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Can every set have a power set ?

Does there exist a set A such that there always is a surjection of A onto B , where B is any arbitrary set?

(note that positive answers to both the questions lead to a contradiction by "Cantor's theorem" )

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    See Axiom of Power set: http://en.wikipedia.org/wiki/Axiom_of_power_set2012-10-08
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    @ Shahab: So , can I say that there does not exist a set A such that there always is a surjection of A onto B , where B is any arbitary set ?2012-10-08
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    You wrote: *note that positive answers to both the questions lead to a contradiction by "Cantor's theorem"*. I don't think the positive answer to the first question leads to a contradiction.2012-10-08
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    @Martin: Souvik means that they cannot **both** have positive answers without a contradiction.2012-10-08
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    @MartinSleziak Rather than "each", if that makes it clearer. English is a little vague on what the use of "both" here actually means.2012-10-08
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    @Souvik: Yes. I think Brian M. Scott has already stated as much.2012-10-08
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    @ Martin Sleziak: What I meant to say is that positive answers to both the questions 'together' lead to a contradiction , as Brian Scott already has said , still sorry for the lack of clarification.2012-10-08

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One of the axioms of ZF set theory is that every set has a power set. There is no set $A$ such that for each set $B$ there is a surjection of $A$ onto $B$.