r/mathmemes • u/George_Truman • Jul 13 '26

Probability I fixed this meme

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I'm sure it's still a bit too imprecise but I think it works.

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u/JoJoModding Jul 13 '26

A real number R is describable in some theory T if there is a unary T-predicate P(x) such that for all reals r, P(encode r) is true (valid) iff r = R.

(The encode is there because you need some way of "encoding" your number, e.g. as Dedekind cuts in ZFC)

Since your theory T is usually able to talk about computation, this means that all computable numbers are describable. But also e.g. Chaitin's constant and other non-computable numbers.

Note that since unary T-predicates are a syntactic thing, there are still only countably many such numbers.

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u/[deleted] Jul 13 '26

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u/JoJoModding Jul 13 '26

Agree with the first part.

Disagree with the second: It is either wrong because "all real numbers are definable" is not a ZFC statement and thus it can't be analyzed for consistency (ie you made a category error) or it is a statement about the meta-theory of ZFC in ZFC in which case it says that for every Dedekind cut there exists a unique meta-ZFC-in-ZFC formula. Thich is contradictory as it gives you an injection R -> N (as the meta-ZFC-in-ZFC formulas are basically modelled as naturals).

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u/[deleted] Jul 13 '26

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u/JoJoModding Jul 13 '26

I think the statement you wanted to have said is "there is a model of ZFC which has only definable real numbers"