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"?
"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 )
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?
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"?
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 )
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?
Right before questions for further thought did you forget the +1? It says "...Numbers is precisely ℕ·(1+ℚ)."
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/