# Can a hyperset be its own powerset? And what is its stream?

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1. May 30, 2015

### Stoney Pete

This is probably a silly question but I am curious: Is it possible in hyperset theory to have a set S such that S = pow S? And wouldn't its stream then be the entire hierarchy V of pure sets?

Last edited: May 30, 2015
2. Jun 4, 2015

### suremarc

The empty set $\emptyset$ is a well-founded set such that $\mathcal{P}(\emptyset)=\emptyset$.

All shenanigans aside, could you elaborate on or point to a reference on the stream of a set? I can't find any articles that mention streams in this particular context.
In any case, $S=\mathcal{P}(S)$ trivially implies that there is a bijection between $S$ and its power set. This contradicts Cantor's theorem which, to my understanding, is independent of the axiom of foundation and thus still applies in hyperset theory.

3. Jun 4, 2015

Incorrect.

4. Jun 4, 2015

### suremarc

Sh-t, I swear I misread. Thanks for the intervention

5. Jun 5, 2015

### Stoney Pete

Here you find some information on the stream of a hyperset (= non-well-founded set): http://plato.stanford.edu/entries/nonwellfounded-set-theory/#1

Of course, when we write S = pow S then obviously we have a version of Cantor's paradox... So I guess this idea can only be formalized in a paraconsistent approach to set theory where contradiction is not explosive. Paraconsistent mathematics have been around for some time now: http://plato.stanford.edu/entries/mathematics-inconsistent/

Apparently such paraconsistent mathematics can circumvent Gödel's incompleteness results. So with a paraconsistent approach, mathematics can be complete.

I guess that if we want a set S such that S = pow S in order to generate the entire hierarchy of sets (and thereby the whole of mathematics, which can be seen as conainted within that hierarchy), we need something like a paraconsistent hyperset theory...

6. Jun 5, 2015

### suremarc

Paraconsistent logic does indeed seem useful, but discarding $\neg (P\land\neg P)$ would render many, many theorems of current mathematics unproven. If one does not wish to do so, then in extensions of ZFC it is possible to speak of a class such as $V$ whose power class (i.e. a class $A$ such that $\forall x. (x\subseteq A\Rightarrow x\in A)$) is itself. I do not know of any related theorems in paraconsistent logic--I will have to do some more reading. Hope this helps

Last edited: Jun 5, 2015