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Introduction to Mathematical Logic

par Elliott Mendelson

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This is a compact mtroduction to some of the pnncipal tOpICS of mathematical logic . In the belief that beginners should be exposed to the most natural and easiest proofs, I have used free-swinging set-theoretic methods. The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with mathematical logic has been obtained. If we are to be expelled from "Cantor's paradise" (as nonconstructive set theory was called by Hilbert), at least we should know what we are missing. The major changes in this new edition are the following. (1) In Chapter 5, Effective Computability, Turing-computabIlity IS now the central notion, and diagrams (flow-charts) are used to construct Turing machines. There are also treatments of Markov algorithms, Herbrand-Godel-computability, register machines, and random access machines. Recursion theory is gone into a little more deeply, including the s-m-n theorem, the recursion theorem, and Rice's Theorem. (2) The proofs of the Incompleteness Theorems are now based upon the Diagonalization Lemma. Lob's Theorem and its connection with Godel's Second Theorem are also studied. (3) In Chapter 2, Quantification Theory, Henkin's proof of the completeness theorem has been postponed until the reader has gained more experience in proof techniques. The exposition of the proof itself has been improved by breaking it down into smaller pieces and using the notion of a scapegoat theory. There is also an entirely new section on semantic trees.… (plus d'informations)

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Indeholder "Foreword by Elliott Mendelson, January 1963", "Introduction", "Chapter 1. The Propositional Calculus", " 1. Propositional Connectives. Truth Tables", " 2. Tautologies", " 3. Adequate Sets of Connectives", " 4. An Axiom System for the Propositional Calculus", " 5. Independence. Many-Valued Logics", " 6. Other Axiomatizations", "Chapter 2. Quantification Theory", " 1. Quantifiers", " 2. Interpretations. Satisfiability and Truth. Models", " 3. First-Order Theories", " 4. Properties of First-Order Theories", " 5. Completeness Theorems", " 6. Some Additional Metatheorems", " 7. Rule C ", " 8. First-Order Theories with Equality", " 9. Definitions of New Function Letters and Individual Constants", " 10. Prenex Normal Forms", " 11. Isomorphism of Interpretations. Categoricity of Theories", " 12. Generalized First-Order Theories. Completeness and Decidability", "Chapter 3. Formal Number Theory", " 1. An Axiom System", " 2. Number-Theoretic Functions and Relations", " 3. Primitive Recursive and Recursive Functions", " 4. Arithmetization. Gödel Numbers", " 5. Gödel's Theorem for S", " 6. Recursive Undecidability. Tarski's Theorem. Robinson's System", "Chapter 4. Axiomatic Set Theory", " 1. An Axiom System", " 2. Ordinal Numbers", " 3. Equinumerosity. Finite and Denumerable Sets", " 4. Hartogs' Theorem. Initial Ordinals. Ordinal Arithmetic", " 5. The Axiom of Choice. The Axiom of Regularity", "Chapter 5. Effective Computability", " 1. Markov Algorithms", " 2. Turing Algorithms", " 3. Herbrand-Gödel Computability. Recursively Enumerable Sets", " 4. Undecidable Problems", "Appendix. A consistency proof for formal number theory", "Bibliography", "Index".

Lærebog i matematisk logik, mængdelære, aksiomsystemer, rekursivt enumerable mængder, uafgørlige problemer og meget andet, der læner sig en del op af datalogi. (

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This is a compact mtroduction to some of the pnncipal tOpICS of mathematical logic . In the belief that beginners should be exposed to the most natural and easiest proofs, I have used free-swinging set-theoretic methods. The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with mathematical logic has been obtained. If we are to be expelled from "Cantor's paradise" (as nonconstructive set theory was called by Hilbert), at least we should know what we are missing. The major changes in this new edition are the following. (1) In Chapter 5, Effective Computability, Turing-computabIlity IS now the central notion, and diagrams (flow-charts) are used to construct Turing machines. There are also treatments of Markov algorithms, Herbrand-Godel-computability, register machines, and random access machines. Recursion theory is gone into a little more deeply, including the s-m-n theorem, the recursion theorem, and Rice's Theorem. (2) The proofs of the Incompleteness Theorems are now based upon the Diagonalization Lemma. Lob's Theorem and its connection with Godel's Second Theorem are also studied. (3) In Chapter 2, Quantification Theory, Henkin's proof of the completeness theorem has been postponed until the reader has gained more experience in proof techniques. The exposition of the proof itself has been improved by breaking it down into smaller pieces and using the notion of a scapegoat theory. There is also an entirely new section on semantic trees.

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