GCH for alephs means the statement that, for any aleph $\kappa$, there are no cardinals $\mathfrak{r}$ such that $\kappa<\mathfrak{r}<2^\kappa$.


Does GCH for alephs imply the axiom of choice?


Remark. Lindenbaum and Tarski assert in ``Communication sur les recherches de la théorie des ensembles'' without proof that GCH for alephs is equivalent to Cantor's aleph hypothesis that $2^{\aleph\_\alpha}=\aleph\_{\alpha+1}$ for all ordinals $\alpha$ (see page 188, No. 96). But as we know (although Lindenbaum and Tarski possibly do not kown) Cantor's aleph hypothesis implies AC. This means that Lindenbaum and Tarski in fact also assert that GCH for alephs implies the axiom of choice. But I do not see how to prove it.



Does GCH for alephs imply the axiom of choice?
The answer is positive, yes.


Note that $2^\kappa\leq 2^{\kappa^+}$, and therefore $\kappa^+\leq\kappa^++2^\kappa\leq 2^{\kappa^+}$. So either $2^\kappa=2^{\kappa^+}$ or $2^\kappa=\kappa^+$.


In the first case $\kappa^+$, $\kappa<\kappa^+<2^\kappa$ is impossible. So the latter case holds.


Therefore the power set of an ordinal can be well-ordered, and the Axiom of Choice follows in $\sf ZF$.


