4 ms·
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 construct
by fdupress 4y ago
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.
- 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.