## The Mathematics of Infinity: A Guide to Great IdeasA balanced and clearly explained treatment of infinity in mathematics. The concept of infinity has fascinated and confused mankind for centuries with concepts and ideas that cause even seasoned mathematicians to wonder. For instance, the idea that a set is infinite if it is not a finite set is an elementary concept that jolts our common sense and imagination. the Mathematics of Infinity: A guide to Great Ideas uniquely explores how we can manipulate these ideas when our common sense rebels at the conclusions we are drawing. Writing with clear knowledge and affection for the subject, the author introduces and explores infinite sets, infinite cardinals, and ordinals, thus challenging the readers' intuitive beliefs about infinity. Requiring little mathematical training and a healthy curiosity, the book presents a user-friendly approach to ideas involving the infinite. readers will discover the main ideas of infinite cardinals and ordinal numbers without experiencing in-depth mathematical rigor. Classic arguments and illustrative examples are provided throughout the book and are accompanied by a gradual progression of sophisticated notions designed to stun your intuitive view of the world. With a thoughtful and balanced treatment of both concepts and theory, The Mathematics of Infinity focuses on the following topics: * Sets and Functions * Images and Preimages of Functions * Hilbert's Infinite Hotel * Cardinals and Ordinals * The Arithmetic of Cardinals and Ordinals * the Continuum Hypothesis * Elementary Number Theory * The Riemann Hypothesis * The Logic of Paradoxes Recommended as recreational reading for the mathematically inquisitive or as supplemental reading for curious college students, the Mathematics of Infinity: A Guide to Great Ideas gently leads readers into the world of counterintuitive mathematics. |

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There are several math errors in the book that clearly not typos. For example on page 61 he states that the mass of the earth is about 10^100 grams. It is actually about 6*10^30 grams.

### Contents

1 Elementary Set Theory | 1 |

2 Functions | 37 |

3 Counting Infinite Sets | 75 |

4 Infinite Cardinals | 103 |

5 Well Ordered Sets | 163 |

6 Inductions and Numbers | 205 |

7 Prime Numbers | 247 |

8 Logic and MetaMathematics | 271 |

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284 | |

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### Common terms and phrases

algebra argument array arrow assume axioms bijection bijection f card(A card(lR CH and notCH codomain completes the proof conclude construct contains Continuum Hypothesis contradiction count countable set decimal expansion deﬁne a function deﬁnition degree 1 term denote divisible equivalent exactly example f is one-to-one f(zr false family of sets ﬁnd ﬁnite sets ﬁrst fractions function f geometric series given googolplex Hence Hilbert Hilbert’s Inﬁnite Hotel implied list Induction Hypothesis Induction Step inﬁnite cardinals inﬁnite family inﬁnite set integer inverse irrational numbers larger Let f limit ordinals logical Mathematical Induction mathematicians natural numbers notation odd number one-to-one function ordered set pattern perfect numbers picture preimage prime factorization prime numbers prove rational numbers reader real numbers room 2 room satisﬁes Sentence Set Theory Speciﬁcally statement subset Suppose symbol Theorem Transﬁnite Induction Trichotomy Property true uncountable uncountable sets write zeros