User blog:Edwin Shade/The Continuum Hypothesis

Except for their inception in the late 1800's, the cardinal infinities denoting the set of real numbers and integers have been assumed to be different. I now say that although there does not exist a bijection between the two sets, I feel there exists an intuitive way to see there does not exist an infinity between these two, and therefore proves the Continuum Hypothesis.

The set of real numbers, $$\mathbb{R}$$, is composed of an infinitude of numbers, which share this common property: 'between any two different real numbers, there exists an infinity of real numbers equal in magnitude to the set of real numbers itself'. The set of integers, $$\mathbb{Z}$$, is composed of a seemingly lesser set of numbers, which if 'zoomed-in-on' far enough, will reveal that there is a discontinuous gap between each integer. It seems clear to me then, that the only differing aspect of the set of integers and real numbers is one of continuity, and dis-continuity, and so therefore an infinity lying between then would be partially continuous, partially discontinuous.

The issue with this is that a set that is partially continuous would by definition have to be at least as large as the set of real numbers, and hence couldn't fall in between that of the integers and the real numbers. Therefore, there must exist no such infinity, and so the Continuum hypothesis is seen to be intuitively true.

Granted, this may not be a formal proof, but I fail to see any flaws in it; perhaps if there any they could be brought to my attention in the comments below, as I do want to seriously true to prove this.