Artículo: AMZ-B09P5NBD5H

Set Theory and Infinity: An Informal Introduction

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This book is an informal exposition of set theory, emphasizing infinite sets. The goal of this document is to develop intuition for as well as mathematical fluency in understanding and manipulating infinite sets. This book should be accessible to anyone with a year of calculus. A great deal of mathematics explicitly requires a background in set theory, and even advanced topics are typically formulated in terms of the kind of informal set theory presented here. This book presents the set theory required for such topics as metric spaces, vector spaces, topology, abstract algebra and analysis up to a first or second year graduate level. Focus is on countably infinite and uncountable sets, including set creation, union, disjoint union, intersection, complement, cartesian product and subset. The crucial use of bijective, injective and surjective functions in analyzing and comparing infinite sets is stressed. There is an introduction to symbolic logic and set theory. The Cantor set, the Hilbert Hotel and Russell's paradox are discussed. The basics of infinite cardinal numbers and infinite ordinal numbers are covered. Other topic include equivalence relations, order relations, the algebra of sets, power sets and the hierarchy of infinities. The formal axioms of ZF and ZFC are not discussed, nor is the axiom of choice. The following are the top-level subjects of the table of contents: introduction; notation; very simple description of a set; some operations on sets; defining more general sets; infinite sets - an introduction; maps (AKA functions); countably infinite sets; real numbers. Uncountable sets; finite unions of uncountable sets; the algebra of sets; equivalence relations and categorization; cartesian product; symbolic logic and set theory; paradoxes and set specification; binary relation; power set; order relations; cardinal number manipulation; hierarchy of infinities; ordinal numbers. As an example, here are the TOC topics under the header "countably infinite sets": countable finite set and cardinality; countably infinite set and cardinality; the set of rational numbers; union of countably infinite sets; other countably infinite sets of rational numbers; another countably infinite set.

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Set Theory and Infinity: An Informal Introduction

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Set Theory and Infinity: An Informal Introduction

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