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I have been trying to reconcile this for a while now, but I'm failing miserably.

I know of Tarski's Undefinability of Truth, that states that there is no formula in the standard model $\overline{\omega}$ of Robinson arithmetic that defines (given a Gödel coding) which sentences hold in $\overline{\omega}$.

Clearly, ZFC interprets Robinson arithmetic. I presume one can now deduce that there is no truth predicate in (the standard model $M$, say, of) ZF simply since any such a predicate would also tell the truth about what happens in $(\overline{\omega})^M$ (which should be just $\omega$ by absoluteness?). There is also the General Undefinability Theorem (which is mentioned on the wikipedia page), which I believe settles the matter; please correct me if I'm wrong here.

Now, suppose we have a model $M$ of ZFC, and an extension $N$ (maybe via forcing, or even an end-extension). Is it possible that $N$ can define a truth predicate for $M$?

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Of course.

If $M$ is a set in $N$ (with the relation $\in_M$, of course), then $N$ knows the truth predicate of $M$. This is the usual definition of the satisfaction relation. It might be, however, that $N$ disagrees with the universe on what is FOL or what are the axioms of $\sf ZFC$, so the truth might be somewhat askew.

But we can blow your mind even further. Say that $\kappa$ is inaccessible. Then $V_\kappa$ is a model of $\sf ZFC$. But it contains all the reals. Include the real which defines, via the suitable coding, the truth of $V_\kappa$. Even more is true, if $M\prec V_\kappa$ is a countable elementary submodel, then $M\in V_\kappa$. So $V_\kappa$ knows all of its small elementary submodels, and their truth predicates (with parameters, this time!), which is truly confusing, since they are all elementarily equivalent to $V_\kappa$.


Nevertheless, the thing about Tarski's theorem is that the truth predicate is that you cannot identify the truth predicate internally using first-order logic. It doesn't mean that the truth doesn't exist. And it certainly doesn't mean that it doesn't exist in a larger universe.

It just states that a model of a theory that can reason about truth, cannot reason about its own truth.

As Noah remarks, a forcing extension will not be able to compute a truth predicate of the ground model, since forcing extensions are in a deep sense "very close" to their grounds. Also, not every end-extension could either, since it might be that the original model is not a set in its end-extension (e.g., if the end-extension is not well-founded, and the well-founded part is $M$, then the extended model will not be able to identify $M$ as a set. We can only do that externally).

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    $\begingroup$ It's worth adding that this cannot happen if $N$ is a forcing extension of $M$ - then $N$ is "too close" to $M$, and Tarski still kicks in. $\endgroup$ Apr 13, 2020 at 9:16
  • $\begingroup$ Thanks, that's a good comment. $\endgroup$
    – Asaf Karagila
    Apr 13, 2020 at 9:34
  • $\begingroup$ What are you alluding to when you say that forcing extensions are close to their ground in a deep sense? $\endgroup$ Apr 13, 2020 at 13:06
  • $\begingroup$ @AlessandroCodenotti: The forcing relationship is "formula-wise definable" so in some sense we know what is the possible truth in the generic extension, and in some situation (homogeneous forcings) we know it entirely. $\endgroup$
    – Asaf Karagila
    Apr 13, 2020 at 13:28
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    $\begingroup$ Plus, the truth of formulas in the ground model is fixed in all forcing extensions, so regardless of the forcing used, if there were such a predicate in the extension, it would be definable in the ground model. $\endgroup$ Apr 13, 2020 at 19:10

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