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>Constructive mathematicians believe the reals exists. I think when people mention the "reals" they are usually referring to the uncountable reals, which you c
by GregBuchholz 9y ago
>Constructive mathematicians believe the reals exists.
I think when people mention the "reals" they are usually referring to the uncountable reals, which you can't construct, compute, name, or know:
https://arxiv.org/abs/math/0404335 https://arxiv.org/abs/math/0404335
...(maybe skip to chapter 5 to get to the meat of it). So constructivists don't believe in those types of numbers, and so get "choice" as a theorem instead of an axiom.
- yequalsx 9y agoConstructive mathematicians definitely do believe the reals exist and are uncountable. Not all reals are computable or definable. The set of computable reals is countable. There are different varieties of constructive mathematics. A very large majority of constructive mathematicians believe the reals are uncountable. In intuitionistic mathematics the reals are uncountable. In some constructive versions of math you get weird things like there being an injection from R to N but there not being a bijection due to diagonalization. I think you are confusing computable/nameable with constructive as that term is used by most mathematicians. Note that Chaitan's Constant is not computable.
- yequalsx 9y agoThis link may be of interest to you: https://mathoverflow.net/a/30694 https://mathoverflow.net/a/30694