman(02:59:28) And nevertheless, when you sit back and look and visualize the thing, some little mini cellular automata civilizations are born and die quickly, and some are very predictable and boring, but some have this rich, incredible complexity.

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See the full transcript and watch the full video episode for more. I shall periodically be posting more excerpts like this one here on Infinitely More—find them in the lex-fridman tag.

Read more about the surreal numbers in my series of essays in surreal-numbers tag, including the introductory essay The Surreal Numbers.

In particular, we shall have a purely order-theoretic account, the merge product α ▪ β, which I prefer to conceive as the principal semantic concept, although in mathematical practice this is less often given as the main definition; next a computational account I shall denote by α ⊗ β, based on the Cantor normal form, along with a closely related formal polynomial account α ⊛ β; after this, we shall have a definition of the natural product α ⊙ β by transfinite recursion; and finally, the multiplication of ordinals that arises in the surreal numbers α • β. Ultimately, we shall prove that all five notions are identical—they are different equivalent ways of looking at the same operation, the natural product, which in the end we shall often denote simply by αβ.

The argument is subtle, certainly not routine, and so I shall be glad to give a slow, careful presentation here. I am especially glad to do so because to my way of thinking, this is a core result about the natural product, but unfortunately, the full result is not commonly available in one place—one finds it piecemeal, stated and proved only partially and indeed it is often stated without any proof.

So let’s get into the fine details of what I regard as a fundamental illuminating result on the nature of the ordinals, regarding one of the most beautiful and natural operations on the ordinals, the natural product.

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Check it out at your favorite booksellers.

• Amazon

• Barnes and Noble

• MIT Press Bookstore

• Other options

From the preface:

Come, let us explore infinity! We shall visit all my favorite paradoxes and conundrums. The ancient puzzles, confounding or intractable, will yield at times to our analysis. And what a joy it is to experience those Aha! moments—a flash of clarity lights the way out of the labyrinth. But alas, having escaped one maze, we shall often find ourselves immediately lost in another—a new paradox with new questions to answer. The puzzles of infinity are endless riddles nestled within one another.

The Book of Infinity was the original motivation for me to begin my substack Infinitely More. When I first arrived a few years ago at the University of Notre Dame from Oxford, I was asked by my new department what course I would most want to teach. My answer was a new course on infinity that I had long dreamed about—what fun it would be to share my ideas and puzzles with enthusiastic students, tracing the concept from ancient times to contemporary issues. I set furiously to work preparing this book, a series of vignettes on infinity, and we offered the course. I serialized the chapters on Infinitely More as they were completed—see the section The Book of Infinity. I’ve since taught the course several more times, and with further polishing and editing, the book is finally completed.

400 pages and 26 chapters:

1. The Book of Numbers

2. The Sand Reckoner

3. Zeno’s Paradox

4. The Method of Exhaustion

5. Supertasks

6. The Infinite Coastline Paradox

7. The Paradox of Giants

8. The Paradox of the Largest Tweetable Number

9. Potential Versus Actual Infinity

10. Equinumerosity and Comparison of Size

11. What Is the Infinite?

12. Hilbert’s Grand Hotel

13. Uncountable Infinity

14. How to Count

15. Transfinite Recursive Constructions

16. Slaying the Hydra

17. The Continuum Hypothesis

18. Throwing Darts at the Real Line

19. The Orders of Infinity

20. The Surreal Numbers

21. The Axiom of Choice

22. Infinitary Hat Puzzles and the Aftermath

23. The Guessing-Box Puzzle

24. We Can Predict the Future

25. Infinite Liars

26. Common Knowledge

Here are a few snippets from the index, to give you an idea of what’s covered…

The book is packed with full-color mathematical figures—over 200 color figures, of my own design, which I produced in LaTeX using TikZ. Here are a few samples:

And many others! Each figure is woven into the text to help explain a mathematical or philosophical idea.

Meanwhile, I am serializing all my other books-in-progress here on Infinitely More—subscribe for full access to all my current work, including the surreal numbers, games, logic, philosophy of mathematics, and more.

