## 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

1 | |

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 |

Chapter 8 ORDINALS AND ORDER TYPES | 209 |

Chapter 9 SPECIAL TOPICS | 241 |

NOTATION LOGIC AND PROOFS | 263 |

269 | |

271 | |

273 | |

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### Common terms and phrases

A G B A U B axiom of choice axiomatic belongs binary relation Chapter concept consider construction continuum hypothesis Corollary countable define definition e-image equation equinumerous equivalence class equivalence relation example Exercise exists extensionality F is one-to-one fact finite set following theorem formula function F Hence holds infinite set isomorphic least element limit ordinal linear ordering mathematics n e o 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 R-minimal element ran f rank rational numbers real numbers replacement axioms set of ordinals set theory Show smaller ordinals Sq(A subset axiom suppose supremum symbol transfinite induction transfinite recursion transitive relation transitive set trichotomy true union unique verify well-ordered structure y-constructed Zorn's lemma