If you adopt the Powerset axiom and the axiom of Infinity, then two infinite sets N and C=PN exist. Let's say we adopt Choice or the Continuum hypothesis, then there is an ordinal of size C, i.e. an uncountable ordinal.
Now adopt transfinite induction for ordinals up to C, then you can apply the powerset operation to C exactly |C| times.
C, PC, PPC, PPPC, ..., P^(|N|)C, P^(|N|+1)C, P^(|N|+2)C, ... P^(|C|)C, ..., P^(|C|+69)C.
All those sets are different, by Cantors result that |PX| > X for any set.
Now of course, I've used a shitton of axioms there. A constructive set theory will only prove that some large but countable ordinal is well-ordered. Say
https://en.wikipedia.org/wiki/Bachmann%E2%80%93Howard_ordinal 