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Notably, the natural sum and product on ordinals are both commutative operations—unlike the standard ordinal arithmetic—and so the natural semiring of ordinals is a commutative semiring. In fact, the natural sum and product operations on ordinals are the same operations that the ordinals exhibit in the surreal numbers, which makes the natural semiring of ordinals a subsemiring of the surreal numbers.

I shall describe several independent and self-standing approaches to the natural sum and product—we shall ultimately have five separate accounts of each operation, which proceed from and express different philosophical perspectives on how we should best undertake mathematical definitions with the ordinals. One account of the natural sum, for example, offers a purely order-theoretic structuralist account, while another can be seen as motivated by essentially computational concerns—how to compute the sum and product values—and still another account adopts in effect a proof-theoretic perspective by presenting a formal transfinite recursion. Ultimately, of course, we shall prove that the various alternative accounts of the natural sum and product are equivalent—they all ultimately define the same ordinal operations of the natural sum and product.

This is a happy situation, therefore, since to have multiple independent accounts of the same underlying mathematical idea is often valuable for mathematical insight. The different but ultimately equivalent approaches to the topic enrich our mathematical understanding by stretching our knowledge in different but fruitful directions. Different perspectives suggest different avenues of generalization, and some perspectives can be more clarifying than others depending on the specific case.

Let’s get into it.

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The first infinite number is ω, but one can always add 1 more.

The next simple limit is ω + ω, which is the same as ω · 2, and so one continues. Each new limit ordinal begins a new block of ordinals of length ω, a new era of infinity.

We proceed to the next limit ordinal ω · 3, starting yet another era of infinity, then ω · 4 after that, and indeed ω · n + k for every finite n and k:

In this way we have counted to ω². The ordinals encountered along the way take the form ω · n + k for finite n and k.

Counting to ω² is rather like counting to 100. When we count to 100 you might notice that within each decade—the teens, the twenties, the thirties, and so on—it is just like counting to 10 again. In counting to 100, which is 10², we thus count to 10 altogether 10 times. Similarly, when we count to ω², we count to ω altogether ω many times. We start with the finite numbers, the original copy of ω, and then proceed from ω to ω · 2, from ω · 2 to ω · 3, and so on. In counting up to ω², we thus encounter ω many eras, each of size ω, in effect counting to ω altogether ω many times. And just as the numbers up to 100 have two digits in base ten, with the form 10 · n + k, similarly the ordinals up to ω² have the form ω · n + k, which is two digits in base ω.

The ordinal ω² is the first compound limit ordinal—a limit ordinal that is a limit of limit ordinals since ω² is the limit of ω · n as n increases in ω. In other words, ω² is a limit ordinal, but there is no largest limit ordinal below it. A simple limit ordinal, in contrast, is a limit ordinal that is not a compound limit—all simple limits take the form α + ω for some ordinal α.

Counting to ω^ω and beyond

But I should truly like us to count much further. We essentially repeat the process of counting to ω² when counting from ω² to ω² · 2, then again when counting further to ω² · 3, and similarly through every successive ω² · n. With ω many repetitions, we thus count to ω² · ω, which is the ordinal ω³. By repeating that process ω many times, we reach ω⁴, and so on. Thus we are on our way to the local peak ω^ω, which is the supremum of ωⁿ for all finite numbers n.

Continuing further, if we count like this to ω^ω altogether ω many times, first to ω^ω · 2, then to ω^ω · 3, and so on, then we shall reach ω^ω · ω, which is the same as ω^ω+1. In light of the difficulty of reaching ω^ω in the first place, however, and having had to do that work ω many times to reach ω^ω+1, we might notice that it was a troublesome burden for us to increase the exponent merely by 1. And we shall have infinitely more such trouble again to reach ω^ω+2, and then still infinitely more trouble to reach ω^ω+3, and so on. Each increase of the exponent by 1 requires an additional infinite duplication of all the preceding difficult work to that juncture. And yet we shall not stop counting. With perseverance we shall reach ω^ω·2 and beyond—every tiny increase in the exponent is an achievement to be celebrated.

