A natural generalization of this question (more specifically, your further question 4) is this: let (P, <) be any poset. We can define a new poset Dict(P) whose elements are finite lists of elements of P, with similar dictionary relations (i.e. (x_n) < (y_n) if some initial segment of (x_n) is less than (in particular, comparable to) some initial segment of (y)). (This construction is functorial, too! Do with that what you will.)
Whenever P is a total order, so is Dict(P), and I believe that's an if and only if. If P is a discrete poset (i.e. no elements are comparable), then Dict(P) is an n-ary tree, where n = |P|. I can't seem to figure out how to describe Dict(P) in the next most basic non-linear example: when P = {a, b, c} with a < b and a < c, no other relations. Do you have any sense of what it looks like?
Further questions then are (a) which posets can appear as dictionaries of other posets and (b) do dictionaries of posets actually show up "in nature"?
Nice questions. If you think about the 2-letter words, you get a copy of what is known as PxP, or P copies of P, in other words, a copy of P where every node is replaced by a copy of P. With 3-letter words, you do it one more time, and so on. The full dictionary has a fractal-like structure in that respect, something like the Sierpinski triangle.
Your question about which posets arise as dictionaries of other posets is interesting, and I don't know the answer. My whimsical sense of nature includes all such kinds of examples, so I would say that yes, dictionary posets do show up in nature.
"For example, the number seven hundred two would appear between seven and seventeen, using the usual dictionary custom that extensions of a short word appear after it, so that zookeeper appears alphabetically after zoo."
HTML, and various other things, would allow you to stack extensions of a short word with that word, instead of in front of or behind it. You'd end up with a list of bifurcating sets (e.g. {seven,{seventeen, seventeenhundred},{{seventy, {seventyone,seventyonethousand},seventythousand}}). It reminds me of multiple displacement amplification from biology ( https://en.wikipedia.org/wiki/Multiple_displacement_amplification )
That is true, and that would make a different and interesting order. I am not quite sure what the order type would be, especially if it were nonlinear, and one would in any case need to specify exactly what was allowed in order for it to be a sensible mathematical question. Meanwhile, the order in question in my essay here has only the official digit-pronunciation names of the numbers, which don't allow for that construction.
I may be confusing the terminology here, but does this mean that the cardinality of the set you enumerated is that of aleph-naught, i.e. that of the natural numbers? Even with the infinitely many"chapters" consisting of infinitely many numbers ending in iterated-8s, the rules of cardinal multiplication still leave the "book" at aleph-naught?
Yes, although each chapter is infinite, still there are only countably many numbers altogether, so aleph_0 many as you say. This is an instance of the phenomenon: a countable union of countable sets is countable.
If I could offer a suggestion for the layman reader, could you publish a substack article on large cardinals and their meanings? To me, reading the literature, that the idea of large cardinal axioms is very technical and non-obvious to the layman mathematician. What are inaccesssible cardinals, Woodin cardinals, Mahlo cardinals, etc. Wikipedia is woefully uninformative. The basic premise I take away is that these are the sizes of large sets that exist outside or beyond the axioms of axiomatic set theory. But what does that mean? Are these describable mathematical objects, super large sets, that set theory posits that they exist but cannot be proved without specific axioms? How does one go about thinking about these objects, or is it simply that the mathematics is so constructed that you have to build them up from simpler principles? Are these artifacts that exist as a result of the various interpretation of the Godel or Von Neumann constructible universes? And can those be explained as well?
Thanks for the suggestion! This is already part of my plans here. I shall be doing an accessible account of Cantor's theorem, of course, but then I shall also have an essay on uncountable infinities, including large cardinals.
A natural generalization of this question (more specifically, your further question 4) is this: let (P, <) be any poset. We can define a new poset Dict(P) whose elements are finite lists of elements of P, with similar dictionary relations (i.e. (x_n) < (y_n) if some initial segment of (x_n) is less than (in particular, comparable to) some initial segment of (y)). (This construction is functorial, too! Do with that what you will.)
