Ripheus23

OK, so one way the argument goes is: given how simple the function 2^aleph-zero is, it's not likely that its output is particularly exotic. That is, the options seem constricted to aleph-1, aleph-aleph-zero, and maybe aleph-2 or aleph-aleph-1. Beyond that, it would be peculiar to go from 2 to the power of aleph-zero, to wherever we had gone.

Now, I also seem to have a disproof of the Generalized Continuum Hypothesis available. Namely, and stipulating that under the circumstances (the countable infinity of the list of aleph-numbers from aleph-zero to aleph-n), it is possible to replace aleph-zero with {n + 1}, so that we read the GCH as

2^[aleph-n] = aleph-[n + 1] = aleph-aleph-zero.

This doesn't imply that any n gives us aleph-aleph-zero, but only n relative to {n + 1}.

Anyway, aleph-aleph-zero is the successor of all the aleph-numbers on the first list. So it should be given from 2 to the power of aleph-[n + 1] if the GCH is true. However, this amounts to saying that 2 to the power of aleph-aleph-zero is aleph-aleph-zero. But now the GCH says that 2 to the power of aleph-aleph-zero should be 2 to the power of aleph-aleph-1. So the GCH leads, here, to a contradiction, it seems, unless we assume that aleph-aleph-zero is supposed to be equivalent in cardinality to aleph-aleph-1, counter to the principle of Cantor's paradise in general.

Now, there could be a map from 2^aleph-zero, to aleph-1, without the GCH being true, in which case 2^aleph-1 might be something like aleph-aleph-zero, for example. But it seems dubious that the question of the CH in particular should be resolved just by means of "which" axiom scheme one uses. Indeed, the principle of ascent is in equilibrium if 2^aleph-zero = aleph-aleph-zero, &c., I think, so...

This is how my argument proceeds so far:

Suppose that 2 to the power of aleph-aleph-zero, equals [aleph-zero]-aleph-zero. Then evaluate 2 to the power of [aleph-zero]-aleph-zero. Now, if 2 to the power of aleph-zero equals aleph-aleph-zero, then from R to the power of aleph-n we can construct all the alephs after aleph-zero and through aleph-aleph-aleph-aleph-aleph-aleph-...-aleph-n. But this exhausts all the points in the infinite-dimensional array of symbols for these numbers, at this stage. [One of the deeper ideas is that we can symbolize all the aleph numbers for {R to the power of aleph-n} using a single object, an infinite-dimensional super-square, if you will [infinite stacks of squares, in an ascending order per stack, at an infinite number of angles perpendicular to the origin].] Accordingly, 2 to the power of [aleph-zero]-aleph-zero does not map back to any point reached from the first series of aleph-n.

However, therefore 2^[aleph-zero]-aleph-zero, would equal something else if 2^aleph-aleph-zero did not equal [aleph-zero]-aleph-zero. And 2^aleph-aleph-zero would not equal that if 2^aleph-zero, did not equal aleph-aleph-zero. Ergo, 2^aleph-zero does not equal aleph-1 but aleph-aleph-zero, therefore the Continuum Hypothesis is false.

Now, my objection to my own reasoning so far is that it seems to "beg the question," if you will. Or, to put it more kindly, the picture in the previous post is the Supercontinuum Hypothesis, which itself must be solved independently to prove my point. Anyway, if the SH is true, though, we have the following interesting series:

• 2^2^2^2^2... = aleph-zero
• 2^aleph-zero = aleph-aleph-zero
• 2^aleph-aleph-zero = [aleph-zero]-aleph-zero
• 2^[aleph-zero]-aleph-zero = ???

This is a consequence of my theory about the Continuum Hypothesis, I think. In fact, I think it allows a proof that 2 to the power of aleph-zero = aleph-aleph-zero, like a proof-from-assumption-by-contradiction or something. There's more but the idea is that aleph-zero with an infinite number of aleph-glyphs, is equivalent to the first aleph-number on the first list succeeding all the lists of aleph-numbers from aleph-zero to aleph-aleph-aleph-aleph-...-aleph-n. And that is supposed to be equivalent to 2 to the power of aleph-aleph-zero.

