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Status Updates posted by Ripheus23

The concept of True Words
This was a concept I liked a lot in fantasy, but couldn't reconcile with the way the concept was executed. My take on it was to suppose a moral codex where each kind of good action corresponded to a letter, so that performing a sequence of good actions meant "spelling out a word" and then forming "sentences" and so on. Anyway, someone with a name in the language of good actions would have that for their True Name, and a True Word would be a word for a thing in this language.

"Since ZFC provides us with the resources to construct ultrapowers, we can construct inner models using mice."

"The souls of the lepidoptera finalitas contain prophecies. Think of them like flying fortune cakes..." "Be gentle! Heed the lepidoptera..."
And they said:
"Hither! Purple whispers creep! The noose is loose! Kitten and a kaboose!"
"There are 9.23 grams of marmalade dried to the base of Mrs. Losensky's blue ceramic bucket..."
"I woke up one morning to find that my nose had detached itself from my face and been transformed into a hideous, lavenderscented aardvark..."
"Apollyon likes bloody omelets..."
"There seem to be a few problems with the argument: it is too long and only quotes The Minstrel of Lettuce..."
"The quotient of a positive shirt and a negative shirt is a set of fourlegged pants..."
"Good grief, it looks like I missed the coronation of my skeleton's last remarks..."
"What a mighty! I! What!"
"My nephews never believed that I used the abacus for firewood to power my mittencrinkling machine..."
"Tsk. As if YOU had ever shaken hands with the Form of Handshakes..."

I realized that there's a difference between proving "x is not a set of anything" and proving "x is a set of nothing," and I needed to prove that the urelement is not a set of no elements of any set, including itself. I think I worked it out* but anyway, another result of the model was that while zero is an empty subset of other sets (a set with no elements of any other sets), so all other sets are empty subsets of zero (sets of no elements of zero). So this underscores why for all natural numbers n besides 0, the powerset is uniquely 2 to the power of n, whereas 0 to the power of 0, 1 to the power of 0, 2 the power of zero... 1 to the power of 1, 1 to the power of 2, 1 to the power of 3... = 0 ^^2 = 1 ^^ 2 = 1. <So zero is the only number before alephzero that has an infinite number of powerset expressions.>
*my idea is that "being a set of no elements of x" means being 0 (in relation to other numbers) and being other numbers (in relation to zero). But the relation is never defined on the urelement. More, then, we can just define the urelement such that it it is not a set of no elements of other sets, nor itself. In conjunction with the urelement's not being a set of anything either, it follows that it is not a set of anything or nothing, which rules out the urelement being a set.

minotaur elements
minotaur ordinals
dragon ordinals
unicorn ordinals
Orpheus surreals
Osirus ordinals
Amaterasu cardinals [again]
Ragnarok cardinals
Michael cardinals
Raphael ordinals
Gabriel surreals
Asmodeus surreals
warlock ordinals
sorcerer ordinals
wizard cardinals
sortilege ordinals
demonic cardinals
leviathan transet
"kenotic transet" <said of the urelement> and "pleromatic set"
happy cardinals
sad ordinals
priestly cardinals
hieratic ordinals
sword cardinals
sword ordinals
nuclear cardinals
nuclear ordinals
nuclear surreals
presbyter cardinals
deacon cardinals
"reformed cardinals" <so large only divine revelation can give knowledge of them>
Ezekiel cardinals
Oberon numbers
Aquinas cardinals
Nicean cardinals
irminsul ordinals
Odysseus ordinals
Aeneas surreals
Milwaukee cardinals ("a kind of joke cardinal")
cheddar ordinal
parmesan cardinal
cola ordinal
hemlock ordinal
Icarus transet
ambrosiac cardinal
cowardly cardinal
cowardly ordinal
helpless surreal
potato surreal
guitar cardinal
piano ordinal [haha]
Deseret cardinal


