Overview
- Authors:
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Ralph McKenzie
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Department of Mathematics, University of California, Berkeley, USA
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Matthew Valeriote
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Department of Mathematics and Statistics, McMaster University, Hamilton, Canada
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Table of contents (15 chapters)
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Front Matter
Pages i-viii
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Introduction
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- Ralph McKenzie, Matthew Valeriote
Pages 5-33
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- Ralph McKenzie, Matthew Valeriote
Pages 35-36
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Structured varieties
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- Ralph McKenzie, Matthew Valeriote
Pages 39-55
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- Ralph McKenzie, Matthew Valeriote
Pages 57-64
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- Ralph McKenzie, Matthew Valeriote
Pages 65-71
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- Ralph McKenzie, Matthew Valeriote
Pages 73-74
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- Ralph McKenzie, Matthew Valeriote
Pages 75-89
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Structured Abelian varieties
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- Ralph McKenzie, Matthew Valeriote
Pages 95-98
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- Ralph McKenzie, Matthew Valeriote
Pages 99-102
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- Ralph McKenzie, Matthew Valeriote
Pages 103-105
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- Ralph McKenzie, Matthew Valeriote
Pages 107-128
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- Ralph McKenzie, Matthew Valeriote
Pages 129-148
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- Ralph McKenzie, Matthew Valeriote
Pages 149-167
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The decomposition
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Front Matter
Pages 169-169
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- Ralph McKenzie, Matthew Valeriote
Pages 171-192
About this book
A mathematically precise definition of the intuitive notion of "algorithm" was implicit in Kurt Godel's [1931] paper on formally undecidable propo sitions of arithmetic. During the 1930s, in the work of such mathemati cians as Alonzo Church, Stephen Kleene, Barkley Rosser and Alfred Tarski, Godel's idea evolved into the concept of a recursive function. Church pro posed the thesis, generally accepted today, that an effective algorithm is the same thing as a procedure whose output is a recursive function of the input (suitably coded as an integer). With these concepts, it became possible to prove that many familiar theories are undecidable (or non-recursive)-i. e. , that there does not exist an effective algorithm (recursive function) which would allow one to determine which sentences belong to the theory. It was clear from the beginning that any theory with a rich enough mathematical content must be undecidable. On the other hand, some theories with a substantial content are decidable. Examples of such decidabLe theories are the theory of Boolean algebras (Tarski [1949]), the theory of Abelian groups (Szmiele~ [1955]), and the theories of elementary arithmetic and geometry (Tarski [1951]' but Tarski discovered these results around 1930). The de termination of precise lines of division between the classes of decidable and undecidable theories became an important goal of research in this area. algebra we mean simply any structure (A, h(i E I)} consisting of By an a nonvoid set A and a system of finitary operations Ii over A.
Authors and Affiliations
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Department of Mathematics, University of California, Berkeley, USA
Ralph McKenzie
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Department of Mathematics and Statistics, McMaster University, Hamilton, Canada
Matthew Valeriote
