What are the real numbers, really?
What are the real numbers, really?
The real real numbers—what are they? Must we answer?
What is a real number?
Let us consider the real continuum. The classical discovery of irrational numbers reveals gaps in the rational number line: the place where √2 would be, if it were rational, is a hole in the rational line. Thus, the rational numbers are seen to be incomplete. One seeks to complete them, to fill these holes, forming the real number line ℝ.
Please enjoy this free extended excerpt from Lectures on the Philosophy of Mathematics, published with MIT Press 2021, an introduction to the philosophy of mathematics with an approach often grounded in mathematics and motivated organically by mathematical inquiry and practice. This book was used as the basis of my lecture series on the philosophy of mathematics at Oxford University.
Dedekind cuts
Dedekind (1901, I.3) observed how every real number cuts the line in two and found in that idea a principle expressing the essence of continuity:
If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions. —Dedekind, 1901
For Dedekind, the real numbers are what we now call Dedekind complete: every cut is filled. In the rational line, some cuts, determined by a rational number, are already filled; but other cuts correspond to holes in the rational line, not yet filled. For any such unfilled cut, Dedekind proposes that we may imagine or “create” an irrational number in thought precisely to fill it. In this way, we shall realize the real number line as the Dedekind-completion of the rational number line.
And if we knew for certain that space was discontinuous there would be nothing to prevent us, in case we so desired, from filling up its gaps, in thought, and thus making it continuous; this filling up would consist in a creation of new point-individuals and would have to be effected in accordance with the above principle. —Dedekind, 1901
Theft and honest toil
Russell explains how one may undertake this creation process explicitly, building the real numbers as a mathematical structure that fulfills Dedekind's completeness property. In a truly elegant construction, he forms the Dedekind-completion of the rational line from the set of all Dedekind cuts themselves, viewing each cut as constituting a single new point. A Dedekind cut in the rational line is a bounded nonempty initial segment of the rationals with no largest element. The no-largest-element requirement ensures that rational numbers are represented uniquely, since otherwise we could place the rational limit point on either side, forming two distinct cuts where only one is wanted.
Toiling under Russell's direction, we form the set of all Dedekind cuts, viewing each as a single new point; we define the natural order upon them (it is just the inclusion order ⊆ on the cuts); we prove easily that this new order is Dedekind complete (the union of any bounded set of cuts is itself a cut that is the least upper bound); we extend the field operations from the rational numbers to the set of cuts, defining what it means to add any two cuts or multiply them; and we prove that these operations and order make the set of cuts into an ordered field. Thus, we construct the real numbers as Dedekind cuts, forming a Dedekind-complete ordered field.
Although one can imagine the Dedekind cuts as arising from the real numbers, to do so is precisely the inverse of the intended logic. Rather, we seek to use the cuts to define what the real numbers are, or at least what they could be. According to this account, a real number is a Dedekind cut in the rational numbers. Indeed, Russell (1919, p. 71) makes a withering criticism of Dedekind's axiomatic approach, by which one postulates that the real numbers are Dedekind complete.
The method of “postulating” what we want has many advantages; they are the same as the advantages of theft over honest toil. Let us leave them to others and proceed with our honest toil. —Russell, 1919
Russell's “honest toil” was to construct the real numbers via Dedekind cuts as described here, proving that the resulting structure is Dedekind complete, rather than merely postulating that the real numbers are already Dedekind complete.
Cauchy real numbers
An alternative continuity concept is provided by Augustin-Louis Cauchy, who was inspired by the idea that every real number is the limit of the various rational sequences converging to it. A sequence of real numbers is a Cauchy sequence if the points in the sequence become eventually as close as desired to one another. The continuity of the real numbers is expressed by Cauchy completeness, the property that every Cauchy sequence converges to a limiting real number.
The rational line, of course, is not Cauchy complete, for there are Cauchy sequences converging to where √2 would be, but there is no rational number there as the limit of this sequence. And it is similar for the other irrational numbers. But one may form the Cauchy completion of the rational numbers by considering all possible Cauchy sequences on them. Two such sequences are equivalent if their members eventually become as close to each other as desired, and we may form the real numbers as the collection of equivalence classes of Cauchy sequences. This admits a natural ordered field structure; it is Archimedean, which means that the finite sums 1 + 1 + ··· + 1 are unbounded; and it is Cauchy complete. According to this account, a real number is an equivalence class of Cauchy sequences.