Anselm cardinals. A transet in V proposed by the "cult of the universal constructor." Its elements are supposed to be "cardinals so large that only divine power could have brought them into being" or give them a reference. Note that this concept therefore requires that Anselm numbers be transfinally ordered, making the first Anselm number into the meridian of a nexus <where the Anselm section of V is the "greatest" section>.
cult of the universal constructor, the. Believers in a transcreationist model of the Godelian universal constructor. Their leader is Cardinal Mahlo [finally!], who is actually secretly working with the Septarch of Commandment (also known as Deonomy). According to the cult, it is not provable whether empirical reality was created by a divine nature, but it is provable whether mathematical reality was transcreated by a divine nature. <The argument goes: there is a possible mathematical world that was transcreated; therefore there is a possible transcreatrix; if something is a transcreatrix, it is this necessarily; therefore the transcreatrix, if it exists for any possible mathematical world, exists for all of them.>
Godelian universal constructor. An entity whose existence is supported by Godel's ontological argument. Similar to the necessary agent.
necessary agent, the. Posit of naive deontic logic: there is an obligation that exists purely from logical grounds (an obligation to "uphold" the law of noncontradiction), wherefore there is always (necessarily) an agent able to discharge the obligation. In ecograph theory, this posit is actually a principle for the manifestation of different kinds of "necessary" agents.
Deonomy. The Septarch empowered by divinecommand theory. He was also the first ecoarch to manipulate the Keyscape in order to the Septatheon. (Note that ecoarchs are already a peculiar kind of "necessary agent.")

<Why are Anselm cardinals not defined as "none greater than which can be conceived"? First, in context, there is no individual such cardinal. Second, this definition is implicit in the notion of a transfinal ordering, which attaches to the base definition by virtue of saying "divine nature" there, since a divine nature is one that transfinally orders things <or so the theory goes>. Of course, those who accepted the existence of Amida cardinals thought otherwise: these were "so large that only a mathematician gifted with divine power could access them." But these mathematicians with this gift need not be divine "inherently," only called that by reference to the power at issues (which power could well be impersonal, as in the socalled Tian model).>



*Regarding zero: Z = {Z}, Z ~= element of ~Z, for all X, if X is not Z then X ~= an element of Z. "Zero is an element of itself, is not an element of any other set, and has no other elements besides itself." By contrast, "The urelement is not an element of itself, is an element of only one set, and is the only element of that set" = 1.


Haha, I finally managed to come up with a thoroughly settheoretic foundation for my theory. The main thing is not to start with the empty set as zero. Rather, we define zero as a Quine atom (a = {a}) and 1 as the equivalent of the empty set. Here zero is the empty subset. I checked the powerset listing under this definition and it works (I couldn't get the classical definition to work, I will add).


"Another dimension of their ingenuity was in using complex functions to single out individual cardinals on the first level of infinity, such that the orders of complexity among those functions could be used to mirror V altogether. The 'slogan' was: the first level of alephnumbers is sufficient for the intuition of 'Cantor's paradise,' which is a thematic image of V (the generic metafinitary description "infinity of infinities" is first satisfied over the first level of the alephs). By this means, the accessibility of some of these complex functions (their relative simplicity, all things considered, nevertheless) could be semiotically interposed with the 'power' of the higher infinities mapped to by this method, so that the Keyscape's mediation of different levels of power as such could be accomplished more easily."

"In other words, an analogy would be formed between the notational permutations and augmentations occasioned by the concepts of the larger and larger types of cardinals overall, and the schematics for computing various finite natural numbers. The rate of a function could be correlated with the generic height of a level, then, and the endless cascade of these finite functions would 'add up to' a semiotic intuition of the totality of the generic intuition of V, i.e. the 'true word for' V. Then the alephs with finite indices would be the signatures of eternity, Godel coordinates of a particular simplicity and advergent stature no less..."


