The Idea of Mapping and its Use in Mathematics
The sentential calculus is an example of a mathematical system for which the objectives of Hilbert’s theory of proof are fully realized. To be sure, this calculus codiﬁes only a fragment of formal logic, and its vocabulary and formal apparatus do not suﬃce to develop even elementary arithmetic. Hilbert’s program, however, is not so limited. It can be carried out successfully for more inclusive systems, which can be shown by metamathematical reasoning to be both consistent and complete. By way of example, an absolute proof of consistency is available for a system of arithmetic that permits the addition of cardinal numbers though not their multiplication. But is Hilbert’s ﬁnitistic method powerful enough to prove the consistency of a system such as Principia, whose vocabulary and logical apparatus are adequate to express the whole of arithmetic and not merely a fragment? Repeated attempts to construct such a proof were unsuccessful; and the publication of Gödel’s paper in 1931
showed, ﬁnally, that all such eﬀorts operating within the strict limits of Hilbert’s original program must fail.