Ripheus23

So I was reading about the von Neumann universe, which the article (Wikipedia's) said was built up by transfinitely iterating the powerset operation. So unfortunately(ish), I had assumed in my version of set theory that the infinite ascension operation (equivalent to the powerset one at least in the simple minimum limit) would not lead to the "apex" of the hierarchy, at least not directly/necessarily/etc. I know there's no given apex to the vNu since the vNu is not supposed to be "the set of all sets" as such, but anyway, I also remember reading (can't find where right now...) that large cardinal axioms (I think) are conceived as "arbitrarily close to vNu" in the sense that a large cardinal is understood in the light of transfinality (my term, but their procedure) first, from which some descension is possible.

Now, this might not make much of a difference one way or another, although it seems to indicate (to me) that simplifying the axiom system so that large cardinal axioms are reduced to theorems of the axioms of transfinality and transcardinality (as such), is a possible "way to go" (as I have been going, to be sure). So in other words, either I'm going to find out that these topics have already been deeply addressed  or that I'm on to something interesting.