# Two naive questions about sets

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" )

-
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
@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. – Matthew Pressland Oct 8 '12 at 12:45

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 $B$.