Comment by rcpt
13 hours ago
I don't know if that's a "flaw". It's what Godel Incompleteness guarantees.
Would be funny if the most famous problem turns out be an example though
13 hours ago
I don't know if that's a "flaw". It's what Godel Incompleteness guarantees.
Would be funny if the most famous problem turns out be an example though
That's not that how that works btw. The incompleteness is a property of the algorithm which does the proofs, not the theorems. If a theorem is true it has a proof and a program which proves it (printf("the proof verbatim")). But no computer program can successfully prove all theorems. The space of potential proofs is too big to search systematically, but you can still get lucky.
That is also not how it works. The theorem says that there exist some true statements for which there is no proof. This is assuming the axioms are consistent.
I think you're right, I'm conflating two separate things.
EDIT: I am wrong (it's been a while) but there is nuance.
I said any computer program fails to prove some true statements. In fact, some 'true' statements cannot be proven by any program, as others have said.
But for those statements G, you can prove 'G is true if our system is consistent', just not 'G is true'.
So when we say G is 'true', we are really just asserting the consistency axiom that our computer program is denied.
Godel says that there are statements that are true but have no proof within that system. It's not about algorithms.
I'm saying it'd be funny if Riemann were true but not proveable.
Though I guess it's actually funnier. If Riemann were false then there would be a counterexample, so it's clearly within existing axioms.
So if one can prove that Riemann isn't proveable in our current axioms then it has to be true.
Can something be both true and unprovable without being a self-referential Godel sentence? I assume most statements that aren't self-referential are independent, like the continuum hypothesis with ZFC.
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