Samuel Horelick

From Counting Numbers to Complete Ordered Fields Samuel Horelick

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.