9 ms·
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 a
by fdupress 4y ago
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.
- fdupress 4y agoJust 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.
- ogogmad 4y agoThere 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/S0273-0979-2016-01556-4/S0273-0979-2016-01556-4.pdf 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.
- mathgeek 4y ago“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.
- ogogmad 4y agoIn this particular foundation, the set of real numbers is countable. It is "the" set of real numbers in that particular foundation.
- fdupress 4y agoSorry, 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.
- ogogmad 4y agoI 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.
- fdupress 4y agoThanks for your comment; learned a couple of things.