My book says
"... If $GCH$ holds, then the notions of strongly inaccessible and weakly inaccessible cardinals coincide, ..."
In $ZFC$ I can prove this. But the paragraph from which I have excerpted this sentence starts with
"... Apparently we have found a set model of $ZF$. ..."
Which suggests that perhaps we have strongly = weakly inaccessible in $ZF + GCH$.
Can one show in $ZF + GCH$ that weakly inaccessible implies strongly inaccessible?
My definition of choice of cardinals in the absence of choice is $|A| = \{ B : B \approx A \text{ and } B \in V_\beta \}$ where $\beta$ is the smallest ordinal such that there exists a $B$ in $V_\beta$ that is in bijection with $A$.
Many thanks for your help.