By A. Shen, Nikolai Konstantinovich Vereshchagin
The most notions of set conception (cardinals, ordinals, transfinite induction) are primary to all mathematicians, not just to those that specialise in mathematical good judgment or set-theoretic topology. easy set conception is mostly given a quick review in classes on research, algebra, or topology, although it is adequately very important, fascinating, and easy to benefit its personal leisurely remedy.
This booklet offers simply that: a leisurely exposition for a assorted viewers. it really is compatible for a wide diversity of readers, from undergraduate scholars to expert mathematicians who are looking to ultimately discover what transfinite induction is and why it truly is regularly changed by way of Zorn's Lemma.
The textual content introduces all major topics of "naive" (nonaxiomatic) set idea: services, cardinalities, ordered and well-ordered units, transfinite induction and its purposes, ordinals, and operations on ordinals. incorporated are discussions and proofs of the Cantor-Bernstein Theorem, Cantor's diagonal technique, Zorn's Lemma, Zermelo's Theorem, and Hamel bases. With over one hundred fifty difficulties, the booklet is an entire and available creation to the topic.
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The set Z (of integers with the usual ordering) is not isomorphic to the set Q (of rational numbers). Indeed, let α : Z → Q be an isomorphism. Take two adjacent integers, say, 2 and 3. The isomorphism α would map them to two rational numbers α(2) and α(3) such that α(2) < α(3) (because 2 < 3). Then rational numbers between α(2) and α(3) should be images (under α) of some integers between 2 and 3—but there are no such integers. • A more complicated example of nonisomorphic sets is Z and Z + Z.
Let m and n be positive integers. Prove that any partially ordered set of cardinality mn + 1 contains either m + 1 pairwise incomparable elements or n + 1 pairwise comparable elements. Problem 90. A sequence consists of mn + 1 numbers. Prove that it is possible to remove some of them so that remaining elements form either an increasing sequence of length m+1 or a decreasing sequence of length n + 1. ) Problem 91. Consider the set of all subsets of the set of nonnegative integers, ordered by inclusion.
26 1. Sets and Their Cardinalities Problem 52. Prove that for any uncountable set A ⊂ R there exists a condensation point a such that any neighborhood of a has an uncountable intersection with A. ) Problem 53. Prove that a closed set A ⊂ R that has no isolated points has the cardinality of the continuum. Problem 54. Prove that any closed set A ⊂ R is either countable (or ﬁnite) or has the cardinality of the continuum. , points b such that every neighborhood of b has an uncountable intersection with A.