Mathematical Logic

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Springer Science & Business Media, 14 mar 2013 - 291 pagine
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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Sommario

Introduction
3
Syntax of FirstOrder Languages 11
10
Semantics of FirstOrder Languages
27
A Sequent Calculus 59
58
The Completeness Theorem
75
PART B 135
78
The LöwenheimSkolem and the Compactness Theorem
87
The Scope of FirstOrder Logic
99
Extensions of FirstOrder Logic
137
Limitations of the Formal Method 151
150
Free Models and Logic Programming
189
An Algebraic Characterization of Elementary Equiva
242
Lindströms Theorems
261
References
277
Subject Index
283
Copyright

Syntactic Interpretations and Normal Forms
115

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