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Ripheus23

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  1. While trying to resolve the Continuum Hypothesis, I accidentally solved an entirely different, and arguably much more important, set of problems, namely the question of justifying the axioms of set theory: the ZFC axioms firstly, and arbitrarily many axioms of higher infinity besides. This is how I did so. (Btw, I happen to wonder how far Brandon Sanderson will take his set-theoretic Easter eggs. There happens to be a form of set theory interwoven with graph theory, which seems relevant to the notion of Spiritwebs. So will these eggs manifest most strongly in stories/substories involving the Cognitive Realm, or is there, after all, a mathematical side to the Spiritual Realm too? And if so, will it be a form of set theory that constitutes this Spiritual aspect? Supposing it is a form of set theory, will Sanderson (obliquely or not) bring up, say, the intricate doctrines of large cardinals? (Will he bring these up even if his set-theoretic Easter eggs reach their apex in the Cognitive Realm only?))

    Preamble: the choice between set, type, and category theory as a foundation of mathematics

    I am choosing set theory as my foundation of mathematics. It is said that category theory and type theory go together very well, in the end, even such as to say that categories are effectively reducible to types. However, in light of the historical fact that set theory won out over type theory, but has not won out over category theory, I am going to assume the following: a term refers to a set if the referent has elements; it refers to a type if the referent has tokens; and it refers to a category if the referent has elements and tokens. That being said, typology adverts more to the logical sphere, whereas elementhood is more distinctly mathematical. So a category is mathematical inasmuch as it has elements. Nothing seems to have actually been gained, then, in providing a foundation of mathematics in category theory instead of set theory.

    The fundamental understanding of set theory's internal justification

    In the pure theory of knowledge, there is a problem, the problem of the regress of reasoning, with four "mathematical" solutions: either the regress ends in self-justified axioms (foundationalism), the regress forms loops (coherentism), the regress is infinite (infinitism), or the problem is unsolved (skepticism, which corresponds to J0 in justification logic). Coincidentally, the elementhood relation can be sequenced in all four of these same ways, viz. there are well-founded sets, looping sets, infinite descending elementhood chains, and then the empty set-theoretic object, that which has no elements. My fundamental claim will, then, be that well-founded, looping, and descending sets are all justifiable modulo the positive solutions to the regress of reasoning. By implication, then, although descending sets are justifiable somehow, it is not permissible to axiomatize this justification. Justification by inference from axioms is per se nota well-founded justification, so that only the well-founded sets are justifiable in terms of the axiomatic method as such. And although I have a model of a justified descending set, my focus for the remainder of this discourse will be the axiomatic hierarchy. This is because it is modulo that hierarchy, that solutions to various other problems of set-theoretic justification with which I am familiar, have appeared.

    Justification values

    Frege proposed that truth is not a predicate of an assertion, but is the reference of that assertion (if it is otherwise factually correct). This is the notion of truth values. Likewise, in my theory of set theory(!), there are justification values. Truth-theoretically, the values are made to coincide with 0 and 1 on the numerical side of things, with fuzzy logic usually also having every other real number between 0 and 1 as a possible "degree of truth." There is no such bracketing required for the doctrine of justification values, and this allows us to formulate the initial axiom of infinity in a novel way, one that wears its justification on its sleeve. This is to have that axiom be, "The assertion that the initial level of infinity exists, has a justification value equivalent to that level." More concisely, have j(S) be the justification function, which takes sentential inputs S and outputs the degree of justification S has. So say: S(j(S) = ω), with the very in question being ω, so that j(ω) = ω.

    This happens to turn the entire question of justifying any axiom of infinity on its head. If every higher infinity makes possible a higher infinite degree of justification, it follows that the stronger and stronger axioms of infinity are all the more justified than the lower ones, down to the axiom of ω. Not that the initial principle is therefore unjustified: it too is infinitely adequate to the question of its own existence, of course, here.

    Specific justifications of large-cardinal axioms

    The above might not be good enough to "explain" the justification of specific large-cardinal axioms, however. Granting that this is so, I would say that we can intrinsically justify, in a Gödelian way, at least some of these axioms, not by analysis of the iterative concept of sets, but by analyzing the concept of justification itself. In other words, replace ZFC's standard background logic with a justification logic. Then you open the door (as far as I know) to at least the following axioms:

    The model-theoretic characterization: every set theory of a certain form has an initial worldly cardinal assigned to it. ZFC with justification logic is such a theory. So there is a justification-theoretic worldly cardinal (and it is justifiable to assert that this cardinal exists).

