Discovering Modern Set Theory: The basics (Google eBook)
From the reviews: Serious graduate students ... would profit from reading the book for the mathematical maturity they would gain in the process. The conversational, almost Socratic, style of exposition is well suited to giving students some insight into the process of doing mathematics as well as to the importance of asking the right questions ... Just and Weese's text would be ideally suited for ... students who are serious about studying set theory. --Journal of Symbolic Logic The careful exposition, written in a lively and very readable style which addresses the reader rather directly, provides (by explanations, comments, and remarks) much information and motivation. Recommended. --Monatshefte fur Mathematik This book is an introduction to set theory for beginning graduate students who want to get a sound grounding in those aspects of set theory used extensively throughout other areas of mathematics. Topics covered include formal languages and models, the power and limitation of the Axiomatic Method, the Axiom of Choice, including the fascinating Banach-Tarski Paradox, applications of Zorn's Lemma, ordinal arithmetic, including transfinite induction, and cardinal arithmetic. The style of writing, more a dialogue with the reader than that of the Master indoctrinating the pupil, makes this also very suitable for self-study.
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assume Axiom of Choice Axiom Schema axiomatic set theory axioms of ZFC binary relation called Chapter Claim CON(ZFC consider contains contradiction Convince Corollary countable Dedekind completeness denote DLONE domain element empty set example EXERCISE exists expression family of sets formal formula free variables function f functional class Godel's holds implies Incompleteness Theorem infinite cardinal initial segment Lemma let f limit ordinal mathematicians mathematics Mathographical Remarks Moreover Mostowski collapse natural numbers nonempty set notion one-to-one function order isomorphism order types pair Paradox partial order relation player power set proof of Theorem proper class Prove Theorem rank function recursive construction rng(F satisfies sentence sequence set-theoretic Show strict partial order strictly wellfounded subset successor ordinal Suppose surjection symbols Theorem 16 theory ZFC transfinite transitive set truth ultrafilter uncountable universe weakly inaccessible cardinal wellfounded relation wellorder wellorder relation winning strategy write
Page xiii - Hurwitz used to say that in teaching on an elementary level one must tell the truth, nothing but the truth, but not the whole truth.
Page xiii - ... set-theoretic prerequisites". Volume II is claimed to be aimed at more advanced graduate students and research mathematicians specializing in fields other than set theory.
Page xiii - ... technique (like the craft of using Martin's axiom). So a technique is occasionally illustrated by proving a somewhat marginal theorem. Although the authors admit that "frequency of use outside set theory" is based on their subjective perception, by the reviewers opinion they were quite successful. Volume I contains the basics of modern set theory. Many graduate texts on analysis, algebra, topology, or measure theory begin with a review of parts of this material as "set-theoretic prerequisites".