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The three other examples show how some very simple examples of infinite computations in alephspace interconnect via the hyperoperator sequence. Namely, the +++ sequence leads into the *** sequence before the exponential sequence, and each can be represented by the next highest operator using alephomega and alephzero: so {alephomega * alephzero}, {alephomega ^ alephzero}, and {alephomega ^^ alephzero}.

The anchor conjecture: that 2 ^a 2 = 4, for all a whatsoever, even ordinals of any transfinite cardinality. This equation is sometimes referred to as the anchor of transfinality. During the apocalypse of the Final Power, it was instead discovered that using the heart operator on {2, 2} goes not to 4, but to 4aleph.*
*Not to be confused with aleph4. 4aleph is the fourth apex number, those that end at the apex of transfinality.

Reverse forcing
In infinite set theory, forcing is a method whereby the axioms are used to arbitrarily construct a model of V such that conclusions in V can be negated in V+. Thus it can be shown that from the axioms of ZFC set theory, the basic continuum hypothesis is arbitrarily decidable over V/V+. Reverse forcing is the construction of a more limited model of V, one that limits the possible answers to questions posed in V. For example, if the axiom of replacement is not assumed, then V is limited to א_{ω}. But since the powerset question can still be asked (given that the powerset axiom can still be used), it follows that the powerset question can be answered under V, which means under א_{ω}. Accordingly, reverse forcing puts a limit on the continuum question. Combined with the formulaic requirement (that if the continuum is some specific alephnumber, its being this number is an instance of a formula), this gives us the prerequisite of the proof in the system that the continuum must be the second alephnumber.
Note that all the hyperoperator questions can be posed under omegaomega, here, but are all advergent from K ^4 K through K ^(omegaomega) K (for K > or = to alephzero as such).

Outline of the argument for the Continuum being aleph1
Note: it is unnecessary to automatically assume both the basic and the generalized Continuum Hypotheses. For example, one might believe that 2^aleph0 = aleph1, but that 2^aleph1 = aleph4, and so on.
Now, the axiom system I'm using replaces the powerset axiom with the axiom of transcension, which roughly says that there is an infinite sequence of operators on the alephnumbers. So although cardinal tetration is obviously indicated in the natural progression of the arithmetic under consideration, we can now go ahead and use the wellordering theorem to fix the idea that a basic Continuum Hypothesis corresponds to a generalized one. I.e. the wellordering theorem, with the hyperoperator sequence in transfinite space, implies that we have to order the values of the operations in general, which implies that the value of an operation using an operator of index n has to be "retrofitted" (profitted?) to a counterpart operator of index n + 1. Accordingly, that 2^aleph0 = aleph0^^2 implies that there is a formula of increase for both operations, and these two formulae have to be expressed so as to coincide.
Without the axiom of replacement, alephomega cannot be proven in the standard model (although to be sure, the model called "standard" always has the AOR). On the other hand, the powerset question can be independently posed under alephomega, then. The erotetic powerset concept therefore applies such that the powerset question is fixed under alephomega. This situation can be illustrated using the interpolation of the basic infinite alephtower with the bethsequence (as I have done elsewhere).
From here, the proof in the system that C = aleph1 proceeds easily enough. But we can then go on to explain the failure of the GCH at alephomega, on the ground that the rewrite of the powerset function in terms of 2^n actually runs out of legitimacy in alephspace. This is because 2^aleph0 = 3^aleph0 = ... = n^aleph0 = aleph0^aleph0, which means the powerset of aleph0 is not completely reducible to 2^aleph0, whereas the powersets of finite numbers can always be reduced to 2^n, indeed they must be (2^3 = P(3), but 3^3 does not = P(3), and so on).

After some more reflection, I have discarded by "epsilonalpha hypothesis" about the interpolation of the finitelyindexed hyperoperator sequence and the aleph numbers. Although this increases the implicit semiotic scale of the universe of sets (on my model), it has the drawback of eliminating a particularly nice example of advergence and also makes my representation of a generalized glyphic index harder to handle (versus the classical/mainstream fixedpoint notation, in this context). Of course, this is really just the fact that the reality is harder to handle, here.
So anyway I have no more solution to the first triangle operation or the first hypertower, to say nothing of my notion of a xatrix cardinal!

