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Suppose that I'm a working mathematician that has just proved a Theorem, say, in Number Theory.

Does Gödel's Incompleteness Theorem imply that I can't know for sure if there exists a proof of the logical negation of my Theorem?

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    Goedel's Second Incompleteness Theorem implies that there is a *formal, finitistic proof* that the logical negation of your theorem cannot be proven from the axioms of Peano Arithmetic if and only if Peano Arithmetic is inconsistent. Goedel's Theorems are not about "truth" and "falsity", they are about provability.2012-05-08
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    I don't know if it will help or not, but [here's an answer](http://math.stackexchange.com/a/16383/742) I wrote some time ago about trying to understand what Goedel's theorems tell us.2012-05-08
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    The Second Incompleteness Theorem shows that a certain sentence $\text{Con}_{PA}$ of PA, which seems to capture the meaning of consistency of PA, is not provable in PA (if PA is consistent!). Not to worry, PA *is* consistent.2012-05-08
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    The second incompleteness theorem implies that you can't prove inside the theory that there is no proof of the logical negation of your theorem.2012-05-08

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