Abstract
In Chapter I we realized that investigations into the logical reasoning used in mathematics require an analysis of the concepts of mathematical proposition and proof. In undertaking such an analysis, we were led to introduce the first-order languages. We also defined a notion of formal proof which corresponds to the intuitive concept of mathematical proof. The Completeness Theorem then shows that every proposition which is mathematically provable from a system of axioms (and thus follows from it) can also be obtained by means of a formal proof, provided the proposition and the system of axioms admit a first-order formulation.
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