From Counting Numbers to Complete Ordered Fields: Set-Theoretic Construction of - Softcover

Horelick, Samuel

Softcover

ISBN 10: 1975629876 ISBN 13: 9781975629878

Publisher: CreateSpace Independent Publishing Platform, 2017

Synopsis

This paper present set-theoretic construction of number sets beginning with von Neumann definition of Natural numbers. Integers are defined in terms of Natural numbers. The set of integers Z is defined to be the set of equivalence classes of ordered pairs (x, y) where x, y are Natural numbers. Integers form a Commutative Ring with Unity. The set of Rational numbers Q is defined to be the set of equivalence classes of ordered pairs (x, y) where x, y are Integers. Rational Numbers form a Field. Rational and Irrational numbers. Dedekind cut. Real numbers form Complete Ordered Field. Further topics include Countable and Uncountable sets, Finite and Infinite sets, the sizes of Infinities, Countable Rational and Uncountable Real numbers, Power Set, Cantor’s theorem, Cantor’s Paradox, Russell’s paradox, Zermelo axioms for set theory, Essentials of Axiomatic method, Continuum Hypotheses, Unlimited Abstraction Principle and Separation Principle, Undecidability of Continuum Hypotheses in Zermelo-Fraenkel system, objections to Zermelo system, and other topics. The paper is aimed at Mathematics and Theoretical Computer Science students.

The article then considers Zermelo-Fraenkel set theory, the independence of the Continuum Hypothesis, and philosophical objections to set-theoretic foundations, especially those concerning actual infinity, impredicative definitions, and the tension between formal consistency and mathematical intuition. Thus, the article is not only an exposition of how number systems may be constructed within set theory, but also a study of the philosophical questions raised by that construction: what mathematical objects are, what it means for them to exist, and how far formal systems can serve as foundations for mathematical truth.

Edition details

• Publisher: CreateSpace Independent Publishing Platform
• Publication date: 2017
• Language: English
• ISBN 10: 1975629876
• ISBN 13: 9781975629878
• Binding: Paperback
• Number of pages: 34