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Gӧdel proved that FOL is complete. The arithmetical theory Q is closed under first-order logical consequence, and yet Gӧdel's theorems establish that Q is incomplete. What's the relationships between the relevant concepts involved? Why these facts are mutually compatible?

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    The term "(in)complete" is unfortunately overloaded: when we say "FOL is complete" we're referring to a totally different notion than when we say "$\mathsf{Q}$ is not complete." There's no tension, only unfortunate terminology. – Noah Schweber Dec 01 '20 at 23:44
  • 'First order logic is complete' means that exactly those formulas are provable from a theory which hold on every model of the theory, while 'incompleteness' of a theory means that there is a formula which has two models for the theory, in one of them it holds but it does not in the other one. – Berci Dec 01 '20 at 23:49
  • This question has been asked several times on this site. Besides the instance linked above, here are some more: 1, 2, 3. If after perusing these there is still some point of interest, you should edit your question to focus on that specifically. – Noah Schweber Dec 01 '20 at 23:50
  • (That said, please don't take my closing this question as an indication that it's a bad question - it's a very good question. But it has been asked and answered multiple times before.) – Noah Schweber Dec 01 '20 at 23:51

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