## Introduction to Modern Set TheoryThis is modern set theory from the ground up--from partial orderings and well-ordered sets to models, infinite cobinatorics and large cardinals. The approach is unique, providing rigorous treatment of basic set-theoretic methods, while integrating advanced material such as independence results, throughout. The presentation incorporates much interesting historical material and no background in mathematical logic is assumed. Treatment is self-contained, featuring theorem proofs supported by diagrams, examples and exercises. Includes applications of set theory to other branches of mathematics. |

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### Contents

Some Mathematical Preliminaries | 1 |

The Axioms Part I | 23 |

Regularity and Choice | 49 |

The Foundation of Mathematics | 67 |

Infinite Nwnbers | 73 |

Two Models of Set Theory | 101 |

Infinite Combinatorics | 111 |

147 | |

153 | |

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### Common terms and phrases

1-1 function antichain Aronszajn tree axiom of choice chapter cofinal combinatorial contradiction corollary define Definition equivalence classes equivalence relation example exercise existence extensionality filter finite subset first-order theory follows formula hence homogeneous implies incompleteness theorem induction hypothesis infinite cardinal infinite set initial segment intervals large cardinals least element Lemma limit ordinal linear order linearly Martin's axiom maximal measurable cardinal minimal element models of set natural numbers nonempty notation Note order type order-isomorphic ordered pairs ordinal arithmetic pairwise disjoint partially ordered set partition power set proof of theorem Proposition prove Theorem reader recursive construction sequence set theory set-theoretic Show singular cardinal splitting tree statement strong limit strongly inaccessible cardinal successor Suppose Suslin line Suslin tree theorem 40 transitive set tree property ultrafilter uncountable union weakly inaccessible well-ordered set

### Popular passages

Page 147 - THEORY (GENERAL) van Dalen, D., Doets, HC, and de Swart, H., Sets: Naive, Axiomatic and Applied, Pergamon, London, 1978. Henle, J., An Outline of Set Theory, Springer- Verlag, New York, 1986. Jech, T., Set Theory, Academic Press, New York, 1978. FORCING Burgess, J., "Forcing," in Handbook of Mathematical Logic, J.