    The proof-theoretic characterization: every set theory of a certain form has a proof-theoretic ordinal assigned to it. Sometimes, to "identify" this ordinal, one has to imagine a much taller, but still countable, ordinal, that figures in what is called a "collapsing function," this function being the one through which the "identification" of the proof-theoretic number is given. Those much taller countable ordinals can be "shadows" of genuine large cardinals. ZFC with justification logic is a theory such that those shadows and their counterpart large cardinals figure in its proof-theoretic analysis. So there is an (otherwise uncharacterized) justification-theoretic large cardinal.

    The infinitary-logic characterization: some standard large cardinal axioms can be formulated in terms of infinitary logic. ZFC can be assigned an infinitary justification logic for its background. So there are large-cardinal characterizations available modulo this assignment. These inherit the intrinsic justification of the logic (again), such that it is sufficiently justifiable to assert that these (they are called "weakly compact" and "strongly compact") cardinals exist. Bonus points: when you introduce strongly compact cardinals, for example, you get some other types of large cardinals below the initial strongly compact one, and you get a sizable amount of those types, too. (You don't get these with the worldly cardinals, and although it is "probable" that the proof-theoretic mirror cardinalities are much greater than the smallest model-theoretic ones, I could tell you nothing about the interim between the mirrors and the worldlies, whereas I could at least attest to measurable and inaccessible cardinals in light of the strongly compact ones.)

    From what I can tell, you can do a lot more with this justificatory template. I've "rambled" long enough for now, though, so I'll leave it to the interested reader (if there are any) to ask me about that "lot more," or to go seeking for it themselves.

  2. I had to change my physics idea significantly due to considerations regarding infinitary logic. The idea is that the laws of physics are infinite conjunctions under an ℒ(κ,λ)-structure that shifts over time, with major cosmological processes constituted by those shifts. For example, at t = 0, let the infinitary logic of a given universe be ℒ(0,0). Over the interval t(0 to 1), the structure shifts to ℒ(ω,ω), which corresponds to the initial expansion, the Big Bang. Further major shifts in expansion dynamics result from further increases in ℒ(κ,λ), so that the accelerated expansion, for instance, is a consequence of these dynamics.

    The model has two grounds: an empirical observation and a major prediction. The first involves the idea that perception of a standard continuum results from existing in a dimensionality that succeeds the cardinality of that continuum, much like "complete" perception of a two-dimensional structure results from existence in three dimensions. Assuming that the cardinality of a continuum is aleph-1, then we assume that ℒ(κ,λ) here has aleph-2 for κ (which is the variable for time's dimensionality in the system), such that we perceive time as continuous (of cardinality aleph-1).

    The prediction the theory makes is that at some point in the future, there will be another major shift. The equation I assigned to the shifts picks out aleph-4 as the value for κ resulting from the next shift. Consequently, empirical consciousness should change accordingly, then. So if we survive to that day, we could receive predictive evidence for the model from specifics of cognitive changes at that time.

  3. Well one prediction came true, one book earlier than I expected, another seems to have been falsified, and my theory about Adonalsium being like a city got some minor evidence behind it, haven't read DAWNSHARD to double-check but yeah, overall, couldn't have asked for a better RHYTHM OF WAR, except maybe he could've used commas when using the word "though," more often.

  4. 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.

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

  6. "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, lavender-scented 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 four-legged 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 mitten-crinkling machine..."

    "Tsk. As if YOU had ever shaken hands with the Form of Handshakes..."

  7. 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 ur-element 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 aleph-zero 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 ur-element. More, then, we can just define the ur-element such that it it is not a set of no elements of other sets, nor itself. In conjunction with the ur-element's not being a set of anything either, it follows that it is not a set of anything or nothing, which rules out the ur-element being a set.

    1. Ripheus23

      Ripheus23

      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 ur-element> 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

  8. 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 non-contradiction), 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 divine-command 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.")

    1. Ripheus23

      Ripheus23

      <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 so-called Tian model).>

  9. Next step/stage.

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    1. Ripheus23

      Ripheus23

      "X is a set" becomes "X has at least one element Y."

