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Comment by defrost | Hacker News Reader

It's a literal assertion that the reals (all reals) ARE countable .. subject to taking issue with aspects of Cantors argument.

It opens:

> In 1874 Georg Cantor published a theorem stating that every sequence of reals is avoided by some real, thereby showing that the reals are not countable.

> Cantor's proof uses classical logic.

> There are constructive proofs, although they all rely on the axiom of countable choice. Can the real numbers be shown uncountable without excluded middle and without the axiom of choice?

>An answer has not been found so far, although not for lack of trying.

> We show that there is a topos in which the real numbers are countable, i.e., there is an epimorphism from the object of natural numbers to the object of Dedekind reals.

> Therefore, higher-order intuitionistic logic cannot show the reals to be uncountable.

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That is an assertion about one particular construction of the reals, in a particular topos, which implies something on the constructed set in that topos.

I'd argue that that set, resulting from carrying out Dedekind cuts in a particular topos, is not in fact the set of real numbers. But I also agree that it means the property of uncountability for the set of real numbers as we understand it in set theoretic terms cannot be proved intuitionistically. And I'm fine with that.

Just a quick addition: note the care taken to avoid using the word "set" in your excerpts. My objection was with the submission's current title ("The countable set of real numbers"). It still stands.

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.

“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.

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.

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