i wrote a mathoverflow answer about godel's incompleteness theorem and nonstandard models of arithmetic that believe arithmetic is inconsistent that some of you might be interested in:
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thinking about the incompleteness theorem is good mental exercise i think. forces you to get really clear about the difference between believing something and something being true ๐ค
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it's very striking that the incompleteness theorem, relativity, and quantum mechanics all emerged in the same few decades around WWI and you have to wonder how much of that had to do with people losing faith in the universe making sense in a general sort of way
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> "It is maybe worth working carefully through why the statement ยฌโn:pโฅn continues to be true in this ultrapower"
Is it because p is still less than p+1?
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yep, that'll do it!
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Some people say that they believe that PA is consistent despite not being able to prove it and I still don't get that. It seems like we can only be in two states of knowledge about it. Either we are uncertain about the consistency of PA, or we know it is inconsistent.
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so if you ask a mathematician who knows a little about this, the standard boilerplate answer is that ZF proves the consistency of PA because you can construct a model of PA in it (the standard natural numbers). of course then you can ask why they believe that ZF is consistent
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