With stoical fortitude, we thus find our way to ω^ω·3 and then to ω^ω·4, on the way to

Eventually, exceeding that we shall arrive at

and then

and so on. We likely find ourselves exhausted at each new height of achievement. Nevertheless, we continue onward to

With enduring heroic dedication, we press on ever upward, successively scaling the towering further summits:

Each new step up with these finite-stack tetrations is a vast increase over the previous instance—remember how difficult it was to increase the exponent just by 1, but here we see huge steps up with vast exponential towers of increase. Nevertheless, with silent resolve and quiet determination we shall climb through these iterated exponential powers. The supremum of these finite-stack tetrations is a vast pinnacle, the ordinal known as ε₀.

Let us get started more seriously.

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The conversation turned at one point to the question of who has been the greatest mathematician of all time. I demurred a bit at the question—explaining that I don’t organize my thinking about mathematicians in such a ranked list, and find insight wherever it might arise, which isn’t always only from the greats—but I did eventually give an answer, which you can find out below. The question was an opportunity to talk about differing mathematical styles, including my own mathematical style, which has served me very well in my mathematical investigations.

Please enjoy this excerpt from our extended conversation. The transcript is below.

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Lex Fridman(03:12:59) Sorry to ask the ridiculous question, but who is the greatest mathematician of all time? Who are the possible candidates? Euler, Gauss, Newton, Ramanujan, Hilbert. We mentioned Gödel, Turing, if you throw him into the bucket.

Joel David Hamkins(03:13:14) So this is, I think, an incredibly difficult question to answer. Personally, I don’t really think this way about ranking mathematicians by greatness. Um…

Lex Fridman(03:13:28) So you don’t have, like… You know, some people have a Taylor Swift poster in their dorm room. You don’t have it.

Joel David Hamkins(03:13:33) I mean, if you forced me to pick someone, it would probably be Archimedes because…

Lex Fridman(03:13:37) Archimedes

Joel David Hamkins(03:13:37) …he had such incredible achievements in such an early era, which totally transcended the work of the other people in his era. But I also have the view that I want to learn mathematics and gain mathematical insight from whoever can provide it and wherever I can find it. And this isn’t always just coming from the greats. Sometimes the greats are doing things that are just first and not… You know, somebody else could have easily been first. So there’s a kind of luck aspect to it when you go back and look at the achievements. And because of this progress issue in mathematics that we talked about earlier, namely we really do understand things much better now than they used to.

Joel David Hamkins(03:14:22) And when you look back at the achievements that had been made, then maybe you can imagine thinking, “Well, somebody else could’ve had that insight also.” And maybe they would have… It’s already a known phenomenon that disparate mathematicians end up proving essentially similar results at approximately the same time. But, okay, the person who did it first is getting the credit and so on.

Lex Fridman(03:14:48) What do you make of that? Because I see that sometimes when mathematicians… This also applies in physics and science, where completely separately, discoveries are made…

Joel David Hamkins(03:14:58) Right. Yeah.

Lex Fridman(03:14:58) …maybe at a very similar time. What does that mean?

Joel David Hamkins(03:15:01) It’s relatively common. I mean, I think it’s like certain ideas are in the air and being thought about but not fully articulated, and so this is the nature of growth in knowledge.

Lex Fridman(03:15:13) Do you understand where ideas come from?

Joel David Hamkins(03:15:16) Not really.

Lex Fridman(03:15:17) I mean, what’s your own process when you’re thinking through a problem?

Joel David Hamkins(03:15:22) Yeah, that’s another difficult question. I suppose it has to do with… My mathematical style, my style as a mathematician, is that I don’t really like difficult mathematics. What I love is simple, clear, easy-to-understand arguments that prove a surprising result. That’s my favorite situation. And actually, the question of whether it’s a new result or not is somehow less important to me. And so that has to do with this question of the greats and so on, whoever does it first. Because I think, for example, if you prove a new result with a bad argument or a complicated argument, that’s great because you proved something new. But I still want to see the beautiful, simple, because that’s what I can understand.

