Ripheus23

My argument that the Continuum Hypothesis is false

First, I would have to present a "reconstruction" of Cantor's original intuition on this matter. Notably, since Cantor would not have been using ZFC, after all, it is for that reason alone independent on Godelian qualifications of things like ZFC. Moreover, thereof, Godel himself thought that the Continuum Hypothesis was decidable, regardless of whether other mathematical propositions are not.

Now, my supposition is that since Cantor's "discovery" of the higher infinite turned on the Diagonal Argument [as an intuitive display of the principle of successive infinities], his intuition about the continuum depended on the appearance of this argument notationwise. In other words, Cantor felt that the continuum was the immediate successor infinity, aleph-1, for aleph-zero, since there was an appearance of the rational numbers being immediately succeeded by the irrational ones. After all, it's just this one variation, repeating vs. nonrepeating decimal notation, that seems to fix the boundary, intuitively, between the countable infinity and the continuum.

Regardless of whether Cantor himself felt/intuited the CH to be true, on such a ground, I personally have believed for a long time that the CH is true for precisely this reason. In fact, and given that I didn't know that many mathematicians think the CH is false, I had spent a decent deal of time trying to come up with a supermathematical argument on behalf of the CH's truth, namely something to do with the physical logic of empirical mathematical notation as used to represent successive infinities and so on.

However, my idea now is that the CH is false, but the way it is makes sense of the possible original intuition that it is true. Viz., the continuum is the immediately succeeding infinite number not by original indexical succession, but by the glyphic index for the aleph-sequence per se nota. In other words, aleph-aleph-zero is the continuum, and the way this infinity "sums over" the entire first-level aleph-n series is a representative of how the infinity of the nonrepeating decimal numbers condenses around the natural numbers so immensely.

Now, if the solution to the CH is, "There are countably infinitely many cardinal numbers 'between' aleph-zero and the continuum," it becomes quite the task, to look for any of those. Given how little I know in particular about such a topic, I'll just toss out the option: what of infinitesimals? Is there a rigorous interpretation of them ZFCwise, for instance, where they might fit into the cardinal hierarchy, as less than the continuum but greater than the countable infinity?