ENGLISH

Mathematical Logic

By J. Flum , W. Thomas , H.-D. Ebbinghaus

Book information

Publisher
Springer
Year
1994
ISBN
978-1-4757-2357-1, 978-1-4757-2355-7
DOI
10.1007/978-1-4757-2355-7
Language
english
Format
PDF
Filesize
5 MB (5560304 bytes)
Series
Undergraduate Texts in Mathematics
Edition
2nd
Pages
289\289
Topic
Mathematics Logic
DPI
600
Orientation
yes
Paginated
no
Scanned
portrait
Time added
2013-08-01 04:00:00
Mathematical Logic and Foundations

Description

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.

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