Ripheus23

Outline of the argument for the Continuum being aleph-1

Note: it is unnecessary to automatically assume both the basic and the generalized Continuum Hypotheses. For example, one might believe that 2^aleph-0 = aleph-1, but that 2^aleph-1 = aleph-4, and so on.

Now, the axiom system I'm using replaces the powerset axiom with the axiom of transcension, which roughly says that there is an infinite sequence of operators on the aleph-numbers. So although cardinal tetration is obviously indicated in the natural progression of the arithmetic under consideration, we can now go ahead and use the well-ordering theorem to fix the idea that a basic Continuum Hypothesis corresponds to a generalized one. I.e. the well-ordering theorem, with the hyperoperator sequence in transfinite space, implies that we have to order the values of the operations in general, which implies that the value of an operation using an operator of index n has to be "retrofitted" (pro-fitted?) to a counterpart operator of index n + 1. Accordingly, that 2^aleph-0 = aleph-0^^2 implies that there is a formula of increase for both operations, and these two formulae have to be expressed so as to coincide.

Without the axiom of replacement, aleph-omega cannot be proven in the standard model (although to be sure, the model called "standard" always has the AOR). On the other hand, the powerset question can be independently posed under aleph-omega, then. The erotetic powerset concept therefore applies such that the powerset question is fixed under aleph-omega. This situation can be illustrated using the interpolation of the basic infinite aleph-tower with the beth-sequence (as I have done elsewhere).

From here, the proof in the system that C = aleph-1 proceeds easily enough. But we can then go on to explain the failure of the GCH at aleph-omega, on the ground that the rewrite of the powerset function in terms of 2^n actually runs out of legitimacy in aleph-space. This is because 2^aleph-0 = 3^aleph-0 = ... = n^aleph-0 = aleph-0^aleph-0, which means the powerset of aleph-0 is not completely reducible to 2^aleph-0, whereas the powersets of finite numbers can always be reduced to 2^n, indeed they must be (2^3 = P(3), but 3^3 does not = P(3), and so on).