Mathematical Logic PDF
by H.-D. Ebbinghaus, J. Flum, Wolfgang Thomas
Part of the Undergraduate Texts in Mathematics series
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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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- Format:PDF
- Publisher:Springer New York
- Publication Date:14/03/2013
- Category:
- ISBN:9781475723557
Information
-
Download Now
- Format:PDF
- Publisher:Springer New York
- Publication Date:14/03/2013
- Category:
- ISBN:9781475723557