    2. Ripheus23

      Ripheus23

      *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 ur-element is not an element of itself, is an element of only one set, and is the only element of that set" = 1.

  10. First-run technical account of the ur-set and the ur-element concept.

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    1. Ripheus23

      Ripheus23

      Or rather "there is a set X for all elements Y such that if Y is not the ur-element, Y is not an element of X."

  11. Even more rambling :P

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  12. Haha, I finally managed to come up with a thoroughly set-theoretic 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).

  13. "By this means, they could make the mental act of resolving the semiotic image into a conduit for the corresponding aleph's power, that is to say the act would be the 'shadow' of evaluations involving the figures themselves thus attained, and so the one act would cause another..."

    aleph.png

  14. "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 aleph-numbers 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."

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    1. Ripheus23

      Ripheus23

      "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..."

    2. Ripheus23

      Ripheus23

      "... and of course..."

      4.png

  15. Rather than focus only on unfathomably faster means of ascending V, I did some more ground-level work on various countable patterns in V, of an elegant nature after all, however. Here are some examples:

     

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    1. Show previous comments  1 more
    2. Ripheus23

      Ripheus23

      The three other examples show how some very simple examples of infinite computations in aleph-space 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 aleph-omega and aleph-zero: so {aleph-omega * aleph-zero}, {aleph-omega ^ aleph-zero}, and {aleph-omega ^^ aleph-zero}.

    3. Ripheus23

      Ripheus23

      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 4-aleph.*

      *Not to be confused with aleph-4. 4-aleph is the fourth apex number, those that end at the apex of transfinality.

    4. Ripheus23

      Ripheus23

      *Further refinements have led me to suspect that aleph-4 tetrated by aleph-4 might in fact be aleph-omega(4)*5.

  16. 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 אω. 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 אω. Accordingly, reverse forcing puts a limit on the continuum question. Combined with the formulaic requirement (that if the continuum is some specific aleph-number, 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 aleph-number.

    Note that all the hyperoperator questions can be posed under omega-omega, here, but are all advergent from K ^4 K through K ^(omega-omega) K (for K > or = to aleph-zero as such).

  17. Outline of the argument for the Continuum being aleph-1

    Note: it is unnecessary to automatically assume both the basic and the generalized Continuum Hypotheses. For example, one might believe that 2^aleph-0 = aleph-1, but that 2^aleph-1 = aleph-4, 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 aleph-numbers. So although cardinal tetration is obviously indicated in the natural progression of the arithmetic under consideration, we can now go ahead and use the well-ordering theorem to fix the idea that a basic Continuum Hypothesis corresponds to a generalized one. I.e. the well-ordering 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" (pro-fitted?) to a counterpart operator of index n + 1. Accordingly, that 2^aleph-0 = aleph-0^^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, aleph-omega 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 aleph-omega, then. The erotetic powerset concept therefore applies such that the powerset question is fixed under aleph-omega. This situation can be illustrated using the interpolation of the basic infinite aleph-tower with the beth-sequence (as I have done elsewhere).

    From here, the proof in the system that C = aleph-1 proceeds easily enough. But we can then go on to explain the failure of the GCH at aleph-omega, on the ground that the rewrite of the powerset function in terms of 2^n actually runs out of legitimacy in aleph-space. This is because 2^aleph-0 = 3^aleph-0 = ... = n^aleph-0 = aleph-0^aleph-0, which means the powerset of aleph-0 is not completely reducible to 2^aleph-0, 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).

  18. After some more reflection, I have discarded by "epsilon-alpha hypothesis" about the interpolation of the finitely-indexed 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 fixed-point 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!

  19. Theory: there is a weak set theory where aleph-omega is a model of V, and this theory was incorporated into the Keyscape to make the finitistic elective interval transpowered by section-sigma.

  20. Examples from the lower main sequence.

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  21. Relevant to "Armirex's equation," the object of a trope in the legend of Ripheus. The equation was a well-formed order of operations in aleph-space 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.

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  22. 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 aleph-zero 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 non-zero 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).

    1. Ripheus23

      Ripheus23

      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)).

    2. Ripheus23

      Ripheus23

      "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 all-consuming'... '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?]

    3. Ripheus23

      Ripheus23

      "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 section-sigma 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."

  23. 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." Aleph-1 can be reached from aleph-zero by the successor operation and is therefore not inaccessible. Aleph-omega 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 aleph-numbers 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.

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