A Resolution to Hilberts First Problem

The continuum hypothesis (CH) is one of and if not the most important open problems in set theory, one that is important for both mathematical and philosophical reasons. The general problem is determining whether there is an infinite set of real numbers that cannot be put into one-to-one correspondence with the natural numbers or be put into one-to-one correspondence with the real numbers respectively. We will devote the first part of this article toward understanding and building upon the mechanical principles involved in the 'binary structure' of irrational numbers. By 'binary structure' we mean, the possible types of binary sequences associated with the fractional portions of irrational numbers. In the second portion, we will use these ideas to build certain sets that are a subset of the real's and develop means of comparing their cardinalities, which is finally used to demonstrate the fallacy of CH. Philosophically and perhaps practically, mathematicians are divided on the matter of a resolution to CH. The uncanny persistence of the problem has led to several mainstream views surrounding its resolution. Discussions on the possibility of a resolution, notably one from the Institute of Advanced study at Princeton gives a vast account of the thoughts on the problem and a detailed summarry of the progress made this far, along with what may constitute a solution see [1]. Most similar discussions express the current main stream thoughts on the matter, and the division that exists amongst mathematicians in the views they hold with regard to a resolution, the nature of the resolution and what a resolution to CH may mean. Some of the mainstream views can be summarized as follows: Finitist-mathematicians believe that we only ever deal with the finite and as such and simply put, we can't really say much about the infinite. Pluralists believe in the plurality that any one of the outcomes of CH are possible, and naturally the non-pluralists are against this idea. Though the efforts of both Cohen and Godel showed the consistency of ZF C+ ̸ CH and ZF C + CH, Cohen held a strong pluralist view that his demonstration that CH cannot be decided from ZFC alone, essentially resolved the matter. Contrary to this however, Godel believed that 'a well justified extension' to ZFC was all that was necessary in the way of deciding CH. Godels program seemingly the promising option going forward aims to find an extension of …