Ripheus23

So I was reading about the von Neumann universe, which the article (Wikipedia's) said was built up by transfinitely iterating the powerset operation. So unfortunately(ish), I had assumed in my version of set theory that the infinite ascension operation (equivalent to the powerset one at least in the simple minimum limit) would not lead to the "apex" of the hierarchy, at least not directly/necessarily/etc. I know there's no given apex to the vNu since the vNu is not supposed to be "the set of all sets" as such, but anyway, I also remember reading (can't find where right now...) that large cardinal axioms (I think) are conceived as "arbitrarily close to vNu" in the sense that a large cardinal is understood in the light of transfinality (my term, but their procedure) first, from which some descension is possible.

Now, this might not make much of a difference one way or another, although it seems to indicate (to me) that simplifying the axiom system so that large cardinal axioms are reduced to theorems of the axioms of transfinality and transcardinality (as such), is a possible "way to go" (as I have been going, to be sure). So in other words, either I'm going to find out that these topics have already been deeply addressed  or that I'm on to something interesting.

Hmm...

What the Continuum Could Or Could Not Be.docx

The Ceryneian unicorn

Obscure unicorn-theoretic being that resembles a deer more than a horse.

My next meme with the same general theme

Could the USAF be an "inherently criminal organization"?

I will admit, my initial reaction to the "Storm Area 51" meme included a peculiar ambivalence. I had become accustomed to thinking of America's nuclear/"national-security" administrations as the major criminal organizations in the government due to e.g. Bomb Power by Gary Wills. So if I was going to think of a particular branch of the military as contaminated by those criminal forces, at this point I would have pointed at the US Navy on the grounds that (a) the Trident submarines are the most demonic weapons the US has (as far as I know) and (b) I live about 45 minutes from the town where the major Trident base is located (the Poulsbo/Bangor area, near the Puget Sound Naval Shipyard in Bremerton)---so the image of the Trident submarines is called to my mind on a regular basis as such. (Unfortunately, also, my dad was a nuclear submariner back in the 80s and one of my two main managers at work is also a former nuclear submariner.)

Now this is actually a little peculiar, otherwise, since my obsession with the Vietnam War USED to prompt me to fixate on the USAF as especially symbolic of what is worst about the US military government. In fact, I had a theory that because the separation of the USAF from the US Army was mediated in part by America's role in the corruption of the Geneva Conventions (the pre-70s exclusion of air-war regulations therein), and because the USAF had so meticulously carried out the holocaust of Vietnam, then by the Nuremberg standards there seemed to be grounds for believing that the USAF is an inherently criminal organization, one where even just joining is a crime, then.

Now, we don't know what kind of higher-end physics weapons-system development is actually going on at Area 51. It could be reverse-engineering of alien technology, I suppose, although until there is a strong working model of FTL (besides the Alcubierre hypothesis, say) I will remain exactly skeptical of the idea. But we know that they are working on higher-end physics-based weapons of some kind there, because that's just what the government does, after all. And since it is a sin to try to use the powers of creation to create these means of destruction as such, we know that some of the people at Area 51 are sinning, and indeed it appears that people there who are exposed to dangerous materials are treated badly (I remember hearing about this years ago, though the details escape me), for example.

It's also worth noting that the Dugway Proving Ground, another (international law-wise) criminal setup (because they work on chemical and biological weapons there), is not terribly far from the Area 51 region. Apparently DPG is administered by the Army, though, but anyway the point is there's a little constellation of darkness in this area of the US, so...

... it would not be unfitting...

*Whistles*

https://en.wikipedia.org/wiki/Infinitary_logic

Very interesting topic as it lines up with my notion of infinite transcension/transfinality sequences...

Now the axiom of transcardinality allows us to apply the concept of the metafinite diamond to the relatively infinite, but it is also possible hereof to conceive of the antifinite. That is, finite-infinite is presence-absence, whereas finite-antifinite/infinite-antifinite are positive-negative. Apollyon does not seem to understand this difference, or seems to wish to "exploit" it (taking the "cardinality of the set X such that X is all the outputs of violations of the Law of Noncontradiction" to be "absolutely infinite"), but not only this, but in the sphere of transmodality, the Destroyer wishes to represent the impossible, not as that which is lacking in possibility, but which is actually contrary to possibility (antipossible), which is an interesting and terrible question of conceptual existence, here...

In setting forth the new picture of set-theoretic axioms, we would be called to justify our appeal to the Law of the Excluded Middle, and double-negation elimination, so that constructive intuitionism is invalidated. In fact, the construct intuitionist's picture of logical knowledge is deeply flawed: we do not start with an interpretation of

A ^ X = X ^ A [where "^" = "and"]

... and proceed to derive the Law of Noncontradiction from this, and then LEM/DNE if we are "lucky," so to speak. Rather, the LEM is itself the foundation of the system, not in the sense that we deduce the Laws of Identity and Noncontradiction from this, but rather in the sense that our symbolic representation or indication of LEM is prior so that otherwise, we know all three logical laws simultaneously in the same act of cognition. There is, therefore, no actually possible logic that involves identity or noncontradiction but adverts away from LEM/DNE.

