## Elements of Set TheoryThis is an introductory undergraduate textbook in set theory. In mathematics these days, essentially everything is a set. Some knowledge of set theory is necessary part of the background everyone needs for further study of mathematics. It is also possible to study set theory for its own interest--it is a subject with intruiging results anout simple objects. This book starts with material that nobody can do without. There is no end to what can be learned of set theory, but here is a beginning. |

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### Contents

Chapter 1 INTRODUCTION | 1 |

Chapter 2 AXIOMS AND OPERATIONS | 17 |

Chapter 3 RELATIONS AND FUNCTIONS | 35 |

Chapter 4 NATURAL NUMBERS | 66 |

Chapter 5 CONSTRUCTION OF THE REAL NUMBERS | 90 |

Chapter 6 CARDINAL NUMBERS AND THE AXIOM OF CHOICE | 128 |

Chapter 7 ORDERINGS AND ORDINALS | 167 |

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### Common terms and phrases

Assume that f axiom of choice belongs binary relation Chapter coﬁnality concept consider construction continuum hypothesis Corollary countable deﬁne deﬁnition e-image epsilon equation equinumerous equivalence class equivalence relation Exercise exists extensionality F is one-to-one fact ﬁnd ﬁnite set ﬁrst ﬁxed following theorem formula function F Hence holds iﬁ iﬂ inﬁnite set isomorphic least element Lemma limit ordinal linear ordering mathematics maximal element natural numbers nonempty set nonempty subset nonzero notation one-to-one correspondence one-to-one function order type ordered pairs ordering relation ordinal number partial ordering Peano system Proof proper subset prove ran f rank rational numbers real numbers recursion theorem replacement axioms satisﬁes set of ordinals set theory Show smaller ordinals subset axiom successor ordinal suppose supremum symbol transﬁnite induction transﬁnite recursion transitive relation transitive set trichotomy true union unique veriﬁcation well-ordered structure y-constructed