Ripheus23

Edit: I mean c^aleph-zero is c, which contradicts my functionality notion...

... unless I add in a specific variable for the dimensionality of the alephs. I found a way to intuitively preserve functionality (one-to-one mapping via the continuum formula) without violating the appearance of c^aleph-zero = c. Namely, when we raise an aleph number given in a higher dimension (of the glyph-lattice of the Ideal Symbol) to one listed as lower, or to the same one otherwise lower (because it was given first with the lower dimension), we have what otherwise might or might not be "visibly" different glyph-sets. So, the uniqueness of the function of aleph-zero^aleph-zero is masqued by inattention to the dimensional variable (in the system of the IS).

In fact, visually, it is here almost self-evident that c = aleph-aleph-1, because if you take the glyph for a point on the first list, and then use a whole line as a glyph on the second list, the conversion of the point of aleph-zero to the line of the continuum literally diagonalizes to aleph-aleph-1.

The mirror of cofinality

So one thing I have tried to do now, as such, is set the axioms of the theory as simply as can be. So rather than the 8 or so in ZFC, I have, basically, just three.

The axiom of transfinity

The axiom of transcension

The axiom of transfinality

The axiom of transfinity is equivalent to the ZFC axiom of infinity, in giving us aleph-zero. It does not necessarily give this as the implicit sum of the infinite iteration of the {zero}-operation, though. There is an erotetic form to the powerset principle at work here, which we will get to shortly.

Now, the axiom of transcension just says that, aside from the introduction rule for the glyphs in general, there must be at least, and possibly a countably infinite number of ways, to ascend the series. It is equivalent to saying that there is a formula by which the Continuum Hypothesis can be determinately represented re: possible solutions. The concept of accessible vs. inaccessible cardinalities is mapped to this possible series of erotetic functions, namely there is some erotetic set Q1 such that the method of transcension proceeds using cardinal arithmetic in Q1 whereas all cardinals mapped relative to Q2 are strictly (stipulatively?) greater than those mapped relative to Q1 and inaccessible from any point in Q1 as such, and so on through Q3 and ... and Qn.

Or, more generally, the axiom of transfinality allows us to start from an arbitrary higher given aleph or k-number and (try to) proceed downwards. So it involves, for example, the concept of cofinality. However, in my system, the absolute infinite is "computable" (not a technical use of the term), so there is a quasi-sense in talk of starting from the absolute finality of the absolute infinite, down towards the absolute cofinal transfinity (aleph-zero). I actually came up with the start of a description of Q1+ transfinity via deontic geometry, among other things, but the basis for the idea is that we could use the concept of an erotetic powerset inside of Q1 such that we define a k-number to be k-index.n were n is the number of steps in the simplest series of cardinal arithmetic between aleph-zero and k. For example, proceeding via the continuum formula is a one-step simplicity, namely taking aleph-zero to itself as a power. So k.1 is c. What of k.1119? IDK what it would be in particular  but anyway, we can then define the nexus of transfinality such that it is k.aleph-zero (in Q1).

There's more to it, a lot of which I've technically gone over before. Like, I made a way to define more or less all those weird symbols I used for "equations" earlier this year. But more on all that later...

[Last but not least: "There is a cardinal number X such that X - n is the interval of quantum renormality, i.e. the cardinality of the set of quantum renormalization operations is [hyper]continuous."]

*EDIT: The simplicity theorem is a deduction from the axiom of transfinality that focuses on descent towards alephic simplicity (aleph-zero) vs. descent from the nexus of transfinality.