God should have his doubts, indeed.
AFAIK, if you do not assume the power set axiom, it is consistent (relative to consistency of ZF) that there are no cardinals bigger then ALEPH_0.
About your weak power set axiom: I don't think it is equivalent to the regular one. My guess is that it is also relatively consistent that the integers have a power set, but the reals don't.

For many interesting axiom systems, one cannot proving consistency, because that would imply consistency of ZFC (which cannot be proved, assuming it's true). Therefore one can only prove relative consistency, i.e. that the system is consistent assuming ZFC is consistent.