Our first goal is Godel's completeness theorem, which shows that the con� sequence relation coincides with formal provability: By means of a calcu� lus consisting of simple formal inference rules, one can obtain all conse� quences of a given axiom system (and in particular, imitate all mathemat� ical proofs).

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This introduction to first-order logic clearly works out the role of first-order logic in the foundations of mathematics, particularly the two basic questions of the range of the axiomatic method and of theorem-proving by machines.

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