The Anomalies in the Ripheus story

[Ripheus23]

My outline for the Ripheus story has the first intended book focused on a weird adventure in a demiplane of evil. This introduces the figure of Apollyon, who is one of the two candidates for "the" Form of Evil itself (the other is already known as the Form of Evil as such, but there is in-world debate about whether this is a misnomer). However, the second book has the adventure of Ripheus proceed from the question of things called the Anomalies. These are random magical phenomena, such as laughing while holding a red crayon resulting in the local color of the sky changing, or some such thing. Some Anomalies are stable enough to endure and even to exist as "living beings" (i.e. characters in the story). However, they are potentially chaotic enough that if enough of them occur, the integrity of physical reality might be undone. And Ripheus has no idea why they are happening.

Eventually, he is supposed to learn that a machine called the Keyscape is producing them. This is a machine that uses transfinite arithmetic to bridge the general source of magic with the average mortal (heretofore only inherently magical beings or objects could use magic, or some such thing). He and his friends long ago were even the engineers of the Keyscape.

There is, though, a form of logic used in the deep parameters of the machine's arithmetic, that is lacking in a sufficient proof-by-programming, and this "error" is the keystone of the Anomalies.

Now, which part of the arithmetic corresponds to the logical gap? My point of departure is dividing by zero. Usually we note that this "can't be done" and move on. But an explanation of why reveals that we actually have two different kinds of cases, here. First, 0/0 actually can be evaluated to 0, or 1, or 2, or 3, or whatever, as by multiplicative inversion these go back to 0 in the right way. By contrast, 1/0 (or 2/0 or so on), just can't be evaluated. (I know, there is an elaborate higher algebraic context where such evaluations can be carried through, but the programming of the Keyscape, here, depends on the lowest context.)

So, if multiplication is repeated addition, exponentiation is repeated multiplication, and so on, then by general inversion division is repeated subtraction, roots/logarithms are repeated division, and so on and on. And we find the same divergence between operations that can equal any number, and those that don't equal any. For example, the logarithm of 0 with base 1 does not go to anything, not even 0, as 1^n never goes to 0 for any number. Let's call the one kind of deviant outputs natural variables and the other antinumbers.

So let's go back to Apollyon and the Form of Evil. At the very least Apollyon is the Form of Destruction. Destruction has a relationship with nothingness, as that which causes nothingness. So Apollyon has a relationship with zero, here. Assuming that the Anomalies arithmetically coincide with natural variables and antinumbers (in the formalism of the Keyscape), let us say that Apollyon is tethered to the antinumbers in such a way that encoding an insufficient analysis of the question whether Apollyon is the true Form of Evil or not, constitutes part of the gap in the logic of the Keyscape: because, then, the interplay of the antinumbers and the natural variables interfaces with the magic system and spits out random outputs (the Anomalies).

There is an ever deeper/greater reason for this: consider set difference or complementarity. This is the pure set-theoretic form of "subtraction." Because of this, it follows that V - V = 0 (where V is the universe of all sets). But 0 - V = V (there isn't a "-V" in itself, although some speak of Apollyon as this). If division is repeated subtraction,* this gives us the interesting fact that 0/V either goes to 0 (if V*0 = 0), or since 0 - V - V = V - V = 0, then 0/V goes to any 2n (but no odd numbers!). So there is a fundamental split in the computation of the Keyscape, one naively algebraic (on the model of multiplicative inversion) and the other hyperoperational, and the natural variables of V, and the antinumber V/0, are examples of the arithmetical source of the Anomalies. (Metaphysically, then, 1/0 is not the same antinumber as V/0, and so on.)

*For example, 4/2 = 2 means that 4 - 2 - 2 = 0. Division is subtraction tending towards zero in particular, then.

EDIT: The interkey of Apollyon and the Keyscape proceeds rather subtly, though. Firstly, we start with the notion of an axiomatic set theory in general. In-universe, the "correct" set theory has a set of axioms known as the axioms of creation. These correspond (roughly) to absolute and relative finitude and infinity. But there is another concept, of the antifinite, in which finitude and infinity are not opposites, but finitude and antifinitude are. This concept corresponds to a "false" axiom of destruction. The axioms and their primordial adjunctions are thought of as introduction rules for different types of mathematical glyphs. The empty case of an axiom system, the number that comes from not using the axioms, so to speak, is always the basic zero of the system. In the "true" set, this is the true number zero, the one we normally think about when we use the word "zero." But the empty case of the axiom of destruction then corresponds to a different zero, the void index. The finitary (antifinitary) case of the destruction axiom introduces the direct index of Apollyon. These two antifinite "numbers" are Ω₀ and Ω_-1. The infernal table lists the corresponding simplified arithmetic, where the void index is set to true zero and the index of Apollyon is the regular -1 (and ↓ is the predecessor relation, the zeroth negative hyperoperation):

0 ↓ 0 = 0; 0 - 0 = 0

0 ↓ -1 = 1; 0 - (-1) = 1

-1 ↓ 0 = -1; -1 - 0 = 0

-1 ↓ -1 = 0; -1 - (-1) = 0

The destruction theorem says that we can then introduce a third glyphset for the values of 1 in the table, Ω_1. This value is restricted in that its infernal counterpart of 1 is only evaluated relative to the next negative hyperoperation (division). This gives us a secondary table:

0/0; 0/-1; 0/1

-1/0; -1/-1; -1/1

1/0; 1/-1; 1/1

So the table gives us the basic natural variable 0/0, and two antinumbers -1/0 and 1/0. Now there is also the empyrean table, which is a similar list of operations involving 0, 1, and V. The restriction of the relevant operations on {0, 1, V} to those up to and including exponentiation, rather than the full complement of the hyperoperators, depends on the fact that tetration and its successors are not "elementary recursive functions." (Now actually, there might be other such hyperoperations, such as the ωth, but I've had to alter my theory of transfinite arithmetic surrounding even pentation and hexation, to say nothing then of "omegation"; so I won't try to decide, here, whether omegation is elementarily recursive or not.) Due to the doctrine of set complements, there is an interpolation of the negative hyperoperator sequence with the {0, 1, V} table, including logarithms, so we end up with the transcendental ratios of V (0/V, 1/V, V/V, and V/0) and quite a few other natural variables and antinumbers. For overly mystical reasons I won't dwell on right now, there are sequences involving these that can be fit to cycles of Apollyon's natural variable and antinumbers. But the ratio of V to 0 and the logarithm of 0 with base V are the "ultimate" simple antinumbers. So the level of destructive power Apollyon ultimately has, is sufficient to endanger the entire physical universe (multiverse), so long as Apollyon can key into the ultimate antinumbers (which it can).

EDIT 2:

Now one question that came to my mind long ago was, if we can simple define a number i into "existence" such that squaring i = -1, why can't we just "define into existence" a set of numbers that when multiplied by zero, go back to some usual number? Now I'm sure there's a set-theoretic derivation of the imaginary unit, but when it was introduced it was by a more purely algebraic method: assume i "exists" and then, if you can come up with an arithmetic for it that is consistent with normal arithmetic, as well as internally, then voila, you've justified i by a sort of "structuralist" argument. Of course, this consistency methodology holds for the question of the axioms to be used in set theory, and V has enough structural reality to itself that you might wonder whether why there would supposedly be a symptomatic drift away from set theory, among structuralists. But IRL this is arguably only a modern (post Benaceraff) tendency, maybe.

Anyway, for worldbuilding's sake (if nothing else), the question of different philosophies of mathematics ends up in the universe of Ripheus as the fact that those different philosophies have consequences for the magic of the Keyscape. The standard level of interface is: assume an axiomatic set theory as such, then vary the theory so that you can use "smaller" cardinals to do the same "magical" work as a larger one would normally have to. So suppose that some inherently magical being is mystically isoquent with, I don't know, a measurable cardinal. These already provide models of ZFC's form of V as they are not provable from within ZFC. Then vary the theory so: alter or even remove the axiom of replacement. Then the theory can't prove aleph-omega, and aleph-omega is a model of V, here. The Keyscape "makes" the theory true (in a magical sense), which in this case allows someone without magic, to become mystically isoquent with aleph-omega, which is then typed to the measurable cardinal in play, and voila, the previously immagical personage now can wield the equivalent of the "power of a measurable cardinal."

But what if you aren't a "Platonic" set theorist? What if you're an intuitionist without sets, or without the paradise of sets (you might grant one or two infinite ones in some sense), or a formalist, or a strict finitist, or a logicist even? Let us clarify what this actually means: each school of mathematical philosophy holds to a different standard of evidence and proof in mathematical reasoning. For the sake of their egalitarianism, the Keyscape's engineers had an ethical motive, if you will, to lean heavily on finitistic proofs in designing the Keyscape, just because having too many infinitely complex proofs would end up making the system unworkable except by those few who would have the independent capacity to go through those proofs. Nevertheless, strict finitism would "annihilate" V, so it was not to be adopted at all, as such, in the construction of the Keyscape.

However, consider, then, a structuralist about i who accepts the algebraic existence of i as sufficient in itself to completely justify or prove our use of the concept. This would be a structuralist who could affect the magic of the Keyscape such that arithmetic could be assigned to that magic on some sort of directly algebraic ground or analogy. In fact, this is exactly part of how Vyrian Armirex silenced Apollyon at the end of the Last War: he took the fact that squaring i goes to -1 to transcreate a new glyphset of Apollyon, namely Ω_i and Ω-_i. This even though i isn't a relatively simple number from the set-theoretic standpoint. Worse, geometrically, the equivalent units of the quaternion set actually form an analogical sphere, so that there are infinitely many such hyperimaginary units. In principle, then, Armirex had to balance the essence of the introduced glyphset with the algebraic progression thereof (part of the transcendental seal on Apollyon's power being that it by itself had only a finite glyphset, and no such glyphset had an infinite index).

The relationship between intuitionism/constructivism and geometrical arguments was also an easy source of higher-level magical interpolation. Not that the rejection of the Law of the Excluded Middle was correct: by erotetic adduction, in fact, this could be ostensively demonstrated false to think. Nevertheless, the intuitive element, here, could be refined, and this is already even a natural principle of economic space generally. ("Witness the power of the isoplexes," as they say.) Formalism might interface too much with Anomalies, I suspect, with the empty randomness of the symbols involved either drawing on various Anomalies to actually have substantive magical effects, or contributing to the growth of Anomalies, or whatever along that line. The logicist paradigm would presumably have an interesting effect on the system... (Hearkening back to the generic list of factions: the Metroarchs and the Artificers might be foils for each other (usually) on this level, for example: say, by the Metroarchs making significant usage of logicist magic, to the chagrin of many an Artificer: due to the irony of the situation, after all (not that the Artificers despise logic!)).

Edited May 10, 2020 by Ripheus23