Theory: there is a weak set theory where alephomega is a model of V, and this theory was incorporated into the Keyscape to make the finitistic elective interval transpowered by sectionsigma.

Relevant to "Armirex's equation," the object of a trope in the legend of Ripheus. The equation was a wellformed order of operations in alephspace whereby Armirex channeled the power of the Form of Evil (among other things) to silence Apollyon at the end of the Last War. Later, Armirex was able to draw on the power of the Septatheon inasmuch as these mirrored the Form of Evil as false conceptions of deontic reality, making use of the equation again. Since no one ever performed the operations except Armirex himself, no one knows how short or long the proof of the equation is, though some speculate that it could be distilled into a relatively simple (finitistic, even) form.

I've had occasion to obsess over infinitesimals, of late, and I came up with a perhaps naive hypothesis about how to characterize them "intuitively." Let us say that every real number has alephzero many digits. This means that we will at least count to a ωth digit if we read out their decimal expansions. Now allow, then, that there is some number for which the ωth digit is 0, but if there is a next digit, this ω+1th is 1. This number is therefore infinitely small but not equal to zero. Voila, an infinitesimal! So think of something like 0.000 ... (ω 0s) ... 01. This is variously the smallest or largest such infinitesimal.
But this model allows us to go on fracturing the continuum. Suppose, then, that after the ω+ωth digit, we go back to an infinite countable sequence of 0s, succeeded by another infinitesimal sequence, and so on. We will have a whole realm of infinitely broken numbers. But we will also have realms of permutations of these options, and so on.
The intuitively maximal case is to go on to imagine a number whose decimal expansion goes at least to ω1, the first uncountable number. That is, it has as many nonzero digits as there are numbers in the Continuum (we're presuming the Continuum Hypothesis). Such numbers, if they are complete (nowhere broken) are each in themselves indecomposable continua, up to the "syrupy" case of the Brouwerian model. In fact, the whole occurrent model appears to allow us to exactly formulate whether and how a continuum is decomposable or not (think back to those fractured sequences), or at least to frame comparable questions relative to a category either akin to or under that of decomposable and indecomposable continua (c.f. the notion of density in this context).

And depending on how far we've counted, we can go on to characterize "small infinitesimals," "very small" ones, ineffable ones, omega cases, etc. much like we have lists of large cardinals with various descriptive names. And so on and on, ultimately to the antifinitesimal numbers. These can then be applied to the description of the Keyscape in the allegory of Ripheus (as playing a role in the mathematics of the Keyscape relative to the hold on Apollyon's potential access to the Final Power, via a relationship between the idea of a "rift in Cantor's staircase" and the complete brokenness or emptiness of an "anticontinuum" or pure void (or whatever as such)).

"Only a few among the Host of Ripheus proved to be versed with antifinitesimals, or even metafinitesimals for that matter. Vyrian Armirex was one, to be sure, but there were others. Another group who later showcased some great aptitude with the system of those numbers were the Dark Metroarchs, who studied this set of transfinitesimals as part of their research into a method to destroy the Shield. The Precentor of Despite, Haller D'Mares, is said to have almost gone insane while performing some functions in this sphere (an attempt to construct an object like the Typhon from separate lesser masses of sincrystal), and most Fallen Artificers are known to have inadequately dealt with them in their application to the transequent order of amendment. Indeed, the relationship between these numbers and the ultimate void of zero by itself is itself an eternal abyss, immensely unfathomable, 'hungry and so wishing to be allconsuming'... 'Only the Form of Evil's own special numbers are as dangerous in principle to reflect on in particular...'" [Can you imagine what happened in Armirex's mind when he drew on the Form of Evil's power to help silence Apollyon at the end of the Last War?]