Joel David Hamkins(03:16:16) Also, I’m kind of naturally skeptical about any complicated argument because it might be wrong. And… …If I can’t really understand it fully, like every single step all at once in my head, then I’m just worried maybe it’s wrong. And so these different styles, sometimes mathematicians get involved with these enormous research projects that involve huge numbers of working parts and… …Different technology coming together. I mean, mathematical technology, not physical technology.

Lex Fridman(03:16:48) And sometimes it actually involves now more and more something like the Lean programming language where some parts are automated, so you have this gigantic…

Joel David Hamkins(03:16:54) Yeah, yeah, I see. Well, that’s another issue because maybe those things are less subject to skepticism when it’s validated…

Lex Fridman(03:17:02) Sure

Joel David Hamkins(03:17:02) …by Lean. But I’m thinking about the case where the arguments are just extremely complicated, and so I sort of worry whether it’s right or not, whereas you know, I like the simple thing. So I tend to have often worked on things that are a little bit off the beaten path from what other people are working on from that point of view.

Lex Fridman(03:17:23) Your curiosity draws you towards simplicity.

Joel David Hamkins(03:17:25) Yeah. I want to work on the things that I can understand and that are simple. Luckily, I’ve found that I’ve been able to make contributions that other people seem to like, in this way, in this style. So I’ve been fortunate from that point of view. My process always, though, and I’ve recommended this always to my students, is just a kind of playful curiosity. So whenever I have…

Joel David Hamkins(03:17:55) Whenever there’s an idea or a topic then I just play around with it and change little things or understand a basic case and then make it more complicated or press things a little bit on this side or apply the idea to my favorite example that’s relevant, and see what happens, or you just play around with ideas, and this often leads to insights that then lead to more methods or more, then pretty soon you’re making progress on the problem. So this is basically my method, is I just fool around with the ideas until I can see a path through towards something interesting… …And then prove that, and that’s worked extremely well for me. So I’m pretty pleased with that method.

Lex Fridman(03:18:47) You do like thought experiments where you anthropomorphize like you mentioned?

Joel David Hamkins(03:18:51) Yeah, yeah. So this is a basic tool. I mean, I use this all the time. You imagine a set-theoretic model, a model of ZFC, as like a place where you’re living, and you might travel to distant lands by forcing. This is a kind of metaphor for what’s going on. Of course, the actual arguments aren’t anything like that because there’s not land and you’re not traveling and you’re not…

Lex Fridman(03:19:13) But you allow your mind to visualize that kind of thing-

Joel David Hamkins(03:19:15) Yeah

Lex Fridman(03:19:15) … in the natural real world.

Joel David Hamkins(03:19:16) And it helps you to understand. Particularly when there are parts of the argument that are in tension with one another, then you can imagine that people are fighting or something. And those kinds of metaphors, or you imagine it in terms of a game theoretic, you know, two players trying to win. So that’s kind of tension. And those kinds of metaphorical ways of understanding a mathematical problem often are extremely helpful in realizing, aha, the enemy is going to pick this thing to be like that because, you know, it makes it more continuous or whatever, and then we should do this other thing in order to… So it makes you realize mathematical strategies for finding the answer and proving the theorem that you want to prove because of the ideas that come out of that anthropomorphization.

Lex Fridman(03:20:01) What do you think of somebody like Andrew Wiles, who spent seven years grinding at one of the hardest problems in the history of mathematics? And maybe contrasting that a little bit with somebody who’s also brilliant, Terence Tao, who basically says if he hits a wall, he just switches to a different problem and he comes back and so on. So it’s less of a focused grind for many years without any guarantee that you’ll get there, which is what Andrew Wiles went through.

Joel David Hamkins(03:20:30) Right.

Lex Fridman(03:20:30) Maybe Grigori Perelman did the same.

