# Consequences of Incompleteness.

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. – Arturo Magidin May 8 '12 at 20:35
I don't know if it will help or not, but here's an answer I wrote some time ago about trying to understand what Goedel's theorems tell us. – Arturo Magidin May 8 '12 at 20:37
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. – André Nicolas May 8 '12 at 20:47
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. – Yuval Filmus May 8 '12 at 22:37

The reason for the confusion is that Gödel's second incompleteness theorem tells us that if a theory T satisfies the above (such as what you use to do number theory) then CON(T) is neither provable or disprovable in T where CON(T) is the formal statement that T is consistent. But a key point here is that CON(T) is undecidable in T not because of some uncertainty of the consistency of T (we are assuming T is consistent) but because it's not clear if the formal statement CON(T) really captures the informal notion of proof. Indeed $T + \lnot CON(T)$ is actually a consistent theory. It just turns out that this theory allows infinite integers which code up fake proofs for a sentence and it's negation causing CON(T) to be false even though the system is actually consistent.