this post was submitted on 12 Aug 2023
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https://en.wikipedia.org/wiki/Continuum_hypothesis
Hmm. Frankly speaking I always assumed Mathematicians had more of an idea about infinities I mean why even have indices if you don't have an inductive rule to descr... oh wait never mind.
That said the reals aren't countable yet we have perfectly reasonable ways to deal with them symbolically, even compute with them, represent ~~every single one of~~^1^ them in finite space, it's just when you want to compare or output them with infinite precision that you might have to wait for eternity. But who needs infinite precision anyway, arbitrary precision is plenty.
^1^ On second thought, after diagonalisation, no we don't. Or we do because there's some magic going on with included transcendental constants that break through that do I look like a numerologist.
there actually is a way to represent the reals with full generality in homotopy type theory -- work is still on-going to implement it in a real programming language/prove type checking is decidable, but the theory is already in place -- via Cauchy sequences.