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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: en.wikipedia.org/wiki/Axiom_of_power_set –  Shahab Oct 8 '12 at 12:37
    
@ 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 ? –  Souvik Dey Oct 8 '12 at 12:42
    
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. –  Martin Sleziak Oct 8 '12 at 12:42
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@Martin: Souvik means that they cannot both have positive answers without a contradiction. –  Brian M. Scott Oct 8 '12 at 12:44
    
@MartinSleziak Rather than "each", if that makes it clearer. English is a little vague on what the use of "both" here actually means. –  Matt Pressland Oct 8 '12 at 12:45

1 Answer 1

up vote 2 down vote accepted

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 $B$.

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