But as has been explained, the LEM admits of a proof by erotetic adduction. That is, adduction of the very form of questions themselves gives us the schematic of LEM, which by Q --> A [every form of question implies a possible form of answer] is equivalent to an assertion of the LEM.

Ok so Conway/Guy say that aleph-zero^aleph-zero = c... But now I wonder if the zero symbol in aleph-zero is usable in a functional formula as such then???

So sinceis true, I figured that the Generalized Continuum Hypothesis should read:

"The set of all sets," is a paradox because we interpret it as, "The set that results from the last use of the powerset operation." But there is no last powerset operation. Accordingly, the concept of the set of all sets doesn't violate the powerset operation, but is founded differently, as in class or category or type or what-have-you theory...

The Forge: magic system involving being magically tied to a specific structure called the Forge, which contains artifacts, the Instruments of the Forge, that can be invoked at a distance by people who are Forge-tied. These are usually large vaults of things like hammers, glowing rings, some swords, etc. Depending on how many people channel an object at once, it can start to resonate more and more violently inside its vault in the Forge, until arcane glyphs/runes manifest around it and close off its channel. [The rings are known to glow reliably even when being drawn on by hundreds of thousands of people per ring at a time; and there are millions of these rings in the vaults.] Forgeswords have some kind of fearsome or inauspicious powers, though IDK what yet. I don't want them to be Shardblade-analogs, to be sure

Anyway, the Messiah of Despite was sealed by the Messiah of Daylight ages ago, by allowing the Servant-avatar to perpetrate two of the incarnations of the Final Sins upon his person. It was ages later when the Princess-goddess, Lavaliere Arestroissa, opened the vaults and the Anvilheart of the Forge to the masses of her and all other lands. When the Messiah of Despite arises again, it is to use the flaw in the seal to fulfill the "prophecy of free will": when the Creatrix was destroyed by the Despite-god during the foundations of the world, She imparted a possibility into the flux of history: that at unknown intervals, the forces of the Final Sins would have the power to end creation entire, and it would always be for the people of the world at those times, in those days, to defy evil unto salvation.

Here's a nifty way to explain how the Dedekind knife maps to higher and higher cardinalities under the auspices of the continuum:

Suppose the erotetic powerset of aleph-zero encodes the first set of functions that can approximate the epistemic decimals of the continuum. Now, this will be the smallest approximation, so it will map to more numbers than are in countable infinity, but strictly less (infinitely less) than in the continuum. [An "epistemic decimal" is just the abstraction over the concept of using decimal/n representations to identify the countable and continuous cardinalities as such.] By aleph-aleph-zero, we will have all of these serrations of the knife at once, in actuated infinity as such. However, these are still only the maps into the real number line, not the real number line itself, so there is still an infinite subset of the real number line that is not contained in the map through the knife, which is additionally constitutive of the continuum as such. I don't know what that subset actually is [unless the axiom of modality and its analogies make sense?], to be sure, if the premises even are to be granted, for that matter...

Anyway, you can also say that each powerset of aleph-zero through -n expresses an increasing implexion in the complete continuum. That is, assign to the continuum a countably infinite order of implexion, where different infinite sets within it are ordered together upwards, towards the completed infinity. By aleph-aleph-zero, we have all the parts in at "adjacent" state, and aleph-aleph-1 is their complete unity in the continuum. So again, c = aleph-aleph-1.

By the way, if it matters, the reason why I'm trying to figure out the Continuum Hypothesis is to make a point about the concept Cantor had of an "absolute infinite." ["Consider that Cantor's intuition told him that a relatively 'small,' by contemporary standards, cardinal such as aleph-aleph-aleph-zero, might be ontologically significant enough to make a theological difference to the question of the Trinity."] But why does that matter? Because my concept of romantic ideality depends on a Kantian gloss of the aleph-series [the ascension through the series, or from the mirror of cofinality [more on that later?!]], and of the application of the concepts of absolutely and relatively finite and infinite values. So, to prove a point about romantic love, I want to resolve the Continuum Hypothesis, a problem that is supposedly unresolvable in the strict sense as such, and for which there is no reward (as with, by contrast, the Riemann hypothesis); so all things considered in the end because of Dean.  [What's worse: I already actually solved a "mathematical" puzzle, that of the liar paradox [I swear I really did and can prove it at any hour of any day], for Dean's sake [back in late 2015], but even this argument would only "add" to the "evidence" for the law of noncontradiction, although it would lead into a resolution of the constructivist rejection of logical bivalence and the value of iterated negation, I suppose, too [technically]. So it would only prove that consistency proofs are meaningful, and proofs-by-contradiction allowable, but not many mathematicians doubt these things, regardless of whether they think there is some "ultimate answer" to the liar paradox [which there is, but again, it's only "ultimate" as far as that goes...].