But so what happens when we take 2 to the power of [aleph-zero]-aleph-zero? Well, that shoots us all the way past all the lists considered just as lists geometrically distinguished. I mean 2 to each successive n number of aleph-glyphs, gives another dimension of an infinite-dimensional procession of aleph-numbers. So once we transit outside of those, we would map to infinite copies of the infinite-dimensional processions, and then infinite differences of copies of these processions, and so on and on. So 2 to the power of [aleph-zero]-aleph-one, as the first infinity succeeding the sum of the primordial geometrical interval [so to speak], opens the door to the index of sound and the chromatic index, maybe...

"... Apollyon sought to present the Law of Noncontradiction as a law of destruction, the destruction of the impossible and purely unreal. By this means, the Destroyer would try to define the act of creation from annihilation first, and have Itself as the Creator in return. For It would present Itself as the force of negation from pure logic, and present this logic as ultimately destructive. And then Apollyon would collapse Its power around all possible sin, and unleash the visio summum malum at last..."

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?

I know prophecies seem malleable enough that people could say "interesting" things in light of them, all the time, but now some "interesting" (to me) ideas about this stuff came to my mind, so...

... firstly, the Book of Daniel is traditionally (and hereof by me) understood as describing the Antichrist as worshiping an unprecedented god, one variously translated as of "forces" or "strongholds." It occurred to me that the development of the American state, in light of the development of nuclear weapons, represents an example of a process that fits the global-domination scenario predicated of the Antichrist's rise. I.e. because nuclear weaponry (and the "national security state" resulting therefrom) weakens the consent conditions required for government to be legitimate, it is possible, just by possessing a large enough nuclear arsenal, for a country to be a de facto ruler of other, even technically perhaps all, nations.

So, the idea would be that the American government, as the Beast from the Sea, worshiped a god of power, unknown to antiquity (since it was unknown until of late, that uranium contained such power as it does), and with the power of this god of power, came to dominate humankind unjustly. And the "ethical theorists" of the US state made an image of this power (the use of nuclear weapons on actual cities), and made for others to worship the image (to believe in the "ethical theory" that justified the use of these weapons on actual cities).

I know that this correspondence is inexact and I tend more to go with the idea that the Beast from the Sea symbolizes patriarchy/rape-culture/et. al. (as a sexuality-derived context of sin, juxtaposed with the harlot-city), but it would be a peculiar fulfillment of prophecy, I suppose...

... secondly, though, then, I did some research and https://www.haaretz.com/archaeology/.premium.MAGAZINE-jewish-god-yahweh-originated-in-canaanite-vulcan-says-new-theory-1.5992072 points to a prehistory of YHWH as a god of metal. Uranium is, after all, a metal, so I wonder if there is a way to map the first idea onto this, sort of. Namely, the end-times deception is so subtle as to involve a semantic reference to the actual one true God YHWH, but under a connotative description that is polytheistic? This would, indeed, be profane (to turn the image of monotheism into a polytheistic one, as such). OTOH this would not match on to the description of the Antichrist's worship as directed to an unprecedented deity (except in the sense that, aside from the prehistorical context, no one worships YHWH as if He were a pagan deity [as it were]).

For most of my relevant life, I have believed the Continuum Hypothesis on the ground that since Cantor proved the distinction between aleph-zero and aleph-zero+n, by referring to the distinction between the cardinality of the repeating vs. the nonrepeating infinite decimal numbers (in the diagonal argument, explicitly/implicitly/w/e), this indicated that our semiotic intuition of the continuum was the immediate alternate for the prior system-set, in which event it would "seem" as if aleph-C = aleph-one.

However, what if the answer is instead that aleph-C = aleph-aleph-zero? I.e. an aleph with a littler aleph with a zero, subscriptwise. This would be intuitive in the sense of being a "successor" set, of aleph-zero, immediately as it were. And indeed, if you compare the difference between "summing up" the first list of aleph-numbers, and the identification of aleph-aleph-zero (as the first of the second list), it seems (to me) that this is an image of the procedure of the continuum itself, i.e. the infinity of infinities on the first list sums to the continuum under aleph-aleph-zero.

This also would mean, though, that 2 to the power of aleph-zero, does not = aleph-1, but equals aleph-aleph-zero, which seems to indicate that 3 to the power of aleph-zero, would be aleph-aleph-aleph-zero, a number Cantor is reported to have believed to exist. {... and so on and on...}

???

Not necessarily in the event the Ripheus story was publishable even possibly, but "just in case"...