Whenever P is a total order, so is Dict(P), and I believe that's an if and only if. If P is a discrete poset (i.e. no elements are comparable), then Dict(P) is an n-ary tree, where n = |P|. I can't seem to figure out how to describe Dict(P) in the next most basic non-linear example: when P = {a, b, c} with a < b and a < c, no other relations. Do you have any sense of what it looks like?
Further questions then are (a) which posets can appear as dictionaries of other posets and (b) do dictionaries of posets actually show up "in nature"?
Nice questions. If you think about the 2-letter words, you get a copy of what is known as PxP, or P copies of P, in other words, a copy of P where every node is replaced by a copy of P. With 3-letter words, you do it one more time, and so on. The full dictionary has a fractal-like structure in that respect, something like the Sierpinski triangle.
Your question about which posets arise as dictionaries of other posets is interesting, and I don't know the answer. My whimsical sense of nature includes all such kinds of examples, so I would say that yes, dictionary posets do show up in nature.
Sorry, I have to comment before reading further:
"For example, the number seven hundred two would appear between seven and seventeen, using the usual dictionary custom that extensions of a short word appear after it, so that zookeeper appears alphabetically after zoo."
HTML, and various other things, would allow you to stack extensions of a short word with that word, instead of in front of or behind it. You'd end up with a list of bifurcating sets (e.g. {seven,{seventeen, seventeenhundred},{{seventy, {seventyone,seventyonethousand},seventythousand}}). It reminds me of multiple displacement amplification from biology ( https://en.wikipedia.org/wiki/Multiple_displacement_amplification )
That is true, and that would make a different and interesting order. I am not quite sure what the order type would be, especially if it were nonlinear, and one would in any case need to specify exactly what was allowed in order for it to be a sensible mathematical question. Meanwhile, the order in question in my essay here has only the official digit-pronunciation names of the numbers, which don't allow for that construction.
I may be confusing the terminology here, but does this mean that the cardinality of the set you enumerated is that of aleph-naught, i.e. that of the natural numbers? Even with the infinitely many"chapters" consisting of infinitely many numbers ending in iterated-8s, the rules of cardinal multiplication still leave the "book" at aleph-naught?
Yes, although each chapter is infinite, still there are only countably many numbers altogether, so aleph_0 many as you say. This is an instance of the phenomenon: a countable union of countable sets is countable.
Dear Joel,
Thanks for the reply.
If I could offer a suggestion for the layman reader, could you publish a substack article on large cardinals and their meanings? To me, reading the literature, that the idea of large cardinal axioms is very technical and non-obvious to the layman mathematician. What are inaccesssible cardinals, Woodin cardinals, Mahlo cardinals, etc. Wikipedia is woefully uninformative. The basic premise I take away is that these are the sizes of large sets that exist outside or beyond the axioms of axiomatic set theory. But what does that mean? Are these describable mathematical objects, super large sets, that set theory posits that they exist but cannot be proved without specific axioms? How does one go about thinking about these objects, or is it simply that the mathematics is so constructed that you have to build them up from simpler principles? Are these artifacts that exist as a result of the various interpretation of the Godel or Von Neumann constructible universes? And can those be explained as well?
I saw that you discussed the issue on your blog, at http://jdh.hamkins.org/multiverse-perspective-on-constructibility/
Thanks for writing this substack. I hope that you gain many readers across the fuield of mathematics.
Thanks for the suggestion! This is already part of my plans here. I shall be doing an accessible account of Cantor's theorem, of course, but then I shall also have an essay on uncountable infinities, including large cardinals.
Right before questions for further thought did you forget the +1? It says "...Numbers is precisely ℕ·(1+ℚ)."
Ah yes, thanks for noticing! I've now fixed this.
I added another question, question 4, which was suggested by Nikita Danilov. Please try to figure it out!
See the discussion happening over on Reddit regarding question 5, where we consider the real numbers in alphabetical order. https://www.reddit.com/r/math/comments/102a032/the_real_numbers_in_alphabetical_order/