The proof of Gödel's incompleteness theorem just sketched is proof-theoretic (also called syntactic) in that it shows that if certain proofs exist (a proof of or its negation) then they can be manipulated to produce a proof of a contradiction. This makes no appeal to whether is "true", only to whether it is provable. Truth is a model-theoretic, or semantic, concept, and is not equivalent to provability except in special cases.
