[Book Cover]

Friendly Introduction to Mathematical Logic, A, 1/e

Christopher C. Leary, SUNY at Geneseo

Coming December, 1999 by Prentice Hall Engineering/Science/Mathematics

Copyright 2000, 240 pp.
Cloth
ISBN 0-13-010705-0


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Summary

For a one-quarter/one-semester, junior/senior-level course in Mathematical Logic. With the idea that mathematical logic is absolutely central to mathematics, this tightly focused, elementary text discusses concepts that are used by mathematicians in every branch of the subject—a subject with increasing applications and intrinsic interest. It features an inviting writing style and a mathematical approach with precise statements of theorems and correct proofs. Students are introduced to the main results of mathematical logic—results that are central to the understanding of mathematics as a whole.

Features


A focus on core topics.

  • Allows instructors to reasonably cover the central topics of first order mathematical logic in a single semester or quarter course. Gives students a smaller, more focused and less expensive text.
Conversational, accessible, and accurate writing style.
  • Invites students to explore the subject. Also makes for a much less intimidating environment that encourages students to read and use the book as they work through the course.
Early coverage of First Order Logic rather than Propositional Logic—Assumes that most students have an intuitive understanding of Propositional Logic.
  • Provides more time for students to digest the material. Gives instructors more time later in the course to spend on more abstract and difficult topics.
A mathematical approach—With precise statements of theorems and correct proofs. An in-depth knowledge of computer science is not assumed, although the applications to computer science are indicated.
  • Exposes students to the core of the field as they work through challenging and technical results.
Coverage of major theorems—e.g., Godel's Completeness Theorem, the Incompleteness Results of Godel and Rosser, the Compactness Theorem, and the Lowenheim-Skolem Theorem.
  • Introduces students to the strength and power of mathematics as well as its limitations.
Helpful asides—Highlight a particular point, provide a pause, restate difficult points, or emphasize important topics that may get lost along the way.
  • Give students another view of a topic.
Exercises and problems of varying difficulty.
  • Provide students with the opportunity to use and apply the material learned.


Table of Contents
    1. Structures and Languages.

      Naïvely. Languages. Terms and Formulas. Induction. Sentences. Structures. Truth in a Structure. Substitutions and Substitutability. Logical Implication. Summing Up, Looking Ahead.

    2. Deductions.

      Naïvely. Deductions. The Logical Axioms. Rules of Inference. Soundness. Two Technical Lemmas. Properties of our Deductive System. Non-Logical Axioms. Summing Up, Looking Ahead.

    3. Completeness and Compactness.

      Naïvely. Completeness. Compactness. The Löwenheim-Skolem Theorems. Summing Up, Looking Ahead.

    4. Incompleteness—Groundwork.

      Introduction. Language, Structure, Axioms. Recursive Sets, Recursive Functions. Recursive Sets and Computer Programs. Coding—Naïvely. Coding Is Recursive. Gödel Numbering. Gödel Numbers and N. NUM and SUB Are Recursive. Definitions by Recursion Are Recursive. The Collection of Axioms Is Recursive. Coding Deductions. Summing Up, Looking Ahead. Tables of D-Definitions.

    5. The Incompleteness Theorems.

      Introduction. The Self-Reference Lemma. The First Incompleteness Theorem. Extensions and Refinements. The Second Incompleteness Theorem. Another Explanation of the Second Incompleteness Theorem. Summing Up, Looking Ahead.

    Appendix: Set Theory.
    Exercises.

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