Ripheus23
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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).
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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)).
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"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?]
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"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."
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