Ripheus23

I came up with what I hope is a super-clever argument for 2^[aleph-zero] = aleph-aleph-1. So, the first thing to do is something like using the aleph-numbers on the first list as Godel numbers for the difference between actual and potential infinity simpliciter. That is, there are finite iterations of the two modality operators for possibility and actuality, as such, and so there is an infinite sequence of further and further iterations for 2^n as such. So, the modal transet ranks of all the aleph-numbers on the first list, are finite, i.e. of finite cardinality (2^n for whichever aleph-n). However, aleph-aleph-zero's modal transet rank should be aleph-zero itself, i.e. countably infinite, as the repeating set of all the sets of repeating such operators,* e.g.

[][][][][][][][][][][][][][][][] ... x

oooooooooooooooo... x

[]o[]o[]o[]o[]o[]o[]o[]o[]o[] ... x

o[]o[]o[]o[]o[]o[]o[]o[]o[]o ... x

But 2^aleph-zero for the modal transets is the successor, then, of aleph-aleph-zero, i.e. is aleph-aleph-1. In other words, it is at aleph-aleph-1 that the cardinality of the modal transet first = c. So c = aleph-aleph-1. [Admittedly, this depends on what might be called "the axiom of modality," which would make the system a "new axiomatization" of set theory in the end. But it's actually worse than that: there's an interpretation of the powerset operation that turns on a representation in erotetic logic. I.e. the powerset of a set of answers is a question that can be computed from a set of answers, but which can't be answered by strict deduction from the set of answers. Now since on my system of things erotetic logic adverts to a sort of "deontic" logic ultimately, or is interpolated with this, or whatever, the distinction between relative and absolute infinity appears here such that the problem of absolute infinity is rendered the problem of the synthesis of the countably infinite number of infinities in Cantor's paradise, with the absolute infinite = to the ideal limit of this series, and unattainable in empirical intuition as such. {But so since Kant says that the deontic value of every individual agent is "without price," and since deontic value is of pure practical reason, as the absolutely infinite synthesis of deontic knowledge, it follows that deontic modality allows us to "access" the absolute infinite.}]

*|||Barcan's formula is: what is possibly actual is actually possible. So for the repeating sets of operators, the cardinality is always the same, even if it should "seem" that actuality contains "more than" the possible. [This is not so, however, as what can be titled with great justice Kant's theorem shows: the concept of a possible x encodes as much internal information as the concept of an actual x.] Their successor, however, is 2^aleph-zero as the infinite permutations of the two operators, so these sum differently in relation to Barcan's formula. I could also bring up the positive and negative imperative operators for imperative-deontic logic ((imperative+erotetic+assertoric)/(deontic-modal) logic) and doubts about Barcan's formula, but I am not going to dwell on those issues, either gladly or hesitantly, right now.