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For sets $x, y$, let $x\approx y$ mean that there is a bijection $\varphi:x\to y$. The Weak Power Hypothesis (WPH) states that

if $x,y$ are sets and ${\cal P}(x)\approx {\cal P}(y)$ then $x\approx y$.

It appears to be open whether (WPH) implies the Axiom of Choice or even the Partition Principle.

Question. Is there a model of ${\sf ZFC}$ where (WPH) holds, but not (CH)?

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    $\begingroup$ Assuming you've spent enough time on this website asking questions related to GCH, you've surely come across Easton's theorem. $\endgroup$
    – Asaf Karagila ♦
    Commented Mar 28, 2022 at 9:26
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    $\begingroup$ See Foreman, Woodin, Annals of Mathematics, 133 (1991), 1-35. $\endgroup$
    – François G. Dorais
    Commented Mar 28, 2022 at 9:29

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