Comment by AdAbsurdum
9 hours ago
I think the relevant quotes are these:
"Dedekind quickly replied that...he’d worked out a proof that the algebraic numbers (the numbers you get as solutions to algebra problems) could be counted.
[...]
Weierstrass had been most excited about the proof that algebraic numbers are countable. (He would later use that result to prove a theorem of his own.) So Cantor chose a misleading title [for his paper] that only mentioned algebraic numbers.
[...]
Writing his paper, Cantor put the proof about algebraic numbers first. Below it, he added his own proof that the real numbers cannot be counted — Dedekind’s simplified version of it, that is."
So the first proof -- the one the article was titled after -- was completely created by Dedekind.
> he’d worked out a proof that the algebraic numbers (the numbers you get as solutions to algebra problems) could be counted
I can't say I'm fully comfortable with that characterization of the algebraic numbers. The definition itself does suggest a proof that they are countable:
1. The number of symbols that can appear in a well-defined algebra problem is finite. (For example, if we define algebra problems as being posed in written English, we can use an inventory of no more than 50 symbols to define them all. If we define "algebra problems" in some other way, the definition will specify how many symbols are available.)
2. The number of possible strings describing algebra problems, created from this finite symbolic alphabet, is necessarily countable, because the strings have finite length.
3. Each algebraic number is the solution to one of those strings, and therefore the algebraic numbers are countable.
But I don't really feel like it's possible to learn anything about the numbers from that proof.
You can also get to computable numbers through a similar argument, substituting something Turing-complete for algebra. You definitely do get to learn some interesting things about numbers from computable numbers. The differences between the computables and the full reals are much more subtle than the differences between the rationals and the reals.