# Reference requestion: theorem guaranteeing self-embeddings of expansions of $\mathit{Ord}$

This is an attempt to locate a theorem I vaguely remember but cannot find (which arose in the context of Consistency of non-trivial elementary embedding $$j : \mathit{Ord} \to \mathit{Ord}$$ and motivated Can $Ord$ have nontrivial second-order elementary self-embeddings?). Specifically, I recall a theorem along the following lines (in contrast of course with the Kunen inconsistency):

Every "reasonably definable" expansion of the class-sized structure $$(\mathit{Ord}; \in)$$ has nontrivial elementary self-embeddings.

Here the relevant notion of "reasonably definable" was broad enough to include the usual ordinal arithmetic operations, and if I recall correctly more complicated operations like $$\alpha\mapsto\omega_\alpha$$. On the other hand, it was a $$\mathsf{ZFC}$$ theorem, so we certainly can't go all the way to second-order logic (see the MO question linked above).

Additionally, I vaguely recall that this was due to either Harvey Friedman or Mostowski and that the relevant definability notion was "least-fixed-point-flavored," but I'm much less confident about either of these memories.

• Per our earlier discussion, I used \mathit on your $\mathit{Ord}$s. I also inlined the titles of the questions you reference. I hope that this is all right. – LSpice Jan 6 at 2:06
• @LSpice Fine by me! (And I totally forgot \mathit was an option ... again. :P) – Noah Schweber Jan 6 at 2:11