Mathematical Logic - 291 (3rd ed. 2021.)

What is a mathematical proof? How can proofs be justified? Are there limitations to provability? To what extent can machines carry out mathe- matical proofs?

Only in this century has there been success in obtaining substantial and satisfactory answers.

The present book contains a systematic discussion of these results.

The investigations are centered around first-order logic.

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).

A short digression into model theory will help us to analyze the expres- sive power of the first-order language, and it will turn out that there are certain deficiencies.

For example, the first-order language does not allow the formulation of an adequate axiom system for arithmetic or analysis.

On the other hand, this difficulty can be overcome--even in the framework of first-order logic-by developing mathematics in set-theoretic terms.

We explain the prerequisites from set theory necessary for this purpose and then treat the subtle relation between logic and set theory in a thorough manner.