The least $\aleph$ that has no surjective map from $m$ to it.

Question

Without $AC$.

Let $\aleph^*(m)$ be the least aleph that $\not\leq^* m$. How to show that $\aleph^*(m)$ exists and $\aleph^*(m)= \{\alpha\in ON\mid\ \alpha\leq^*m\}$.

$ON$ is the class of all ordinal.

$a \leq^* b$ means there is a surjective map from $b$ to $a$.

Answer 1

HINT: If $A$ can be mapped onto $B$, then there is an injection from $B$ into $\mathcal P(A)$. Now apply Hartogs theorem.