What is the consistency strength of existence of a nontrivial elementary embedding $j:L_δ→L_δ$ for some ordinal $δ$?
The consistency strength is strictly between totally ineffable and $ω$-Erdős cardinals, but I do not know where it should be placed relative to completely ineffable, 1-iterable, remarkable, 2-iterable, $n$-iterable cardinals ($n<ω$).
The least such $δ$ is also the least such that there is $S⊂δ$ with $|S|=ω$ and $S$ being indiscernibles for $L_δ$. (The latter formulation makes it a deceptively weak-looking statement.) Moreover, we can pick $S$ such that $S$ are good indiscernibles and also that every element of $L_δ$ is definable in $L_δ$ from a finite subset of $S$. $L_δ$ will satisfy ZFC with every $κ∈S$ (if we used good indiscernibles) totally ineffable and $L_κ≺L_δ$.
A theory with slightly weaker consistency strength, using language $(V,∈,j)$, is ZFC + $j$ is a nontrivial elementary elementary embedding $V→V$, with transfinite induction (least ordinal principle), but without separation and replacement for formulas involving $j$. (This avoids Kunen's inconsistency; see for example The spectrum of elementrary embeddings j : V → V by Paul Corazza.) I am also curious about the strength of that theory.
Ramsey-like cardinals and Ramsey-like cardinals II would suggest that the strength is above $n$-iterable. If that is correct, then for $n<ω$, existence of $α$ and $β$ with a nontrivial elementary embedding $j:L_α→L_β$ with $j^n(\mathrm{crit}(j))<α$ is likely interleaved in $Σ^1_2$ strength between $n-1$-iterable and $n$-iterable cardinals (if I am not off by 1). However, none of this is explicitly stated, and I do not know if I am missing some assumptions.
