## Axiomatic Set TheoryThis clear and well-developed approach to axiomatic set theory is geared toward upper-level undergraduates and graduate students. It examines the basic paradoxes and history of set theory and advanced topics such as relations and functions, equipollence, finite sets and cardinal numbers, rational and real numbers, and other subjects. 1960 edition. |

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### Contents

INTRODUCTION | 1 |

GENERAL DEVELOPMENTS | 14 |

RELATIONS AND FUNCTIONS | 57 |

EQUIPOLLENCE FINITE SETS AND CARDINAL NUMBERS | 91 |

FINITE ORDINALS AND DENUMERABLE SETS | 127 |

RATIONAL NUMBERS AND REAL NUMBERS | 159 |

TRANSFINITE INDUCTION AND ORDINAL ARITHMETIC | 195 |

THE AXIOM OF CHOICE | 239 |

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### Common terms and phrases

A X B arithmetic asserts axiom for cardinals axiom of choice axiom of extensionality axiom of regularity axiom schema cardinal numbers Cauchy sequence Chapter continuum Dedekind deﬁne deﬁnition DEFINrrIoN denumerable domain empty set equipollent equivalence relation exercise ﬁnite cardinals ﬁnite sets ﬁrst element formulation fractions function f given inﬁnite cardinal inﬁnite set integer introduced intuitive JC(A limit ordinal logic mathematics maximal element mooF natural numbers non-negative rational numbers notation notion object language obvious ordered pairs ordinal addition ordinal arithmetic paradox partition PnooF power set primitive formula proof of Theorem Prove Theorem quantiﬁer real numbers reﬂexive schema of replacement schema of separation sequence of real sequences of rational set theory special axiom Suppose Tarski Theorem 12 Theorem 28 THnoHEu THnonsu transﬁnite cardinal transﬁnite induction transﬁnite recursion unique upper bound variables virtue of Theorem well-ordered sets whence Zermelo Zermelo-Fraenkel set theory