Joel David Hamkins(03:20:32) I mean, Wiles proved an amazing theorem, Fermat’s Last Theorem result is incredible. This is a totally different style than my own practice, though, of working in isolation. For me, mathematics is often a kind of social activity. I have… I counted, I mean, it’s pushing towards a hundred collaborators, co-authors on various papers and so on. And, you know, if anybody has an idea they want to talk about with me, if I’m interested in it, then I’m going to want to collaborate with them and we might solve the problem and have a joint paper or whatever. You want to have a joint paper? Let me-

Lex Fridman(03:21:06) Yeah, exactly. Let’s go.

Joel David Hamkins(03:21:08) So my approach to making mathematical progress tends to involve working with other people quite a lot rather than just working on my…

Joel David Hamkins(03:21:17) …own, and I enjoy that aspect very much. So I, personally, I couldn’t ever do what Wiles did. Maybe I’m missing out. Maybe if I locked myself, you know, in the bedroom and just worked on whatever, then I would solve it. But I tend to think that no, actually, being on MathOverflow so much and I’ve gotten so many ideas, so many papers have grown out of the MathOverflow conversations and back and forth. Someone posts a question and I post an answer on part of it, and then someone else has an idea and it turns into a full solution, and then we have a three-way paper coming out of that. That’s happened many times. And so for me, I enjoy this kind of social aspect to it. And it’s not just the social part.

Joel David Hamkins(03:22:01) Rather, that’s the nature of mathematical investigation as I see it, is putting forth mathematical ideas to other people and they respond to it in a way that helps me learn, helps them learn, and I think that’s a very productive way of undertaking mathematics.

Lex Fridman(03:22:20) I think it’s when you work solo on mathematics, from my outsider perspective, it seems terrifyingly lonely. And because you’re, especially if you do stick to a single problem, especially if that problem has broken many brilliant mathematicians in the past, that you’re really putting all your chips in. And just the torment… …The rollercoaster of day to day. Because I imagine you have these moments of hopeful break, mini breakthroughs, and then you have to deal with the occasional realization that, no, it was not a breakthrough, and that disappointment.

Lex Fridman(03:23:00) And then you have to go, like, a weekly, maybe daily disappointment where you hit a wall, and you have no other person to brainstorm with. You have no other avenue to pursue. And it’s, I don’t know, the mental fortitude it takes to go through that. But everybody’s different. Some people are recluse and just really find solace in that lone grind. I have to ask about Grisha Grigori Perelman. What do you think of him famously declining the Fields Medal and the Millennial Prize? So he stated, “I’m not interested in money or fame. The prize is completely irrelevant to me. If the proof is correct, then no other recognition is needed.” What do you think of him turning down the prize?

Joel David Hamkins(03:23:52) I guess what I think is that mathematics is full of a lot of different kinds of people. And my attitude is that, hey, it doesn’t matter. Maybe they have a good math idea, and so I want to talk to them and interact with them. And so I think the Perelman case is maybe an instance where, you know, he’s such a brilliant mind and he solved this extremely famous and difficult problem, and that is a huge achievement. But he also had these views about, you know, prizes and somehow, I don’t really fully understand why he would turn it down.

Lex Fridman(03:24:33) I do think I have a similar thing, just observing Olympic athletes that are, in many cases, don’t get paid very much, and they nevertheless dedicate their entire lives for the pursuit… … Of the gold medal. I think his case is a reminder that some of the greatest mathematicians, some of the greatest scientists and human beings do the thing they do, take on these problems for the love of it, not for the prizes or the money or any of that. Now, as you’re saying, if the money comes, you could use it for stuff. If the prizes come, and the fame, and so on, that might be useful. But the reason fundamentally the greats do it is because of the art itself.

Joel David Hamkins(03:25:13) Sure, I totally agree with that. I mean, I share the view. That’s, you know, that’s why I’m a mathematician is because I find the questions so compelling and I’ve spent my whole life thinking about these problems. But, you know, but like if I won an award…

Lex Fridman(03:25:32) Yeah, it’s great. It’s great. I mean, I’m pretty sure you don’t contribute to MathOverflow for the wealth and the power. That you gain. I mean, it’s, yeah, genuine curiosity.

Joel David Hamkins(03:25:46) Well, you asked who the greatest mathematician is, and of course if we want to be truly objective about it, we would need a kind of an objective criteria…

Lex Fridman(03:25:55) Criteria, yeah.

