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Abstract
We provide a canonical construction of the natural numbers in the universe of sets. Then, the power set of the natural numbers is given the structure of the real number system. For this, we prove the co-finite topology, C o f ( N ) , is isomorphic to the natural numbers. Then, we prove the power set of integers, 2 Z , contains a subset isomorphic to the non-negative real numbers, with all its defining structures of operations and order. We use these results to give the power set, 2 N , the structure of the real number system. We give simple rules for calculating addition, multiplication, subtraction, division, powers and rational powers of real numbers, and logarithms. Supremum and infimum functions are explicitly constructed, also. Section 6 contains the main results. We propose a new axiomatic basis for analysis, which represents real numbers as sets of natural numbers. We answer Benacerraf’s identification problem by giving a canonical representation of natural numbers, and then real numbers, in the universe of sets. In the last section, we provide a series of graphic representations and physical models of the real number system. We conclude that the system of real numbers is completely defined by the order structure of natural numbers and the operations in the universe of sets.
Highlights
In building the continuum, we make use of properties of integers and sets
We prove that the real number system is isomorphic in structure to 2N
Previous expositions of axiomatic set theory for analysis begin with a description of the natural numbers in two main forms [7]
Summary
We make use of properties of integers and sets. Apart from this, we assume the basic concepts of category theory. That is why real numbers are usually presented axiomatically as a set satisfying certain rules, without specifying the nature of the set nor proving its existence. Constructions of the real number system are rarely taught, even in advanced courses of analysis This leads to a certain gap in the learning of the student. We apply the same method to define order and operation on 2−N ; the continuum [0, 1] is isomorphic in structure to the power set of −N. These constructions are generalized to obtain R+ , the set of non-negative real numbers. The structures Co f (N), 2−N are naturally embedded into Z−
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