  Gödel had, it seems, not thought of giving a consistency proof  of arithmetic through the use of non-finitary but still constructive  principles. In the late thirties, at least from 1938 on, he developed as a response to Gentzen's proof his own special interpretation of  intuitionistic logic and arithmetic, what came to be known as the  Dialectica interpretation. It uses computable functionals to  interpret the proofs of intuitionistic arithmetic. Gödel  published the interpretation only in 1958, even though he had  presented it in lectures in 1941. It is unknown if he discussed the  matter when he met Gentzen in December 1939.