This classic introduction to the main areas of mathematical logic provides the basis for a first graduate course in the subject. It embodies the viewpoint that mathematical logic is not a collection of vaguely related results, but a coherent method of attacking some of the most interesting problems, which face the mathematician. The author presents the basic concepts in an unusually clear and accessible fashion, concentrating on what he views as the central topics of mathematical logic: proof theory, model theory, recursion theory, axiomatic number theory, and set theory. There are many exercises, and they provide the outline of what amounts to a second book that goes into all topics in more depth. This book has played a role in the education of many mature and accomplished researchers.
22 pages matching elementary extension in this book
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The Nature of Mathematical Logic
Theorems in FirstOrder Theories
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axiom of constructibility axiom system axiomatized benign calculable Card(x cardinal closed formula conservative extension consistency proof constructible continuum hypothesis corollary countable decision method defining axiom disjunction element elementarily equivalent elementary extension equality axioms equality theorem equivalent explicit definition expression numbers extension by definitions F is defined F is recursive finitary follows function F function or predicate function symbol functions and predicates Hence Higman homomorphism hyperarithmetical identity axioms implies induction hypothesis infinite interpretation introduce isomorphism Lemma means n-ary function n-type natural numbers negation nonlogical axioms nonlogical symbols obtain occurrences ordinal partial functional predicate symbol prenex form provable prove Q(xi quantifiers recursive function recursive partial functional recursively enumerable set relation replace result special constants strongly undecidable structure subgroup subset suppose syntactical variables tautological consequence tautology theorem theory transfinite induction truth values unary predicate valid variable-free