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Stoney Pete
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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?
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suremarc said:The empty set ##\emptyset## is a well-founded set such that ##\mathcal{P}(\emptyset)=\emptyset##.
Sh-t, I swear I misread. Thanks for the interventionmicromass said:Incorrect.
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 helpsStoney Pete said: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...
A hyperset is a mathematical concept that refers to a set that contains other sets as its elements. It is a generalization of the concept of a set, which can only contain individual elements.
No, a hyperset cannot be its own powerset. A powerset is a set that contains all possible subsets of a given set. Since a hyperset contains sets as its elements, it cannot also contain all possible subsets of itself.
As mentioned before, a hyperset cannot be its own powerset because it already contains sets as its elements. If it were to also contain all possible subsets of itself, it would lead to a paradoxical situation.
The main difference between a hyperset and a powerset is that a hyperset contains sets as its elements, while a powerset contains all possible subsets of a given set.
The stream of a hyperset refers to the sequence of elements within the hyperset. Since a hyperset can contain sets as its elements, the stream can be infinitely long and complex, depending on the number of sets and their elements within the hyperset.