Classic Set Theory: For Guided Independent StudyRoutledge, 2017. gada 6. sept. - 296 lappuses 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 definition of the real numbers in terms of rational numbers and ultimately in terms of natural numbersDefining natural numbers in terms of setsThe potential paradoxes in set theoryThe Zermelo-Fraenkel axioms for set theoryThe axiom of choiceThe arithmetic of ordered setsCantor's two sorts of transfinite number - cardinals and ordinals - and the arithmetic of these.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. |
Saturs
The Real Numbers | |
The Natural Numbers | |
The ZermeloFraenkel Axioms | |
The Axiom of Choice | |
Cardinals without the Axiom of Choice | |
Ordered Sets | |
Ordinal Numbers | |
Set Theory with the Axiom of Choice | |
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argument arithmetic assuming AC axiom of choice axiom of replacement axiom of separation bijection Cantors Cauchy sequences chapter construction contradiction Dedekind left sets define a function Definitions Let disjoint equal equinumerous equivalence classes exploit finite sets formal language formula function f give Hint inductive step infinite sets initial ordinal initial segment instance integers 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 oneone order-embedding order-isomorphic ordered pairs partial order prove Range(f rational numbers real analysis real numbers result holds result of Exercise Schröder-Bernstein theorem SegX(x set theory Solution subset of ℕ successor Suppose transfinite induction unique well-ordered sets Zorns lemma ℵγ