[FOM] powerset {0} = {0,{0}} in intuitionistic set theory?

[Florian Rabe]

Hello,

I've recently done some formalization of set theory. I wasn't sure whether I should use classical or intuitionistic ZF. So I decided to start with intuitionistic ZF and see how far I'd get.

My axioms are essentially those of IZF at http://plato.stanford.edu/entries/set-theory-constructive/axioms-CZF-IZF.html. I introduce symbols for empty set, pair, union, powerset, and {x in A | phi(x)} after proving that those sets exist uniquely.

Then, writing 0 for the empty set, I tried to prove
  (*) powerset {0} = {0,{0}}
and was surprised to find that (*) seems to be equivalent to excluded middle.

Intuitively, I feel that (*) has to be true. Now I wonder whether that makes me a classical mathematician or whether there are intuitionistic set theories in which (*) holds.

I'd appreciate any answer or pointer to the literature.

Thanks,
Florian