# Godel's 1st incompleteness theorem - clarification.

This should be a trivial question for people who know Gödel's 1st incompleteness theorem. I quote the statement the theorem from wikipedia: "Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory."

My question is: what is the meaning of 'true' in the last sentence? Let me elaborate: the only (introductory) proof of the theorem that I know starts with a specific model of the theory and constructs a sentence which is true in that model, but not provable from the theory.

So does true arithmetical statement' in the statement of the theorem meantrue in the (implicitly) given model', or true in EVERY model?

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It means true in the usual model. For 1st order logic we have Godel's Completeness theorem, which guarantees that if something is true in every model, then it is actually provable in the theory.

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Thanks, I guess was confusing true' with valid'. –  auniket Nov 6 '09 at 21:59