# 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

### micromass

Staff Emeritus
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