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The anchor of transfinality

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So I had an intuition some time ago, that there was a fundamental reason of arithmetic, why the number 4 kept showing up in the axiom system for my model of set theory. Either I discovered or rediscovered the role of 2 and 4 as such in the hyperoperator sequence, so anyway for the sake of the Ripheus story, I decided to have part of the "programming" of the Keyscape turn on a proof regarding this point. Namely, since the programmers wanted the arithmetic to be highly accessible to the general public (of an entire multiverse!), they looked for simple but impressive examples of arithmetic transfinality, and voila, here was the perfect case.

Now, another part of the equation: the model of set theory, here, doesn't have the axiom of foundation as an axiom, but as a theorem: and not a theorem of all of V simpliciter, but of a certain meta-level of V. So there is an entire other meta-level of V that exists, in a way, in reverse order, descending from a largest set to an asymptotic limit of infinitesimally smaller ones. In other words, varying the axiom of foundation allows us to mathematically transcreate the image of a largest infinity as such, the caveat being instead that there is no smallest one, on this meta-level, and indeed therefore this meta-level never determinately intersects the ascending order of V as such.

But so on the "other end of the universe," the apex numbers are absolute mirrors of the first finite numbers, after a fashion. This is why one of them allows us to "break the symmetry of eternity" by reprogramming the ultimate arithmetic of metafinity. Rendering the apex numbers accessible was therefore of preeminent importance to the Keyscape project (was it not?). Or: they are even at the nexus of the Final Power, the meaning and order of its intended advent as well as the transcendental magnitude of their self-apocalypse.





"Some objected that this made a game of V. The reply was to concede the point entirely, but to say that it was the very Form of Games itself with which we had to deal. In other words, there was a logic of accomplishment built into the image of V; the axioms were basic moves in a game, then, yes, but ultimately there was nothing else to do but accept it: and moreover, anyway, would it not prove true that to get the highest score in the Game was the same as to get the highest score on the Test?"


Edited by Ripheus23

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