"The demonic numbers were defined in a starkly different way. They were virtual interpolations with specific other numbers given from the axiom system per se nota, such as sectionsigma or the void index and the Apollyon index. They only concerned these special numbers, including the whole last staircase. But in fact this amounts here to a reduction to the finite numbers used in the general combinatorics of the final offenses, or rather they 'run out' of evaluation past those numbers. Thus there is no question of infinitesimal forms, or even relatively infinite ones otherwise at all (as such). There is no such infinite sequence of evil in itself as such. Rather, if evil is taken to the power of infinity, it cancels itself out: its intrinsically negative essence negates itself, transforming into constructive reality. So the Form of Evil had no motive to form a standard cardinal game (in the metafinite order), nor a subset thereof in that way."


In the standard model of set theory, an inaccessible cardinal is loosely defined as one that cannot be reached from smaller cardinals "by the usual operations of transfinite arithmetic." Aleph1 can be reached from alephzero by the successor operation and is therefore not inaccessible. Alephomega can be expressed as the sum of all the cardinals smaller than itself, so it is not inaccessible. Depending on how one evaluates the powerset operation's outputs, many other small alephnumbers are not inaccessible.
In the hyperoperational model of transfinite arithmetic, however, a much more general and perhaps exact definition of inaccessibility can be supplied. Here, we say that a cardinal K is inaccessible if it cannot be reached from smaller cardinals via hyperoperations indexed by ordinals smaller than the initial ordinal of K. For example, suppose k is < K. Have the initial ordinal of K be O. Then K is inaccessible if no operation k ↑a x goes to K for a < O. So cardinals are inaccessible not in an absolute sense as such, but only relative to different levels of the hyperoperator sequence in transfinite space.

