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

Thomas Forster T.Forster at dpmms.cam.ac.uk
Mon Oct 19 22:48:14 EDT 2009

```This is known to be equivalent to excluded middle.  However the proof
relies on the fact that in ZF separation holds for *all* formulae. If you
work in a fragment of ZF which has separation on for formulae in some
class Gamma, then it implies excluded middle on;y for formulae in Gamma.
And there is nothing special about the (singleton on the) empty set here,
either.  Any singleton will do.

On Mon, 19 Oct 2009, Florian Rabe wrote:

> 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
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