## 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 |

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### Contents

The intuitive background | 1 |

The beginnings of formalization | 39 |

The development of axiomatic systems | 143 |

Strengthening the foundations | 249 |

259 | |

### Common terms and phrases

addition argument arithmetic axioms becomes bijection called cardinal number chapter classes commutative complete concept consider construction contradiction corresponding decimal deduce define definition denote described elements equal equation equivalence example exists expansion express fact factor false finite follows formal function give given Hence ideas implies induction infinite injection instance integers intuitive inverse involved irrational isomorphic least lemma less limit logical look mathematician mathematics means multiplication namely natural numbers negative notation operations ordered ordered field ordered pairs particular picture places positive possible precisely predicate prime problem proof properties proposition prove rational numbers reader real numbers reason relation ring satisfies sense sequence solution square statement structure subset Suppose surjective symbol theorem theory thing true truth unique upper bound usual write written