Hmm, "SET THEORY, is dead", I like it, Adam.
It's fair of you to say "here's a theory where I've axiomatized away my model of
a trust in set theory, I must make myself another way to trust", the set theory,
then you constructively bring up what you want, while then pointing out a
crankish argument in set theory, that shows something you've hobbled yourself, from.
Powers of 2?
What you get is ordering, numbering, and counting, and when numbering invoves counting.
Set theory models these sufficiently all their "regular" way, "well-founded", for example,
the regular set theory.
You can make inconsistent models of set theory and show how they're inconsistent.
It's not considered constructivist, say, insofar as formal rigor and "can't not trust it".
So, you want to square away your Aleph numbers, cardinals, and the Omega-many ordinals.
The Aleph, is the counting infinity, while the Omega, is moreso the numbering infinity
and the ordering infinity, in ordinals.
The counting infinities the Aleph numbers, their arithmetic builds the orders of the spaces,
above each constructive, regular, ordinary, ..., theory of words like sets, here elementary
objects.
That's one reason why cardinals and ordinals are different, different infinities.
Anyways usually insofar as any mistake you write here someone will point it out to you.
Anyways what results I enjoyed this for some time, currently looking at my own slates,
I sort of organize analysis in continuum mechanics.
"Infinitely-many", ....
So, what you want to do, I think to really get an understanding of the cardinal and ordinal
numbers, and, the cardinal and ordinal infinities, is give yourself axioms for example "inverse",
but for example "counting" or whatever other results "infinity" axioms, then figuring out
where their sameness and differences, do or strongly do or don't or strongly don't, hold,
what do.
"It's a continuum mechanics, ...", just saying, Adam, that if you're looking for a theory that
really digs up set theory, I made one with both cardinals and ordinals and their infinities
what otherwise sometimes aren't "extra" enough to be real.
I've even gone so far as to stand up letting a simplest mathematical infinity, back into
the philosophy, of the theory, the one that science is missing.
So, when I suggest, "Powers of 2?", I suggest that you're thoroughly familiar with them,
all the powers of 2, then for the two infinities you call "w", omega, and "2^w", 2 to the omega,
which as a number, is an ordinal, but also results when writing ordinals regular-ly, is the
space "2^w", each of the infinite sequences of zeros and ones, with a beginning.
It's set theory, ....