Classic Set Theory: For Guided Independent Study
Designed for undergraduate students of set theory, Classic Set Theory presents a modern perspective of the classic work of Georg Cantor and Richard Dedekin and their immediate successors. This includes:
The book is designed for students studying on their own, without access to lecturers and other reading, along the lines of the internationally renowned courses produced by the Open University. There are thus a large number of exercises within the main body of the text designed to help students engage with the subject, many of which have full teaching solutions. In addition, there are a number of exercises without answers so students studying under the guidance of a tutor may be assessed.
Classic Set Theory gives students sufficient grounding in a rigorous approach to the revolutionary results of set theory as well as pleasure in being able to tackle significant problems that arise from the theory.
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The Real Numbers
The Natural Numbers
The Zermelo Fraenkel Axioms
The Axiom of Choice
Cardinals without the Axiom of Choice
Set Theory with the Axiom of Choice
argument assuming AC axiom of choice axiom of replacement axiom of separation bijection Cantor reals Cantor's Card(A cardinal arithmetic cardinal numbers Cauchy sequences chapter codomain construction contains contradiction corresponding countably infinite Dedekind left sets define a function Definitions Let disjoint equal equinumerous equivalence classes exploit finite sets fixed formal language formula function g Further exercises Exercise give given Hint inductive step infinite sets initial ordinal initial segment instance integers intuition isomorphic least element least upper bound limit ordinal limit point linear linearly ordered set look mathematics maximal element means multiplication natural numbers non-empty set non-empty subset notation one-one function order-embedding order-isomorphic ordered pairs partial order proper subset prove rational numbers real analysis real numbers represent result holds result of Exercise Segx(x set theory Solution successor Suppose transfinite induction uncountable union unique well-ordered sets write Zorn's lemma