You might say "but wait, haven't we just proven that it's true? So isn't that also a contradiction?" This would be a disaster, because it would prove that the axioms of arithmetic are inconsistent! Now 1+1=3 for all we know.

The catch is that when we proved that the sentence is not false, we used proof by contradiction, and for proof by contradiction to be a valid method of proof, we need to assume that the axioms we are working with are consistent (and therefore can't produce a contradiction). So really all we have proved is that either:

- the sentence is true

Or

- the axioms of arithmetic are inconsistent

We can't prove that the axioms of arithmetic are consistent, so we haven't actually proven that the sentence is true. Contradiction avoided.

This issue is actually a major part of Gödel's theorem; we can only avoid a paradox of the axioms of arithmetic can't prove their own consistency. These theorems apply to any system of axioms that are rich enough to state the liar's paradox.