Introduction to Mathematical Logic

ISBN-13: 9781461572909 / Angielski / Miękka / 2012 / 342 str.

Elliot Mendelsohn

Opis

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.

Spis treści

One 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.- Two Quantification Theory.- 1. Quantifiers.- 2. Interpretations. Satisfiability and Truth. Models.- 3. First-Order Theories.- 4. Properties of First-Order Theories.- 5. Additional Metatheorems and Derived Rules.- 6. Rule C.- 7. Completeness Theorems.- 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.- 13. Elementary Equivalence. Elementary Extensions.- 14. Ultrapowers. Nonstandard Analysis.- 15. Semantic Trees.- Three Formal Number Theory.- 1. Axiom System.- 2. Number-Theoretic Functions and Relations.- 3. Primitive Recursive and Recursive Functions.- 4. Arithmetization. Gödel Numbers.- 5. The Fixed Point Theorem. Gödel’s Incompleteness Theorem.- 6. Recursive Undecidability. Church’s Theorem.- Four 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.- 6. Other Axiomatizations of Set Theory.- Five Effective Computability.- 1. Algorithms. Turing Machines.- 2. Diagrams.- 3. Partial Recursive Functions. Unsolvable Problems.- 4. The Kleene-Mostowski Hierarchy. Recursively Enumerable Sets.- 5. Other Notions of Effective Computability.- 6. Decision Problems.- Answers to Selected Exercises.- Notation.