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

This is an attempt to locate a theorem I vaguely remember but cannot find (which arose in the context of this MSE questionConsistency of non-trivial elementary embedding $j : \mathit{Ord} \to \mathit{Ord}$ and motivated this MO questionCan $Ord$ 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 $(Ord; \in)$$(\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 abovethe 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.

Reference requestion: theorem guaranteeing self-embeddings of expansions of $Ord$

This is an attempt to locate a theorem I vaguely remember but cannot find (which arose in the context of this MSE question and motivated this MO question). 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 $(Ord; \in)$ 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.

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.

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Noah Schweber
Noah Schweber
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Reference requestion: theorem guaranteeing self-embeddings of expansions of $Ord$

This is an attempt to locate a theorem I vaguely remember but cannot find (which arose in the context of this MSE question and motivated this MO question). 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 $(Ord; \in)$ 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.