Set Theory and Logic   Evaluating f(x) = C for Infinite Set Domains

Authors: Ron Ragusa

In a previous paper, The Function f(x) = C and the Continuum Hypothesis, posted on viXra.org (viXra:1806.0030), I demonstrated that the set of natural numbers can be put into a one to one correspondence with the set of real numbers, f : N → R. In that paper I used the function f(x) = C to create an indexed array of the function’s real number domain d, the constant range, C, and the index value of each iteration of the function’s evaluation, i, for each member of the domain di. The purpose of the exercise was to provide constructive proof of Cantor’s continuum hypothesis which has been shown to be independent of the ZFC axioms of set theory. Because the domain of f(x) = C contains all real numbers, evaluating and indexing the function over the entire domain leads naturally to the bijective function f : N → R. In this paper I’ll demonstrate how the set of natural numbers N can be put into a one to one correspondence the power set of natural numbers, P(N). From this I will derive the bijective function f : N → P(N). Lastly, I’ll propose a conjecture asserting that f(x) = C can be employed to construct a one to one correspondence between the natural numbers and any infinite set that can be cast as the domain of the function.

Comments: 5 Pages. email: ron.ragusa@gmail.com