Ripheus23

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...]

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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...