Ripheus23

"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."

The universe strikes again: http://logika.ff.cuni.cz/radek/papers/misclogVII.pdf

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:

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 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).

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).

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!

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.

Examples from the lower main sequence.

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.

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).

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.

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 Zermelo-Fraenkel 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 set-theoretic 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 set-theoretic transconstruction, vitiate the idea of a well-ordered 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 already-established 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.