Metafinity as the foundation of mathematics
Let the finite and the infinite be relative or absolute. By absolute is meant a universal relation: x is absolutely y if in all relevant relations, x is y. For example, if x is to the left of all y, x would be absolutely leftwards (though this is not really possible, let us suppose). Accordingly, we have an order of four metafinite predicates: absolutely finite, relatively finite, relatively infinite, and absolutely infinite.
Next, allow that there are two fundamental numerical questions: “Which one?” and, “How many?” A number is either an answer to some form of these questions, or eventually derives from such answers. Let us refer to numbers as answers to the first question by the term ordinals, the second cardinals. Numbers as answers, in some way, to both questions will be denoted surdinals (with reference, as will be explained far below, to the concept of surreal numbers). Numbers as answers to neither question, used purely to differentiate between some x and y, will be indexicals. Thence, to use the number 1 as a name for someone, and the number 3 as a name for someone else, is to use these numbers indexically; they are not subject to arithmetic as such, which is to be considered an ordering of numbers in themselves. So while we might add 1 and 3 when these are used as ordinal, cardinal, or surdinal numbers, we would never add them when using them indexically. —The four positive metafinite predicates are then generally correlated with four basic uses of numbers.
Our axiom system is a set of axioms and their schemas used in mathematical proofs. The most commonly used such system is ZermeloFraenkel set theory with the axiom of choice, which has up to nine basic extralogical principles, some of which can perhaps be reduced to others or waived altogether. Now the attempt to provide a foundation for mathematics, using the concept of metafinity, involves the question (roughly enough put), “Assuming that there is some set of axioms and axiom schemas, why is the number of basic principles what it is?” So if ZFC set theory is true, the question is, “Why are there (as many as) nine such principles?” And according to the theory of metafinity, there would not be nine, but only four, generally correlated with the pairings of metafinite predicates and the numerical erotetic dyad. Indeed, this theory is telling us that we know how many axioms and axiom schemas to look for, without directly knowing as such what the system is.
Let a logically irreducible number be a number occurring in the logical principles of the system. For example, if 0 and 1 are mapped to FALSE and TRUE,[1] then it is unnecessary to derive either number from the other using extralogical principles. Rather than by transconstructing 1 as the simplest successor of 0, for example, we have it immediately, here. Moreover, if this is true, then the classical settheoretic definition of the ordinal 1 is false: being the simplest successor of 0 characterizes 1, but does not define it. 1 is not actually reducible to the set containing the empty set, or any such thing.
If there are four axioms or schemas, here, what are they to be expressed as? Axiomatic structures are assertoric: the basic case is subjects and predicates, which in mathematics comes to numbers and operations thereon. At least one axiom will give us some numbers to work with, and at least one axiom will give us operators for them. Now in the metafinite context, we immediately know that we have numbers for all the metafinite predicates, so the classical axiom of infinity is encoded into the context. This gives us our basic case of a relatively infinite number, here as an ordinal (and surdinal) and as a cardinal. Arithmetical intuition, and settheoretic transconstruction, vitiate the idea of a wellordered sequence of operators, so that our second axiom will be one that gives us the hyperoperation sequence as such.[2] This sequence will be represented under the heading of the axiom of transcension.
The existence of relevantly basic cases of the other metafinite numbers will generally be referred to under the heading of the axiom of transcardinality. This has it that there are relations among the metafinite predicates such that, starting from the transfinite cardinals, we can go on to situate the absolutely and relatively finite numbers fairly exactly. Until actual examples of what this situation means are provided, of course, this remark is not even quite programmatic but might appear rather mystical (at best).
Lastly, the axiom of transfinality collates all the relations of finality that appear in the mathematics of the numbers and their sets referred to here, and adds to them, in light of the metafinite context in total: therefore, in quintessential relation to the absolute infinite. Expressing the concept of an absolutely infinite number in a consistent way is the challenge to be met in this case, as the standard vectors of approach give rise to simple contradictions. By way of a very preliminary remark, the axiom of transfinality involves an extreme reimagining of the principle of foundation in set theory. In other words, though a naively infinite and descending sequence of sets is not to be presented, some infinite descending numerical sequence will be. As far as alreadyestablished mathematical systems go, a clear example of an analogy with what we will be looking for is the surreal sequence . But a transfinal sequence is meant to be advergent under absolute infinity, which is an absolutely strong relation to enter into, and so we will have cause to analyze phrases that appear in the literature such as “a set that might as well be as large as the universe V of sets” as well as the equation V^{V} = V (and its cognates, e.g. “”).
Think of the axioms, then, as introduction rules for types of glyphs. The axiom of infinity introduces the aleph glyphset in general; the axiom of transcension introduces the ascending arrow glyphset; the axiom of transcardinality introduces glyphs for juxtaposing the relatively infinite with the absolutely and relatively finite numbers; and the axiom of transfinality gives us notation for the idea of advergence under the absolute infinite. Logical conjunctions of two or more axioms or schema then yield further analogous or derivative glyphsets. The infinitary context also allows us to apply the axiomatic propositions in an infinitary way. There are therefore 23 such adjunctions of the axioms and their schemes to consider, which gives us much to work with, as will be seen.
[1] For technical reasons, the logical system in use here will actually map FALSE to 1. Though in assertoric space, for a sentence to lack truth is for it to be contrary to the true, in simpler predicative space this is not so (that is, absence and opposition are different kinds of difference). Accordingly, though, 1 is also logically irreducible in the metafinite theory of mathematical foundations; it is not to be irrecursively defined, though it may be expressed, as the additive inverse of 1.
[2] To try to consolidate and simplify matters as much as possible, I will be using a variant of Knuth arrow notation for this sequence throughout the text. As will be seen, if the aleph numbers can be assigned a minimum of two indices—a base and a compounding index—then it will be an elegant symmetry for the same assignment to hold of the transcension operator. That this symmetry obtains in fact will be illustrated by the overall system of notation as it is developed

https://math.stackexchange.com/questions/2313576/ordinaltetrationtheissueofepsilon0omega has more on the subject.