Using tetration to judge the Continuum Hypothesis

[Ripheus23]

For a long time now, I've suspected there must be a direct way to compute the value of 2 to the power of aleph-zero. The original proposal was aleph-one, which I just spent a month a half trying to disprove (not in a strict regimen of ZFC, btw, since "you can't do that"). However, for peculiar reasons I realized this is false, and in fact ended up with what appears to be a basic, nearly direct arithmetic proof that the Continuum is in fact aleph-one, and in fact the formula of the Continuum is the Generalized Continuum Hypothesis.

I've written it out in Word so I'll screenshot the calculations:

EDIT: The tetration formula equivalent to the GCH is that aleph-n ^^ (m+1) = aleph-m.

Edited October 12, 2019 by Ripheus23

[I think I am here.]

7 minutes ago, Ripheus23 said:

The tetration formula equivalent to the GCH is that aleph-n ^^ (m+1) = aleph-m.

I mean, of course. I thought that was obvious?

Spoiler

 I jest. Seriously good work man, although I must admit I understand none of it. 

 

[Ripheus23]

On 10/12/2019 at 3:48 PM, I think I am here. said:

I mean, of course. I thought that was obvious?

I'm so sorry, augh... The way it works is the GCH goes from 0 to 1. In the tetration formula, you add n and m first, like aleph-0 up-up 2 gives you aleph-(0+2), but to get aleph-1, then, parallel to the GCH, you have to suppose that you subtract 1 from the other result. However, for formal reasons you shouldn't have to subtract, and the need to do so comes from tetration being higher than exponentiation, but so anyway if you switch things around, you can say aleph-n up-up m + 1 = aleph-(n + m), and that gives you the right result.

That being said...

I'm going to plow ahead with more images because this is so storming fun.

Note: besides "aleph-megaliths," the sets in question also sound cool (to me) if denoted aleths.

EDIT: Although I cordially say "respondents," I mean critics. My initial post on the subject there involved an allergic reaction to omega-arithmetic (on my part), so I had to go through the meticulous process of having my fear of ordinals disproved. No problem, ultimately, as natural points of entry for the dismissed ordinals, had shown up already in the hyperoperation method of transfinite arithmetic, but by the end of the discussion, no one cared about the rest of my theory (AFAIK) but had lost interest in my proposal due to the massive failure in the initial presentation.

Edited October 16, 2019 by Ripheus23

[Ripheus23]

Normally I wouldn't post a reply to my own post but I have to acknowledge something, namely a lot of the functions that I thought I discovered, are already known. Hypertowers seem to be Veblen functions, for example. Nevertheless, my intuition about the epsilon numbers and the hyperoperation sequence is therefore much more sound than I thought.

[IllNsickly]

There are times I consider myself at least Reasonably intelligent...

 

and then there are these times.
 

I wish I had enough math to even begin following your math.