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-$\iota$) 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)-$\iota$ implies (WPH).

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

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

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).

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-$\iota$) 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)-$\iota$ implies (WPH).

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

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

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-$\iota$) 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)-$\iota$ implies (WPH).

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

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

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-$\iota$) 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)-$\iota$ implies (WPH).

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

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

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-$\iota$) 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)-$\iota$ implies (WPH).

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

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