# Truth Predicate in Models of ZFC

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 $$M$$?

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).

• 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. Apr 13, 2020 at 9:16
• Thanks, that's a good comment. Apr 13, 2020 at 9:34
• What are you alluding to when you say that forcing extensions are close to their ground in a deep sense? Apr 13, 2020 at 13:06
• @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. Apr 13, 2020 at 13:28
• 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. Apr 13, 2020 at 19:10