no, we doesn't include non-mathematicians. Mathematics is a culture. This is what gödel's thms actually show: there are provably infinitely many possible independent axioms forming a hierarchy of logics by relative consistency, and at the top assuming all of them is inconsistent
Yes, but the point is that in principle (and to a surprisingly large degree in practice) mathematicians can know and agree that this is the case.
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and because mathematicians agree that they can prove that they can't prove that they are consistent, the agreement is cultural and not mathematical. Statements such as "we don't condone pedophilia" also hold to a surprisingly large degree in practice and are cultural
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The difference is that societies have large degrees of freedom about whether they condone pedophilia, but mathematicians don't have many degrees of freedom about whether they accept a proof.
End of conversation
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