A Course in Mathematical Logic
This book is a text of mathematical logic on a sophisticated level, presenting the reader with several of the most significant discoveries of the last 10 to 15 years, including the independence of the continuum hypothesis, the Diophantine nature of enumerable sets and the impossibility of finding an algorithmic solution for certain problems. The book contains the first textbook presentation of Matijasevic's result. The central notions are provability and computability; the emphasis of the presentation is on aspects of the theory which are of interest to the working mathematician. Many of the approaches and topics covered are not standard parts of logic courses; they include a discussion of the logic of quantum mechanics, Goedel's constructible sets as a sub-class of von Neumann's universe, the Kolmogorov theory of complexity. Feferman's theorem on Goedel formulas as axioms and Highman's theorem on groups defined by enumerable sets of generators and relations. A number of informal digressions concerned with psychology, linguistics, and common sense logic should interest students of the philosophy of science or the humanities.
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abbreviated notation algorithm alphabet arithmetic assertion atomic formula axiom of choice axiom of extensionality axioms of equality basic benign Boolean algebra cardinality Chapter closed with respect coincides computable consider consists contains Continuum Hypothesis contradiction coordinates corresponding countable defined definition denote Diophantine Diophantine sets elements embedding enumerable sets equivalent example exists expression extensionality f.p. group fact finite formal language free variable give Godel's graph Hence homomorphism induction assumption infinite integer intersection intuitive isomorphism L-true L,Set Lemma limit ordinal logical mathematics obtain obvious occur operations pair partial function polynomial primitive enumerable primitive recursive function problem projection prove quantifier random class real numbers relations result semi-computable sequence set of formulas set theory subgroup subset Suppose Tarski's theorem tautologies tion translation true formulas truth function verify versal family von Neumann universe Zermelo-Fraenkel axioms
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Mathematics, Education, and Philosophy: An International Perspective
No preview available - 1994