The Logical Foundations of Mathematics: Foundations and Philosophy of Science and Technology SeriesThe Logical Foundations of Mathematics offers a study of the foundations of mathematics, stressing comparisons between and critical analyses of the major non-constructive foundational systems. The position of constructivism within the spectrum of foundational philosophies is discussed, along with the exact relationship between topos theory and set theory. Comprised of eight chapters, this book begins with an introduction to first-order logic. In particular, two complete systems of axioms and rules for the first-order predicate calculus are given, one for efficiency in proving metatheorems, and the other, in a "natural deduction" style, for presenting detailed formal proofs. A somewhat novel feature of this framework is a full semantic and syntactic treatment of variable-binding term operators as primitive symbols of logic. Subsequent chapters focus on the origin of modern foundational studies; Gottlob Frege's formal system intended to serve as a foundation for mathematics and its paradoxes; the theory of types; and the Zermelo-Fraenkel set theory. David Hilbert's program and Kurt Gödel's incompleteness theorems are also examined, along with the foundational systems of W. V. Quine and the relevance of categorical algebra for foundations. This monograph will be of interest to students, teachers, practitioners, and researchers in mathematics. |
Contents
| 1 | |
Chapter 2 The Origin of Modern Foundational Studies | 68 |
Chapter 3 Freges System and the Paradoxes | 76 |
Chapter 4 The Theory of Types | 103 |
Chapter 5 ZermeloFraenkel Set Theory | 135 |
Chapter 6 Hilberts Program and Godels Incompleteness
Theorems | 190 |
Chapter 7 The Foundational Systems of W V Quine | 213 |
Chapter 8 Categorical Algebra | 237 |
| 313 | |
| 317 | |
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Common terms and phrases
abstract algebra axiom of choice axiom of extensionality axiom of infinity cardinal numbers category of sets category theory Chapter codomain consistent constant letters constructions contradiction defined Definition denumerable domain dummy constant equality equivalent example Exercise existence expression extensionality fact finite first-order system first-order theory formal system formula free variables Frege's function letters functor given global elements Gödel number hypothesis induction interpretation intuitive isomorphic language mathematics means model for ST mono monomorphism morphisms natural deduction natural numbers nonempty normal model notion object obtain operation ordered pair ordinal paradox Peano postulates predicate calculus predicate letter principle proof proper axioms properties provable pullback quantifier real numbers recursive relation restriction result rules of inference satisfies Section sentence sequence set theory stratified subset system F Taut tautology term tion topos true type hierarchy type symbol type theory typical ambiguity unique