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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Syntax of FirstOrder Languages 11
Semantics of FirstOrder Languages
A Sequent Calculus 59
The Completeness Theorem
PART B 135
The LöwenheimSkolem and the Compactness Theorem
The Scope of FirstOrder Logic
Syntactic Interpretations and Normal Forms
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applied arbitrary assignment axiom system binary relation binary relation symbol chapter clauses Compactness Theorem consistent set contains witnesses countable define definition derivable dom(p domain elementarily equivalent elementary elements enumerable equality-free example Exercise finite subset first-order language first-order logic following holds free(p function symbol Gödel H-derivation hence induction hypothesis infinite isomorphism K1 and K2 Let q logical system logically equivalent Löwenheim-Skolem Theorem mathematical natural numbers negation complete nonempty normal form notion obtain ordered field partial isomorphism Peano structures prenex normal form procedure properties propositional logic prove quantifier quantifier-free R-decidable R-enumerable real numbers S-formula S-sentences S-structures 21 S-terms Sar-sentence satisfiable second-order second-order logic sequent calculus set of formulas set of sentences set theory set-theoretic Suppose symbol set system of axioms unary uncountable universal Horn formulas valid