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

  1. May 30, 2015 #1
    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. jcsd
  3. Jun 4, 2015 #2
    The empty set ##\emptyset## is a well-founded set such that ##\mathcal{P}(\emptyset)=\emptyset##. :wink:

    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.
     
  4. Jun 4, 2015 #3

    micromass

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    Incorrect.
     
  5. Jun 4, 2015 #4
    Sh-t, I swear I misread. Thanks for the intervention
     
  6. Jun 5, 2015 #5
    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...
     
  7. Jun 5, 2015 #6
    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
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