this post was submitted on 12 Aug 2023
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Computable. There are countably many computable numbers since there are only countably many possible programs. Non-computable numbers can't be exactly referred to / described / constructed by a program, so if your point of view is that everything is a program, you would say they don't exist.
homotopy type theory has a way to describe the reals in full generality via cauchy sequences, so if type checking ever gets proven to be decidable, this won't be true any longer. it's chapter 11 in the hott book.
I guarantee you're misunderstanding or accidentally misrepresenting something here. The fact that there are only countably many computable numbers is a simple consequence of the fact that there are only countably many programs, which is bounded above by the number of finite sequences of letters from a finite alphabet, which is countably infinite.
There may be more finitistic/computable models for the real numbers or something, but "the computable real numbers" are countable.
the model provided is a lazy cauchy sequence so any given real number can be computed to arbitrary precision. the theorems about real numbers are directly provable and potentially machine checkable, assuming decidable type checking works out.
Nothing about what you said contradicts what I said. You can either change the definition of the computable real numbers, or agree that they are countable.