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Let $x,y$ be sets. We use the following notation:

  • $x\simeq y$ means that there is a bijection $\varphi:x\to y$, and
  • $x\leq y$ means that there is an injection $\iota:x\to y$.

The Weak Power Hypothesis says

(WPH) Whenever ${\cal P}(x)\simeq {\cal P}(y)$ then $x\simeq y$.

Consider the statement

(WPH$_\leq$) Whenever ${\cal P}(x) \leq {\cal P}(y)$ then $x\leq y$.

Using the theorem of Cantor-Bernstein-Schroeder we can show in ${\sf (ZF)}$ that (WPH$_\leq$) implies (WPH).

Question. In ${\sf (ZF)}$, does (WPH) imply (WPH$_\leq$)?

Notes.

  1. As Joel David Hawkins notes in the comment section, (WPH) and (WPH$_\leq$) are equivalent in ${\sf (ZFC)}$.

  2. There are models of ${\sf (ZFC)}$ in which (WPH) does not hold - but interestingly, it appears to be open whether (WPH) implies (AC).

Improve this question
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    $\begingroup$ Who's to say... $\endgroup$
    – Asaf Karagila ♦
    Dec 17, 2022 at 13:15
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    $\begingroup$ You might mention that in ZFC the two principles are equivalent. $\endgroup$
    – Joel David Hamkins
    Dec 17, 2022 at 14:49
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    $\begingroup$ As a general advice, never link to your Google search. It may not work the way you expect with other people. Either install one of the browser extensions that cleans out Google search result links, or open the link, then copy-paste. Even more so in the case of an academic paper that has a DOI, it is good practice to just use the DOI link. $\endgroup$
    – Asaf Karagila ♦
    Dec 17, 2022 at 16:21
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    $\begingroup$ The DOI link to DOI XXX is doi.org/XXX . $\endgroup$
    – Emil Jeřábek
    Dec 17, 2022 at 18:10
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    $\begingroup$ Have you seen the remarks on mathoverflow.net/questions/81204/…? $\endgroup$
    – Asaf Karagila ♦
    Dec 21, 2022 at 17:47

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