The well known proof of Cantor's theorem (stating that $A<2^A$ for any set $A$) does not make any use of the axiom of choice. I have now spent some time wondering if the analogous result $A^2<3^A$ can be proved by similarly explicit means. It is straightforward enough to produce an injection from $A^2$ into $3^A$ (just send $(a,b)$ to $\chi_{\{a\}}+\chi_{\{a,b\}}$). But trying to mimic the proof to show that there can be no surjection gives me trouble and I am wondering if there even is some explicit construction.
Note that the strict inequality is a theorem of $\sf ZF$ alone, but the only proof I know (an adaptation of Lemma 11.10 in Jech's "The Axiom of Choice") instead shows that there can be no injection going the other way, using (for instance) the Hartogs Aleph of $A$. But apart from not giving that much insight into a more explicit method, this does not even rule out the existence of such a surjection.
So my question is this: Can you give a proof without $\sf AC$ that there is no surjection from $A^2$ onto $3^A$? Alternatively, are there models of $\sf ZF$, where there is such a surjection for some set $A$ (much to my surprise)? Any insights into this are well appreciated
It's consistent relative to ZF that ZF holds and there's a surjection of this kind; the argument of Peng, Shen, Wu linked to in the comments can be slightly modified to give a model in which there is a set $A$ and a surjection $\Phi:[A]^2\to\mathcal{P}(A)\times\mathcal{P}(A)$, and hence a surjection $\pi:A^2\to 3^A$.
For this, following their notation, modify their map $h_n$ so that $$h_n:[\omega_{n+1}\backslash\omega_n]^2\twoheadrightarrow\big(\mathcal{P}(\omega_n)\times\mathcal{P}(\omega)\big)^2.$$ (This part is done using choice and enough GCH, so there is no difficulty here.) In adapting Lemma 3.1, the functions $g$ are of course of the form $$g:\alpha\to\big(\mathcal{P}(\omega_n)\times\mathcal{P}(\omega)\big)^2.$$
A partial automorphism of $\mathcal{A}_n$ is then a map $\pi:D\to\omega_{n+1}$ with domain some $D\subseteq\omega_{n+1}$, satisfying conditions as in the paper, except that in the second bullet point, if $$h_n(\{\xi,\eta\})=\big(\left<B_0,r_0\right>,\left<B_1,r_1\right>\big)$$ where $B_0,B_1\subseteq\omega_n$ and $r_0,r_1\subseteq\omega$, then $$h_n(\{\pi(\xi),\pi(\eta)\})=\big(\left<\pi``B_0,r_0\right>,\left<\pi``B_1,r_1\right>\big).$$
Lemma 3.2 adapts correspondingly, and the definition of automorphism of $\mathcal{A}$ adapts analogously to that for $\mathcal{A}_n$. Lemma 3.3 is just the same. The model $\mathcal{V}$ is constructed from $\mathcal{A}$ as before. Lemma 3.4 goes through. And finally for Lemma 3.5, define the surjection $$\Phi:[A]^2\twoheadrightarrow\mathcal{P}(A)\times\mathcal{P}(A)$$ via the obvious adaptation: set $$\Phi(\{\xi,\eta\})=\left<B_0\cup\bigcup_{k\in r_0}\omega_{n+k+1}\backslash\omega_{n+k},B_1\cup\bigcup_{k\in r_1}\omega_{n+k+1}\backslash\omega_{n+k}\right>$$ where $h(\{\xi,\eta\})=\big(\left<B_0,r_0\right>,\left<B_1,r_1\right>\big)$.