Joel David Hamkins(03:25:55) …about how to evaluate the relative, you know, strength and the reputation of various mathematicians. And so, of course, we should use MathOverflow score… …Because…

Lex Fridman(03:26:06) That you’re definitively… I mean, nobody’s objectively the greatest mathematician of all time.

Joel David Hamkins(03:26:10) Yes, that’s true. I’ve also argued that tenure and promotion decisions should be based…

Lex Fridman(03:26:15) Based on MathOverflow.

Joel David Hamkins(03:26:16) …Yeah. So my daughter introduced me to her boyfriend. …And told me that she had a boyfriend. And I, um…

Lex Fridman(03:26:25) Asked him what his MathOverflow…

Joel David Hamkins(03:26:26) I wanted to know, first of all, what is his chess rating, and secondly, what is his MathOverflow score?

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See the full transcript and watch the full video episode for more. I shall periodically be posting more excerpts like this one here on Infinitely More—find them in the lex-fridman tag.

Let’s get into it, learning how to do ordinal arithmetic easily in the Cantor normal form. There are some surprising tricks that allow whole parts of the expressions to simplify away to nothing—they just disappear, leaving only the correct simplified expression behind.

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If you add two finite numbers together, then the resulting sum, of course, remains finite. Thus, the finite numbers are closed under addition. We may view this phenomenon as expressing a closure property of the ordinal ω itself, the first infinite ordinal. Namely, ω is closed under addition—the sum of any two numbers smaller than ω, that is, the sum of any two finite numbers, has a result that is still below ω. This is what it means to say that the ordinal ω is additively indecomposable.

Which other ordinals have this closure property? For example, what is the next additively indecomposable ordinal after ω?

Think about it…

Well, if an ordinal above ω is closed under addition, it would have to be bigger than ω + ω, and bigger than ω + ω + ω, and so forth. It would have to be bigger than ω · n for every finite number n. So it will be at least ω², the first compound limit ordinal.

Meanwhile, we can observe that ω² itself is closed under addition. Namely, if ordinals α, β are both below ω², then they are both below some ω · n for some large enough finite n, and consequently, we may bound the sum α + β as follows:

In other words, if α and β are smaller than ω², then α + β also is smaller than ω²—so the ordinal ω² is closed under addition. That is, ω² is additively indecomposable.

Additively Indecomposable Ordinals

In the general case, we define that an ordinal λ is additively indecomposable if every finite sum of ordinals below λ remains below λ. The ordinal ω, for example, is additively indecomposable, since the sum of finitely many finite numbers is finite, and we just observed above that ω² is additively indecomposable. The number 1 also is additively indecomposable, since every finite sum of ordinals below 1 adds up to 0, which remains less than 1.

What about 0 itself? Is 0 additively indecomposable? One might be inclined to say that 0 is additively indecomposable in a vacuous manner, since there are no smaller ordinals and therefore no finite sums of smaller ordinals. But this is not actually quite right according to the letter of the definition in light of the empty sum, which after all is a finite sum of ordinals, vacuously all less than 0, but the empty sum has value 0, which is not less than 0. On this technicality, therefore, 0 does not officially count as additively indecomposable. But actually, this is the answer we shall ultimately want, for this outcome makes for a smoother theory overall. The situation is similar to the question in number theory of whether the number 1 counts as prime. Sure, the only factors are 1 and itself, and yet mathematicians have agreed that we should not count 1 as prime. The number 1 after all is the value of the empty product, which is vacuously a product of smaller numbers. So 1 can be factored as a product of smaller numbers, the empty product, just as 0 is the value of a finite sum of smaller numbers, the empty sum.

Meanwhile, this consideration about the empty sum can also simply be absorbed into the statement by defining equivalently that an ordinal λ is additively indecomposable if 0 < λ and α + β < λ whenever α, β < λ. The stipulation that 0 < λ handles the empty sum, and the other finite sums are generated from ordinals below λ by adding extra terms one at a time. And so one often sees this latter definition, avoiding any need to consider the empty sum.

