"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.
Yeah, that was a weird way to describe the algebraic numbers. The formal definition (those numbers for which a finite polynomial p exists with p(x)=0) is not that complicated, is it?
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.
> The differences between the computables and the full reals are much more subtle than the differences between the rationals and the reals.
How so?
Using the definition of computable numbers where you provide input of n and the output is every digit of the number up to n places past the decimal point, we can rephrase that definition like so:
A computable number c is one with the following property:
A Turing machine exists which, provided with a tolerance δ, will exhibit a rational number q < c such that c - q < δ
Clearly, a suitable rational will always exist, since rationals can be found within any distance of any real.
But for some particular real, we might not be able to find that rational through the use of a fixed Turing machine, in which case the real would be noncomputable. This suggests to me that there is a wider gap between the computables and the reals, where the approximation of a real number is limited by the need to describe it with a Turing machine, than there is between the rationals and the reals, where we can use the same approximation, but without that limitation.
(Obviously the rationals are a subset of the computables, but if we're considering a relationship to the real numbers, the rationals seem to have one that is closer and more direct...? The relationship of a computable number to a real number is defined through intermediary rational numbers.)
"This suggests to me that there is a wider gap between the computables and the reals"
I didn't say "wider". I said more subtle. It doesn't take much mathematical intuition and training to understand the rationals versus the reals. Understanding the computables versus the reals is a lot more tricky and takes a lot more thought. The simple arguments that show the difference between the rationals and the reals require a lot of very careful adjustment if you want to translate them to the computables versus the reals.
"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.