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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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up vote 5 down vote accepted

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

As already pointed out here by Tom Leinster, "true" doesn't make sense without some sort of model. It is a technical word.

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That assumes a contentious position in the philosophy of mathematics. Some people think that there are no such things as numbers - all there are are various models of arithmetic, and in one context or another we can choose one of these models to be what we're talking about when we use the word "number". (This view gets into a sort of regress problem though when we ask what a model is rather than a number.) –  Kenny Easwaran Nov 6 '09 at 6:19
    
The other view is that the word "number" has an ordinary meaning, and what it means to say that a statement like "every number has property X" is true is just to say that every number actually has property X. On this view, it seems that every arithmetical statement (that is grammatically well-formed) must be either true or false. There may be some model in the set-theoretic sense such that truth-in-that-model coincides with plain old truth, but that's not needed for the notion of truth to make sense. –  Kenny Easwaran Nov 6 '09 at 6:19
    
You raise interesting points. I wasn't meaning to get too deep into the philosophical tangent, but at least in the sense of the question above there's no reason to think of Gödel's "true" as anything other than the strict definition. –  Jason Dyer Nov 6 '09 at 16:05

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