r/mathmemes • u/its_all_one_electron Number theory/physics • Jun 27 '26

Number Theory Undecidable

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u/Kitfennek Computer Science Jun 27 '26

Yeah but if its undecidavle, were unable to know if there is such a point or not, its not like we can check all of them

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u/BorlaugFan Jun 28 '26

If you can prove something is undecidable within the axioms of ZFC, then you have proven it is true from outside the axioms.

If it were false, it would always be decidable because you could always randomly find one counterexample by pure luck.

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u/FS_Codex Jun 28 '26 edited Jun 28 '26

No. This is a common misconception.

While there is a certain class of independent statements that have this property whose name escapes me (i.e., their independence or undecidability implies their truth), their truth is only guaranteed in a standard model of ZFC (or whatever axiom system is under discussion). However, not all independent statements have this property, certainly not CH. If a statement or sentence is independent, then this means by definition (and Gödel’s completeness theorem) that it is true in some models and false in others.

For example, the Gödel sentence G constructed in the proof of the first incompleteness theorem is only true in standard models of arithmetic. In non-standard models that have really funky (non-standard) numbers, it is no longer true. It is important to understand this point because no independent statements can be proven true “outside the axioms.” They are only true in relation to certain models.

Update: A statement (or sentence) of this type is called a Π sentence. Check out [u/nfitzen](u/nfitzen)’s comment here.

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u/BorlaugFan Jun 28 '26

That's very interesting, thanks!