Real numbers as geometric continuum
The ancient Greek conception of the continuum, in contrast, persisting through the ages, was inherently geometric: a real quantity is a length, area, or volume. According to the classical number line conception of number, advanced by René Descartes and taught in primary schools everywhere, a real number is a point on the number line, specified by an origin and a unit length.
One problem with this conception is that if a real number x is a length, a product xy is an area and xyz is a volume, then how are we to conceptualize expressions such as x + xy + xyz, which mix quantities of different dimensions? Can we add a length to an area or a volume? Quadratic expressions ax2 + bx + c become problematic. We all agree that 2 × 3 = 6, but if 2 × 3 is an area and 6 is a length, what does that mean? One can solve this, of course, by considering 6 = 1 × 6 also as an area, and similarly in higher dimensions.
Another problem is that one wants to express the idea that the geometric continuum itself is continuous. Dedekind does this by means of his cuts, asserting that every cut is filled.
Categoricity for the real numbers
David Hilbert identified the essential natural properties that we want to be true of the real numbers, which, it turns out, characterize the field of real numbers up to isomorphism. He specified that the real numbers are a maximal Archimedean ordered field—maximal in the sense that they cannot be extended to a larger Archimedean ordered field. This is a form of completeness precisely because the Dedekind completion of any Archimedean ordered field remains Archimedean. In modern terminology, the definition amounts to saying that the real numbers are a complete ordered field, using the Dedekind formulation of completeness, since the least-upper-bound property implies the Archimedean property, as I shall argue in the proof of Huntington’s theorem below. Indeed, one can prove that the real numbers construed as Dedekind cuts or as equivalence classes of Cauchy sequence are complete ordered fields and thereby fulfill Hilbert's axioms.
What is a real number? What is the number π, for example, as a mathematical object? Is it a certain Dedekind cut? Is it an equivalence class of Cauchy sequences? A geometric length? Something else? The structuralist answers these questions by pointing to the categoricity result, asserting that there is only one complete ordered field up to isomorphism. The real numbers are a complete ordered field, and all such fields are isomorphic.
Theorem. (Huntington, 1903) All complete ordered fields are isomorphic.
Proof sketch. I claim first that every complete ordered field R is Archimedean—there is no number in R that is larger than every finite sum 1 + 1 + ··· + 1. If there were such a number, then by completeness, there would have to be a least such upper bound b to these sums; but b - 1 would also be an upper bound, which is a contradiction. So every complete ordered field is Archimedean.
Suppose now that we have two complete ordered fields, ℝ0 and ℝ1. We form their respective prime subfields, that is, their copies of the rational numbers ℚ0 and ℚ1, by computing inside them all the finite quotients ±(1 + 1 + ··· + 1)/(1 + ··· + 1). This fractional representation itself provides an isomorphism of ℚ0 with ℚ1, indicated below with blue dots and arrows:
Next, by the Archimedean property, every number x in ℝ0 determines a cut in ℚ0, indicated in yellow, and since ℝ1 is complete, there is a counterpart x̄ in ℝ1 filling the corresponding cut in ℚ1, indicated in violet. Thus, we have defined a map π:x↦x̄ from ℝ0 to ℝ1. This map is surjective, since every y in ℝ1 determines a cut in ℚ1, and by the completeness of ℝ0, there is an x in ℝ0 filling the corresponding cut. Finally, the map π is a field isomorphism since it is the continuous extension to ℝ0 of the isomorphism of ℚ0 with ℚ1. □
This result characterizes the structure of the real numbers in the same way that Dedekind's arithmetic axioms characterize the structure of the natural numbers. We found the fundamental principles for the real continuum and proved that they determine that structure up to isomorphism. Thus, we have identified the real numbers ℝ as a mathematical structure.
According to structuralism, it is not necessary to pick out a particular complete ordered field, an official copy, since the only mathematically relevant property of the real numbers—the only property that should be used in a mathematical argument—is that they constitute a complete ordered field. Individual real numbers are comprehended by their roles within such a structure. As we noted earlier, √2 is the unique object in whichever complete ordered field you have selected, that happens to be positive and to square to the number 2 in that field, where 2 is the number 1 + 1 in that field, where 1 is the unique multiplicative identity in that field. This is the structural role played by √2. In any complete ordered field, every rational number is algebraically definable, and every real number is characterized by the cut that it makes in the rational numbers. It follows that the real field ℝ is a Leibnizian structure: any two real numbers are discernible in the language of fields.