Also I know the semantics for deontic logic impeccably well, but...]

_____________________________________

Kant's theorem, or the theorem of modal cardinality, says that the set of the possible is equal in cardinality to the set of the actual. [See the section of the first Critique on the "ontological argument for the existence of God," and the discussion of a hundred possible or real dollars, and of the necessity of God.] Accordingly, the transit of modal cardinality can only occur once we have the infinite permutations of the infinite sequences of modal operation, i.e. between aleph-aleph-zero and aleph-aleph-1...

OK, so now as I understand it, 3^aleph-zero would = 2^aleph-zero, and so on, up to aleph-zero^aleph-zero. So my method of constructing the first transquadrant of aleph-glyphs using transfinite arithmetic, would not go through, as such. However, putting together all the "known" constraints on the cardinality of the continuum, I've come up with this nifty little "graph":

I came up with what I hope is a super-clever argument for 2^[aleph-zero] = aleph-aleph-1. So, the first thing to do is something like using the aleph-numbers on the first list as Godel numbers for the difference between actual and potential infinity simpliciter. That is, there are finite iterations of the two modality operators for possibility and actuality, as such, and so there is an infinite sequence of further and further iterations for 2^n as such. So, the modal transet ranks of all the aleph-numbers on the first list, are finite, i.e. of finite cardinality (2^n for whichever aleph-n). However, aleph-aleph-zero's modal transet rank should be aleph-zero itself, i.e. countably infinite, as the repeating set of all the sets of repeating such operators,* e.g.

[][][][][][][][][][][][][][][][] ... x

oooooooooooooooo... x

[]o[]o[]o[]o[]o[]o[]o[]o[]o[] ... x

o[]o[]o[]o[]o[]o[]o[]o[]o[]o ... x

But 2^aleph-zero for the modal transets is the successor, then, of aleph-aleph-zero, i.e. is aleph-aleph-1. In other words, it is at aleph-aleph-1 that the cardinality of the modal transet first = c. So c = aleph-aleph-1. [Admittedly, this depends on what might be called "the axiom of modality," which would make the system a "new axiomatization" of set theory in the end. But it's actually worse than that: there's an interpretation of the powerset operation that turns on a representation in erotetic logic. I.e. the powerset of a set of answers is a question that can be computed from a set of answers, but which can't be answered by strict deduction from the set of answers. Now since on my system of things erotetic logic adverts to a sort of "deontic" logic ultimately, or is interpolated with this, or whatever, the distinction between relative and absolute infinity appears here such that the problem of absolute infinity is rendered the problem of the synthesis of the countably infinite number of infinities in Cantor's paradise, with the absolute infinite = to the ideal limit of this series, and unattainable in empirical intuition as such. {But so since Kant says that the deontic value of every individual agent is "without price," and since deontic value is of pure practical reason, as the absolutely infinite synthesis of deontic knowledge, it follows that deontic modality allows us to "access" the absolute infinite.}]

*|||Barcan's formula is: what is possibly actual is actually possible. So for the repeating sets of operators, the cardinality is always the same, even if it should "seem" that actuality contains "more than" the possible. [This is not so, however, as what can be titled with great justice Kant's theorem shows: the concept of a possible x encodes as much internal information as the concept of an actual x.] Their successor, however, is 2^aleph-zero as the infinite permutations of the two operators, so these sum differently in relation to Barcan's formula. I could also bring up the positive and negative imperative operators for imperative-deontic logic ((imperative+erotetic+assertoric)/(deontic-modal) logic) and doubts about Barcan's formula, but I am not going to dwell on those issues, either gladly or hesitantly, right now.

Continuum Hyperthesis

This idea is that the value of c [ = the continuum] is somehow either the second of a series or the place after the second move across a series. So it would be either aleph-one (a), aleph-two (b), aleph-aleph-zero (c), aleph-aleph-one (d), aleph-aleph-two (e), or aleph-aleph-aleph-zero (f), aleph-aleph-aleph-one (g), or aleph-aleph-aleph-two (h). I'm going to assume for the time being that (a) and (c) are ruled out.

Now, if (d) is true, which is my new belief, then 2^[aleph-0] = aleph-aleph-one, which means X^[aleph-n] = [as many alephs as X]-(n+1). If this is so, then all the alephs from aleph-aleph-one through aleph-aleph-n can be given from the first list of alephs. Accordingly, X^[aleph-aleph-n] would map outside of the set of lists, i.e. out of the first transquare. If this is so, and if the first set on the next set of lists is equivalent to [aleph-zero]-aleph-zero [i.e. an infinity of aleph-glyphs subscripted by zero], then 2^[aleph-aleph-zero] maps to the next dimension of lists, 2^[aleph-aleph-aleph-zero] to another dimension still, and so on and on...

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...}