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.

Author: Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas
Binding Type: Hardcover
Publisher: Springer
Published: 05/29/2021
Series: Graduate Texts in Mathematics #291
Pages: 304
Weight: 1.5lbs
Size: 9.40h x 8.00w x 0.80d
ISBN: 9783030738389
3rd 2021 Edition

About the Author
Heinz-Dieter Ebbinghaus is Professor Emeritus at the Mathematical Institute of the University of Freiburg. His work spans fields in logic, such as model theory and set theory, and includes historical aspects.

Jörg Flum is Professor Emeritus at the Mathematical Institute of the University of Freiburg. His research interests include mathematical logic, finite model theory, and parameterized complexity theory.

Wolfgang Thomas is Professor Emeritus at the Computer Science Department of RWTH Aachen University. His research interests focus on logic in computer science, in particular logical aspects of automata theory.