So I think that the Beast, the False Prophet, and the Dragon are also the lust of the flesh, the lust of the eyes, and the pride of life. That is, they are three kinds of sin, viz. the final offenses. The early Christian theorists did, if faintly here, recognize the existence of this problem...

[A tropey-sounding story, haha!]

Long ago, when the Morning-god, Ramn Surdur, tried to destroy the world, He was bound into the Shadow of Heaven by the power of the Firelights. For thousands of years, in the City Under Heaven alone, these lights burned to uphold the Shadow. But a crisis of war arose that threatened our protection with extinction, whereafter the Firelights ignited themselves anew elsewhere, to strengthen their purpose to the last. So it was that the City Under Heaven lost splendor and influence, until this very day...

[Deviant element: the doctrine of the antagonist, here, revolves around a quasi-stigmatic image of the Atonement. I.e. there is a supreme evil, the actual nature of, or precursor to, the Morning-god, but this was not originally some primordial malice as such, but instead the theory was that the Atonement wrought by the ancient Savior required the Savior to suffer the bridge-analogies of the final offenses themselves, to wit rape, torture, and murder, at the hands of at least one individual person. So, the deviant element is that to save the world by suffering in its place, the ancient Savior had to be tortured and murdered, but also raped, and the one who raped the Savior was transformed by this action into Ramn Surdur or its predecessor/source/lord... How to integrate the Christian theme here with the prologion-theme of the Shadow-binding, IDK yet. I don't even have any actual characters haha!]

"... outwitting the wicked,

One sin of theirs

And one grace of mine

At a time..."

❣❀

You must remember that the final offenses are not united as the Trinity is. Their "unity" in the entanglement theorem is enough of an illusion to tempt us to the profane belief, here. However, it is true that due to their entanglement, romantic injustice and recursive disgrace can be intercomputed: the four forms of amendment "go with" the square of metafinity, so that for the problem

• Find the appropriate degree of punishment for the sin of romantic injustice.

... the solution goes for: [absolute finite: A; relative finite: B; relative infinite: C; absolute infinite: D] so that the priority of redemption over retribution gives us A = punishment, B/C = forgiveness/apologies, and D = redemption (atonement, salvation even). But if even the worst punishable sin (note: recursive disgrace is proto-talionic at least, but it does not seem as if transcendental delusion is the kind of thing that can be externally "avenged") merits absolute finite retribution at most, does it follow not that no sin ought to be punished so gravely as is required in the system of damnation?

[Of course, to tempt the victims [subjects] of this injustice, with the motive of vengeance, seemingly justified, is of the will of transcendental evil, here. For the rage of those violated in the first place is translated into the power of damnation itself, is it not?]

Hmm...

• []()[]()()()()[][]()()[][][][][][][][][][]()[][][]()()()()[]...
• [][]()()[][][]()()()[][]()[][]()[][]()()()[]()()()[][]()[]()...
• &c. ...

Don't these potentially map to nonrepeating decimals, except represented in the equivalent of binary? That also suggests a gloss on binary, I suspect: instead of 0/1::true/false, we have 0/1::possible/actual. I think that's probably even more fitting since false does not = 0, but -1 (I know in propositional space by itself, the absence of truth+ is equivalent to the presence of truth-, and true/false is a propositional opposition, but...).

At any rate, this would imply that the cardinality of the set of sigma-actuations is equivalent (as such) to the cardinality of the continuum. Allowing that every sigma-actuation represents a possible world, this would imply the abstract existence of at least aleph-C worlds. [This would be a far more massive multiverse than the string-theoretic one by itself, then... Not sure about Everett's or Tegmark's, though. Tegmark's seems to contain universes whose component cardinalities (if that description makes sense!) are whichever aleph-X there could be, in which event these self-transfinite universes would have to contain infinite stacks of systems of multiverses or what, individually. So who knows!] Now I'm in the camp that believes aleph-C = aleph-1, and so that 2 to the power of aleph-0 = aleph-1. [I think those beliefs go together...] So I don't have to suppose that the cardinality of the known universe's mathematical equations (the level of infinity from which the numbers for the equations are drawn) is greater than aleph-1, at least not as such. Indeed, it appears that irrational numbers occur in physics equations so it would appear that the universe is thus condensed at least from the ether of the continuum, so to speak. [Btw, this means both (A) finite/mortal/w/e beings can grasp infinity but also that (B) this turns out to be a potential let-down as there are still infinite numbers of infinities that we can't (yet/easily) grasp, and so on.]