Sets, Logic and CategoriesSet theory, logic and category theory lie at the foundations of mathematics, and have a dramatic effect on the mathematics that we do, through the Axiom of Choice, Gödel's Theorem, and the Skolem Paradox. But they are also rich mathematical theories in their own right, contributing techniques and results to working mathematicians such as the Compactness Theorem and module categories. The book is aimed at those who know some mathematics and want to know more about its building blocks. Set theory is first treated naively an axiomatic treatment is given after the basics of first-order logic have been introduced. The discussion is su pported by a wide range of exercises. The final chapter touches on philosophical issues. The book is supported by a World Wibe Web site containing a variety of supplementary material. |
Contents
I | 1 |
II | 2 |
III | 4 |
IV | 7 |
V | 11 |
VI | 15 |
VII | 20 |
VIII | 24 |
XXIV | 101 |
XXV | 109 |
XXVI | 113 |
XXVII | 114 |
XXVIII | 118 |
XXIX | 124 |
XXX | 130 |
XXXI | 133 |
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Common terms and phrases
assertion Axiom of Choice bijection binary cardinal number cartesian product category theory Chapter choice function colours Completeness Theorem consistent constant symbols construction containing contradiction countable sets deduced Deduction Theorem defined definition empty set equivalence classes equivalence relation example Exercise exists finite set finite subset first-order language first-order logic follows formal system function f functor given Gödel number holds homomorphism hypothesis identity induction infinite injective function integers isomorphic least element Lemma limit ordinal mathematics Modus Ponens morphisms n-ary natural numbers non-empty objects ordered pair ordered set Peano arithmetic power set proof propositional logic propositional variables provable prove rational numbers real numbers ring rules of inference Russell's Paradox satisfies sequence set of formulae set theory Show Soundness and Completeness structure Suppose tautology totally ordered true union unique valuation vector space well-ordered set zero