An Outline of Set Theory
A problem-oriented introduction to set theory, this volume is intended for undergraduate courses in which students work on projects in groups and present their solutions to the class. With all necessary assistance from the instructor, students complete their weekly assignments, write short papers for their classmates, and lecture on their results. This method offers a deeper understanding of the material, as well as a clearer view of what it means to do mathematics.
An Outline of Set Theory is organized into three parts: the first presents definitions and statements of problems, the second offers suggestions for their solutions, and the third contains complete solutions. Topics include standard undergraduate set theory, as well as considerations of nonstandard analysis, large cardinals, and Goodstein's theorem. Drawn from the author's practical experience asáProfessor of Mathematics at Smith College, this text offers a novel and effective approach to teaching and learning the fundamentals of set theory.
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adding numbers aÚ Q Axiom of Choice Cantor chapter commutative contradiction countable Dedekind ﬁnite Dedekind inﬁnite deﬁnable subset deﬁne a function Deﬁnition difﬁculty domain of g equivalence relation example false ﬁnd ﬁrst follows function f G6del’s Goodstein’s Theorem greatest element hence hyperreal inﬁnite numbers inﬁnite sets integers least element least upper bound Let f map limit ordinal linear ordering maps n one-to-one mathematicians mathematics maximal element natural numbers neg numbers new|I(n nonempty Notation numbers is pos one-to-one function order-type Pair Set positive problem PROJECT 28 proof of Theorem proper subset prove range of g rational numbers real numbers Regularity Replacement satisﬁes satisfying schnitt sequence set by Comprehension set theory slot steps strongly inaccessible successor ordinal suggestions superbase Theorem 8.2 transﬁnite induction transitivity trichotomy true iff ultraﬁlter well-deﬁned Well-ordering Theorem write