The question of existence of sets $x,y$ such that


$$|x|<|y| \wedge |P(x)|=|P(y)|$$


is known to be independent of $\text{ZFC}$!


But are there known examples of sets fulfilling the above condition
that necessitates violation of choice?



Are there known examples of sets whose power set is equal in size to power set of larger sets only in absence of choice?
If there is an infinite Dedekind-finite set, then there are two infinite Dedekind-finite sets which have equipotent power sets, and one is larger than the other.


This is due to the fact that the existence of an infinite Dedekind-finite set implies that there are two sets $X$ and $Y$ such that $|X|<|Y|$ and $|Y|\leq^\*|X|$, namely there is a surjection from $X$ onto $Y$. This immediately implies that there are injections between the two power sets, and Cantor–Bernstein implies now there is a bijection.


And of course, assuming the axiom of choice Dedekind-finite sets are always finite, so the above contradicts the axiom of choice.


