The Foundations of Mathematics"There are many textbooks available for a so-called transition course from calculus to abstract mathematics. I have taught this course several times and always find it problematic. The Foundations of Mathematics (Stewart and Tall) is a horse of a different color. The writing is excellent and there is actually some useful mathematics. I definitely like this book."--The Bulletin of Mathematics Books |
Contents
The intuitive background | 1 |
The beginnings of formalization | 39 |
The development of axiomatic systems | 143 |
Strengthening the foundations | 249 |
| 259 | |
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Common terms and phrases
a₁ addition and multiplication algebraic argument arithmetic axioms bijection bijection f binary operations cardinal number Cauchy sequence chapter codomain commutative complete ordered field complex numbers concept context contradiction countable decimal expansion decimal places deduce define definition denote elements equation equivalence classes equivalence relation example exists factor false finite follows formal function f give given Hence ideas implies induction infinite injection instance integers intuitive involved irrational least upper bound lemma logical look mathematician mathematics means N₁ natural numbers negative numbers notation number systems order isomorphic order relation ordered field ordered pairs ordered ring p₁ positive precisely prime proof properties proposition prove quaternions rational numbers reader real numbers right inverse satisfies Schröder-Bernstein Theorem set theory set-theoretic solution subset Suppose surjective symbol theorem true truth table unique VxES write zero