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bryjnar comments on Proofs, Implications, and Models - Less Wrong
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I think you're confusing syntax and semantics yourself here. An undecidable proposition has no proof, that does not mean it is not semantically valid. So the second theorem shows that for a consistent theory T, the statement ConT that T is consistent is not provable, but it is true in all models.
No, it isn't. When T includes arithmetic, ConT is not provable in T (provided that T is consistent, the usual background assumption). Therefore by the completeness theorem, there are models of T in which ConT is false.
ConT can be informally read as "T is consistent", and by assumption the latter statement is true, but that is not the same as ConT itself being true in any model other than the unique one we think we are talking about when we talk about the natural numbers.
Okay, I think this is still because I'm thinking about the second-order logic result. I just went and looked up the FOL stuff, and the incompleteness theorem does give some weird results in FOL! You're quite right, you do get models in which ConT is false.
I think my point still stands for SOL, though.