I am considering the construction in [Peng—Shen—Wu] in which the authors show the consistency of a set $X$ such that there is a surjection from $X^2$ onto the power set of $X$ (henceforth $\mathscr{P}(X)$).

First, some terminology. Given sets $X,Y$, I shall write $|X|\leq|Y|$ to mean that there is an injection $X\to Y$, and $|X|\leq^*|Y|$ to mean that $X$ is empty *or* there is a surjection $Y\to X$.

Given a set $X$, the *Hartogs number* of $X$, denoted $\aleph(X)$, is the least ordinal $\alpha$ such that $|\alpha|\not\leq|X|$. Similarly, the *Lindenbaum number* of $X$, denoted $\aleph^*(X)$, is the least ordinal $\alpha$ such that $|\alpha|\not\leq^*|X|$. These ordinals always exist and are cardinals.

The construction in [Peng—Shen—Wu] uses permutation models, but the translation to $\mathsf{ZF}$ from $\mathsf{ZFA}$ is clear. I noticed that the set constructed with this process satisfies $\aleph(X)=\aleph^*(X)=\aleph_\omega$. My question is thus: Is having limit Hartogs/Lindenbaum number necessary?

Really, this is (at least) four questions: Is it consistent with $\mathsf{ZF}$ to have a set $X$ such that $|\mathscr{P}(X)|\leq^*|X^2|$ and...

- $\aleph(X)$ and $\aleph^*(X)$ are successor cardinals?
- $\aleph(X)=\aleph^*(X)$ is a successor cardinal?
- $\aleph(X)$ is a successor cardinal?
- $\aleph^*(X)$ is a successor cardinal?

**References**

*Peng, Yinhe; Shen, Guozhen; Wu, Liuzhen*, **A surjection from square onto power** (ArXiv).