# Consistency strength of $j:L_δ→L_δ$ for some δ

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.