Weak Power Hypothesis with injections instead of bijections

Asked Dec 17 '22 at 12:56

Active Dec 21 '22 at 16:56

Viewed 198 times

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

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.

As Joel David Hawkins notes in the comment section, (WPH) and (WPH$_\leq$) are equivalent in ${\sf (ZFC)}$.
There are models of ${\sf (ZFC)}$ in which (WPH) does not hold - but interestingly, it appears to be open whether (WPH) implies (AC).

edited Dec 21 '22 at 16:56

asked Dec 17 '22 at 12:56

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Who's to say... – Asaf Karagila Dec 17 '22 at 13:15
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You might mention that in ZFC the two principles are equivalent. – Joel David Hamkins Dec 17 '22 at 14:49
Thanks @JoelDavidHamkins, will do. – Dominic van der Zypen Dec 17 '22 at 14:53
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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. – Asaf Karagila Dec 17 '22 at 16:21
Thanks @AsafKaragila . I found the DOI number, but not the DOI link - but the projecteuclid.org link which I hope, is also acceptable? – Dominic van der Zypen Dec 17 '22 at 17:48
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The DOI link to DOI XXX is https://doi.org/XXX . – Emil Jeřábek Dec 17 '22 at 18:10
Thank you @EmilJeřábek, I modified the link in my question accordingly. – Dominic van der Zypen Dec 17 '22 at 18:43
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Have you seen the remarks on https://mathoverflow.net/questions/81204/does-2x-2y-rightarrow-x-y-imply-the-axiom-of-choice? – Asaf Karagila Dec 21 '22 at 17:47

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