An Algebraic Introduction to Mathematical Logic

1. Front Matter

   Pages i-ix

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2. Universal Algebra

   • Donald W. Barnes, John M. Mack

   Pages 1-10

3. Propositional Calculus

   • Donald W. Barnes, John M. Mack

   Pages 11-17

4. Properties of the Propositional Calculus

   • Donald W. Barnes, John M. Mack

   Pages 18-25

5. Predicate Calculus

   • Donald W. Barnes, John M. Mack

   Pages 26-37

6. First-Order Mathematics

   • Donald W. Barnes, John M. Mack

   Pages 38-51

7. Zermelo-Fraenkel Set Theory

   • Donald W. Barnes, John M. Mack

   Pages 52-61

8. Ultraproducts

   • Donald W. Barnes, John M. Mack

   Pages 62-73

9. Non-Standard Models

   • Donald W. Barnes, John M. Mack

   Pages 74-84

10. Turing Machines and Gödel Numbers

    • Donald W. Barnes, John M. Mack

    Pages 85-104

11. Hilbert’s Tenth Problem, Word Problems

    • Donald W. Barnes, John M. Mack

    Pages 105-113

12. Back Matter

    Pages 115-123

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This book is intended for mathematicians. Its origins lie in a course of lectures given by an algebraist to a class which had just completed a sub­stantial course on abstract algebra. Consequently, our treatment of the sub­ject is algebraic. Although we assume a reasonable level of sophistication in algebra, the text requires little more than the basic notions of group, ring, module, etc. A more detailed knowledge of algebra is required for some of the exercises. We also assume a familiarity with the main ideas of set theory, including cardinal numbers and Zorn's Lemma. In this book, we carry out a mathematical study of the logic used in mathematics. We do this by constructing a mathematical model of logic and applying mathematics to analyse the properties of the model. We therefore regard all our existing knowledge of mathematics as being applicable to the analysis of the model, and in particular we accept set theory as part of the meta-Ianguage. We are not attempting to construct a foundation on which all mathematics is to be based--rather, any conclusions to be drawn about the foundations of mathematics come only by analogy with the model, and are to be regarded in much the same way as the conclusions drawn from any scientific theory.

• algebra
• mathematical logic
• set theory

• Department of Pure Mathematics, The University of Sydney, Sydney, Australia

  Donald W. Barnes, John M. Mack

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