This is from Kunen's book Set Theory, 2011 edition, Exercise II.4.24.
Assume that $\gamma>\omega$ and that Replacement Axiom holds in $R(\gamma)$. Prove that $\gamma=\beth_{\gamma}$ and that $\{\delta<\gamma\mid \delta=\beth_\delta\}$ has size $\gamma$.
* In this book, $R(\alpha)$ denotes the von Neumann Hierarchy, whose more standard notation is $V_\alpha$.
I believe that I can show that $\gamma=\beth_\gamma$: Axiom of Replacement in a universe with at least 2 elements implies Axiom of Pair, so $\gamma$ must be a limit ordinal. Therefore, $\gamma$ is the class of all ordinals in $R(\gamma)$, and so is $\beth_\gamma$, so they are equal.
Unfortunately, I have no idea on how to tell the size of $\{\delta<\gamma\mid \delta=\beth_\delta\}$. I think the limit of $\beth_0, \beth_{\beth_0}, \beth_{\beth_{\beth_0}}, \ldots$ should be an example of $\delta=\beth_\delta$, and one may start with other ordinals to obtain other ordinals with this property. However, I don't know how to count them.
Best Answer
I don't quite follow your argument, to be honest. Perhaps a better argument is to note that the function $\alpha\mapsto\beth_\alpha$ is a definable function. If $R(\gamma)$ satisfies Replacement, then for any $\alpha<\gamma$, $\beth_\alpha$ exists inside $R(\gamma)$. But that means that $\beth_\alpha<\gamma$. Therefore $\gamma$ is the limit of $\{\beth_\alpha\mid\alpha<\gamma\}$, so $\gamma=\beth_\gamma$.
Similarly, we can repeat the same proof by noting that the function mapping $\alpha$ to the $\alpha$th $\beth$-fixed point.
Alternatively, you can note that if $R(\gamma)$ satisfies Replacement, then it actually a model of $\sf ZFC$, and it is an initial segment of the actual universe. So any kind of proofs that work in $\sf ZFC$ are going to work "inside" $R(\gamma)$, and by the fact it is an initial segment of the universe, these must also agree with the actual universe. Now, if you prove, in $\sf ZFC$, that the class of ordinals is the limit of $\beth_\alpha$s, then this holds in $R(\gamma)$, so $\gamma=\beth_\gamma$. And if you prove, in $\sf ZFC$, that there is a proper class of $\beth$-fixed points, then this holds in $R(\gamma)$, etc.