After having completed a few courses in logic, even at a graduate level, at no point has it been discussed what the proof system for proving statements about formal systems actually is. It seems that using regular set-theoretic mathematics is allowed, so I believe it's grounded in set theory, but it would be nice to have a clarification.
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Yes, mostly set-theory, and also 'pure logic'. Lots of proofs by structural induction! – Bram28 Jul 23 '17 at 21:46
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I'd like to see a motivating example, more intuitive than the discussions on Godel's theorems. – reuns Jul 23 '17 at 21:49
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A related post of mine on MathOverflow: https://mathoverflow.net/q/11699/1916 – Zev Chonoles Jul 23 '17 at 21:53
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@reuns I was thinking of a Gödel example ;). But let's stick to something more down to earth. Consider proving that there is a set of sentences that is satisfiable only in an uncountable domain (but not in countably infinite domain), (the formula has uncountable length). So now we are assuming that the proof system is so strong that it can prove that there are more real numbers ... we are also assuming there are formal systems with formulas that are uncountably long, with uncountable relations. I can see someone not being able to coax up this proof without realizing the strength of axioms. – Tony Jul 24 '17 at 13:10