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By pointwise definable models, it's meant that every element of those models is definable after a formula in a parameter free manner, but that defining formula is in the language of that model, i.e., has all of its variables ranging over elements of that model with the membership relation restricted to that model also.

For the sake of this posting, I'd call those internally pointwise definable.

Can we have externally pointwise definable models, where the parameter free defining formulas come from worlds beyond the models themselves. For example, coming from the power sets of those models, or further up iterative power sets of them, or even from upper stages of the cumulative hierarchy, or even unleash them totally to come from $V$.

Would the externally pointwise definable models of $\sf ZFC$ be subject to all of the consequences of the usual internally pointwise definable models? For example, it appears to me that they'd still be hereditarily countable, but would they necessarily also satisfy $\sf V=HOD$, or $\sf GC$, or $\sf C$?

If the answer to the above is to the negative, then would it be possible to have every consistent first order theory extending $\sf ZF$ having an externally pointwise definable model satisfying it?

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Obviously not.

Any model living inside a pointwise definable model is going to be "externally pointwise definable" for obvious reasons.

Now take any generic, symmetric, or otherwise extension of the model which still exists inside the pointwise definable one, and it will still be "externally pointwise definable".

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  • $\begingroup$ Ok, but that doesn't answer the second question? $\endgroup$ Commented Jan 1, 2023 at 11:58

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