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Is it generally true that if $|P(A)|=|P(B)|$ then $|A|=|B|$? Why? Thanks.

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I didn't understand how this helps –  benjamin Mar 27 '11 at 21:43
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A very closely related question on MathOverflow: mathoverflow.net/questions/17152/… –  Jonas Meyer Mar 27 '11 at 21:48
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Don't use $\Rightarrow$ together with the word "if". That makes the entire implication the "if" clause of a statement (for which you have no consequent). To each worded "if" there should be a worded "then". Also, your use of ASCII would easily have been confused with you asking if $|P(A)|=|P(B)|\geq|A|=|B|$ was possible. –  Arturo Magidin Mar 27 '11 at 21:58

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Your question is undecidable in ZFC. If you assume the generalized continuum hypothesis then what you state is true. On the other hand Easton's theorem shows that if you have a function $F$ from the regular cardinals to cardinals such that $F(\kappa)>\kappa$, $\kappa\leq\lambda\Rightarrow F(\kappa)\leq F(\lambda)$ and $cf(F(\kappa))>\kappa$ then it's consistent that $2^\kappa=F(\kappa)$. This of course shows that it's consistent that we can have two cardinals $\kappa<\lambda$ such that $2^\kappa=2^\lambda$.

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Surely you need the generalized continuum hypothesis, not just CH itself? –  Chris Eagle Mar 27 '11 at 21:54
    
@Chris: Of course, thanks for noticing it. :) –  Apostolos Mar 27 '11 at 21:56
    
You mean you have to assume the greater continuum hypothesis. –  Ross Millikan Mar 27 '11 at 22:09
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@Ross: I'm not sure I get your question but you don't need to assume the generalized continuum hypothesis. Easton's Theorem is enough to show that the sentence is independent from the axioms of ZFC. I just used the generalized continuum hypothesis as a more concrete example (and since it's the first thing that came to my mind when I read the question). –  Apostolos Mar 27 '11 at 22:18
    
In your first line you say the OP's question is true if you assume CH, but that only solves it for $\aleph_0$. If you want it generally true, don't you need GCH? Your later statement that without GCH you can't be sure is correct. –  Ross Millikan Mar 27 '11 at 22:25

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