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Comment by ogogmad | Hacker News Reader

There is only a "the" set of real numbers if there is a "the" foundation of mathematics. https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016...

I chose that title out of a combination of deliberate clickbait, and because I felt that the original title was confusing to people who hadn't heard the abbreviation "the reals" for the set (or "space"?) of real numbers.

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Sorry, I was more careful in my second mention of "the set of real numbers". Thoughout, I meant "the set of real numbers that arise from set theoretic constructions".

In other words, the object on which the existing proofs of uncountability hold and the object constructed in the talk are not necessarily the same object. In fact, the care taken by Bauer in clarifying "the object of Dedekind reals" in stating his main results leads me to believe the topos in which the Dedekind reals are countable is also a topos in which the Dedekind reals are not equivalent to other constructions of the reals.

I don't know if you know this: In a topos in which Countable Choice holds, the Cauchy reals are isomorphic to the Dedekind reals. In other words, it doesn't matter whether the reals are constructed via Cauchy sequences or Dedekind cuts. But in some toposes where Countable Choice fails, the two objects Dedekind Reals and Cauchy Reals may become non-isomorphic. In the sheaf topos Sh([0,1]) for instance, the Dedekind reals are (externally) the sheaf formed out of the continuous functions [0,1]->R, and the Cauchy reals are the sheaf formed out of the constant functions [0,1]->R.

Thanks for your comment; learned a couple of things.

“A countable set of real numbers” is more appropriate as there isn’t a single countable set of real numbers. Much like the set of real numbers is “a real set of numbers” but not the only such set.

In this particular foundation, the set of real numbers is countable. It is "the" set of real numbers in that particular foundation.

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