Let us assume that there is a non-trivial elementary embedding $j \colon L\_\gamma \to L\_\gamma$ and $\gamma \geq \omega\_1^V$. Can we conclude that $0^{\#}$ exists?


In general, it is known that if there is an elementary embedding $j \colon L\_\alpha \to L\_\beta$ such that $\mathrm{crit}\, j = \delta$, $(\delta^{+})^L \leq \alpha$ and $\delta \geq \omega\_2^V$ then one can conclude that $0^{\#}$ exists, but can we get something better by starting with the assumption that the embedding is from $L\_\gamma$ to itself?



$0^{\#}$ and self embeddings of $L_\gamma$
No, such embeddings can consistently exist in $L$. In the following, $\omega\_1$ and $\omega\_2$ will denote these cardinals as computed in $V$.


Let us assume that $0^\sharp$ exists in $V$. There is an increasing sequence $\langle \xi\_n\mid n<\omega\rangle\in V$ of order indiscernibles over the structure $(L\_{\omega\_2};\in, \alpha\mid\alpha<\omega\_1)$, for example take the first $\omega$-many Silver indiscernibles $\geq\omega\_1$. In $L$, we can define the tree $T$ which attempts to find such a sequence. As $T$ is illfounded in $V$, $T$ is illfounded in $L$, i.e. there is such a sequence $\langle \xi\_n\mid n<\omega\rangle$ in $L$. Now let $X=\mathrm{Hull}^{L\_{\omega\_2}}(\omega\_1\cup\{\xi\_n\mid n<\omega\})$ and let $\pi\colon X\rightarrow L\_\gamma$ be the transitive collapse of $X$. Then $\omega\_1<\gamma$. Let us write $\bar \xi\_n=\pi(\xi\_n)$. We now have


* $L\_\gamma=\mathrm{Hull}^{L\_\gamma}(\omega\_1\cup\{\bar \xi\_n\mid n<\omega\})$ and
* the $\langle \bar\xi\_n\mid n<\omega\rangle$ are order indiscernible over $(L\_\gamma;\in,\alpha\mid\alpha<\omega\_1)$.


This allows us to build elementary embeddings $j\colon L\_\gamma\rightarrow L\_\gamma$ similar to how one builds elementary embeddings $j\colon L\rightarrow L$ from Silver indiscernibles. Here is the construction:


Let $i\colon\omega\rightarrow\omega$ be any strictly increasing function. We can define an elementary $j\colon L\_\gamma\rightarrow L\_\gamma$, $j\in L$, by
$j(\tau^{(L\_\gamma;\in, \alpha\mid\alpha<\omega\_1)}(\bar\xi\_0,\dots,\bar\xi\_n))=\tau^{(L\_\gamma;\in,\alpha\mid\alpha<\omega\_1)}(\bar\xi\_{i(0)},\dots,\bar\xi\_{i(n)})$ for any term $\tau(x\_0,\dots, x\_n)$ in the language with the $\in$-relation and a constant for every ordinal below $\omega\_1$. It is not difficult to check that $j$ is well-defined, elementary and, in case $i\neq\mathrm{id}\_{\omega}$, even non-trivial.