The additive indecomposability of an ordinal λ turns out to be equivalent to λ being additively irreducible, meaning that it cannot be expressed as a finite sum of smaller ordinals. Since 0 is the empty sum, an ordinal λ is additively irreducible if and only if 0 < λ and it is impossible that λ = α + β for some α, β < λ. (The reader is asked to prove the equivalence of additive indecomposability and irreducibility in the questions for further thought.)

Characterizing additive indecomposability

Which ordinals exactly are additively indecomposable? Can we characterize them? And what about multiplicative indecomposability? Is that the same as multiplicative irreducibility? And what about exponential indecomposability?

Think about it...

Let’s get into it…

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In this series of essays—you can find them in the ordinals tag—I shall cover all the basics of ordinal arithmetic, starting with the standard addition, multiplication, and exponentiation operations, but eventually getting to indecomposable ordinals, irreducible ordinals, Cantor normal form, binary ordinal representation, and more. Those who are new to the ordinals might want to start with How to Count.

A foundation for what is coming

I plan to lay down a solid foundation on these ordinal matters, with the aim in subsequent posts to grow the discussion into several deeper matters beyond, which require mastery over these concepts.

In particular, we shall continue on with my essays on the surreal numbers (in the surreal numbers tag), since our further work with that requires grounding in the ordinals. To foreshadow the coming topics, I shall subsequently introduce and investigate the so-called natural ordinal arithmetic (also known as the Hessenberg operations), which unlike the standard classical ordinal arithmetic are commutative operations. With the natural operations, the ordinals form a commutative semiring, the natural semiring of ordinals. The relevance of the natural ordinal arithmetic for the larger project of this book is that these are the same operations the ordinals experience in the surreal number field. After this, I shall introduce and develop the theory of what I call the natural ring of ordinals ⟨Ord⟩, which is the commutative ring generated by the ordinals under these operations, in which you can form such numbers as ω³ · 5 - ω² + ω - 7. This is precisely the subring generated by the ordinal numbers in the surreal field. How does the natural ring of ordinals compare with the Omnific integers? We saw that √2 is rational in Oz, for example, but what is the situation in the natural ring of ordinals ⟨Ord⟩? Do we have unique factorization in ⟨Ord⟩? Those are the questions at which we shall aim in the coming posts.

But first, in this essay, we shall put down a proper reliable foundation for the standard order-theoretic operations and basic theory of the ordinals. Everything coming later will build upon this.

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Namely, in this final essay I should like to discuss and defend what I see as an underlying potentialist nature to ultrafinitism. Specifically, I propose that we may fruitfully view the ultrafinitist perspective in a potentialist light, which will help illuminate its philosophical commitments, whilst also enabling a formal treatment of various ultrafinitist theories. What is more, I believe that the potentialist perspective brings to light certain fundamental issues on the nature of mathematical existence on which differing ultrafinitists might disagree, but which are most naturally discussed and adjudicated in a potentialist setting.

At the 2025 conference on ultrafinitism at Columbia University, Sam Buss mentioned that Ed Nelson had expressed ideas having a certain affinity with a potentialist outlook, in particular, the idea that things become true in arithmetic as you develop the theory—perhaps the twin primes conjecture could become true or the negation, depending on how the theory develops.

From the point of view of this essay, however, my entry into potentialism arises instead from a semantical perspective regarding the models of arithmetic, including models of the weak or the ultrafinitist theories. In a previous post, we saw, for example, how every model of finite arithmetic M ⊨ FA extends to taller models M⁺ and M⁺⁺ and so forth, with which it is bi-interpretable, and in another post, we saw that M extends ultimately to the limit model M* ⊨IΔ₀ of bounded induction. My view on this is to take it directly as a form of potentialism. Even if a finite-arithmetic ultrafinitist does not agree with the M* limit construction, nevertheless it seems to be in accordance with ultrafinitism to allow the move from M to M⁺, and this kind of move already leads exactly to the potentialist picture for arithmetic that I would like to paint.

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