Kevin Buzzard (2019) highlights the question of structuralism by inquiring: How do we know that a theorem proved using the Dedekind-cut real numbers is also true of Cauchy-completion real numbers? Why is it that a mathematical assertion involving the real numbers, even if only incidentally, when true for the Dedekind real numbers, must also be true when one uses the Cauchy real numbers? There would seem to be an enormous pile of mathematical material that would have to be proved isomorphism-invariant in order to make such sweeping general conclusions, and has this work actually been done?
As a community, mathematicians in current practice are highly structuralist, often insistently so. It would be considered very strange to prove a theorem involving the real numbers by insisting that one is using the Dedekind real numbers as opposed to the Cauchy real numbers, for example, unless one were specifically concerned with the additional structural features that those formulations of the real numbers involved. Because of this widespread practice, the vast bulk of mathematical development is indeed structuralist and follows the structuralist imperative with regard to the central mathematical structures, including the natural numbers, the integers, the real numbers, and so on. Therefore, the enormous pile of isomorphism-invariant material that Buzzard claims must be undertaken has in fact already been undertaken—this is the standard practice of normal mathematics—and this is why we may deduce that mathematical statements involving the real numbers do not depend on which particular copy of the real numbers we are using.
Categoricity for the real continuum
We characterized the real numbers above by the fact that they form a complete ordered field and all such fields are isomorphic. This categoricity argument, therefore, uses the algebraic properties of the real numbers—the fact that they form an ordered field—as an essential part of the characterization. It turns out, however, that we may also characterize the real number line purely by its order properties rather than its algebraic properties as an ordered field.
Specifically, let us consider the real number line ⟨ℝ, < ⟩ with only the order structure. Viewed as a topological space under the order topology, this is known as the real continuum. What can we say about it? Well, this is a linear order, of course, since any two real numbers are comparable; and it is endless, meaning that there is neither a largest nor a smallest real number; and it is densely ordered, meaning that between any two real numbers, there is another; and it is Dedekind complete, meaning that every cut in the real number line is filled. Thus, the real number line is an endless, complete, dense linear order. This is not yet enough to characterize the real number line, however, for there are other such orders, not isomorphic to the real number line, such as the endless long line, for those who are familiar with it.
One additional property, however, will enable a characterization to go through. It suffices to add that the real number line has a countable dense subset. That is, there is a subset ℚ ⊆ ℝ, the set of rational numbers, which is (1) dense in the real number line in the sense that every nontrivial interval (a,b) of real numbers contains elements of the subset; and (2) the subset ℚ is countable, as discussed in chapter 3. All these properties together now determine the real numbers order up to isomorphism.
Theorem. Any two complete endless dense linear orders with countable dense sets are order isomorphic.
The essence of the proof is Cantor's back-and-forth method, which shows that the two countable dense suborders are isomorphic, for indeed, Cantor shows that any two countable, endless, dense linear orders are isomorphic. One can then lift this isomorphism from the suborders to the whole order using the completeness of the orders, just as we did in the case of the complete ordered fields.
I mention this categoricity result in part because a fascinating foundational issue arises when one considers a small variation of it, weakening the countable-dense-set requirement to what is called the countable chain condition, which asserts that every family of nonoverlapping intervals is countable. The real number line has the countable chain condition, since if we have a family of nonoverlapping intervals in the real number line, then inside each one we may pick a rational number, and we will never pick the same rational number twice since they do not overlap; so the family must have been countable.
Question. Are all complete endless dense linear orders with the countable chain condition isomorphic?
In other words, do these properties characterize the real number line? The answer is subtle and fascinating. A positive answer is known as Suslin's hypothesis, while a counterexample order, a complete endless dense linear order with the countable chain condition, but which is not isomorphic to the real number line, is called a Suslin line. The extremely interesting situation is that this question cannot be settled using the standard axioms of set theory; Suslin's hypothesis is an independent statement, neither provable nor refutable from the axioms of set theory; it is consistent either way. In particular, the question of whether the real numbers are categorically characterized by the property of Suslin's hypothesis is itself independent, neither provable nor refutable from the axioms of set theory. We shall discuss the independence phenomenon at length in chapter 8.
Continue reading more about this topic in the book:
Lectures on the Philosophy of Mathematics, MIT Press 2021