### Killing truth

It is not difficult to construct models of $\mathsf{ZFC}$ whose truth predicate is not strongly amenable, where $X \subseteq M$ is strongly amenable if the structure $(M,X)$ satisfies the Collection and Separation schemata in the expanded language. Namely, this can be accomplished by considering a pointwise definable model of $\mathsf{ZFC}$, such as the Shepherdson–Cohen least transitive model. The truth predicate for the model $M$ reveals that it is countable, allowing one to define a surjection $\omega^M \to M$. This witnesses a failure of Replacement (in the expanded language).

On the other hand, one can find models whose truth predicate is strongly amenable. If $(M,\mathcal X) \models \mathsf{GBC}$ contains the truth predicate for $M$ then its truth predicate must be strongly amenable, due to the Class Replacement and Class Comprehension axioms of $\mathsf{GBC}$. In particular, if $(M,\mathcal X)$ satisfies Kelley–Morse set theory (or any other second-order set theory which proves the existence of the truth predicate) then truth will be strongly amenable for $M$.

Let’s consider those models whose truth is strongly amenable. If we have a $\mathsf{GBC}$-realization for such a model can we always add more classes  (but not sets!) to expand it to a $\mathsf{GBC}$ model which contains the truth predicate?

The answer is no. For countable models we can kill truth with the right $\mathsf{GBC}$-realization, so that no matter what new classes we add we cannot have the truth predicate without violating Class Replacement.

Proposition: Let $M \models \mathsf{ZFC}$ be a countable $\omega$-model. Then there is a $\mathsf{GBC}$-realization $\mathcal X \subseteq \mathcal P(M)$  for $M$ so that no $\mathsf{GBC}-realization$ $\mathcal Y \supseteq \mathcal X$ can contain the truth predicate for $M$.

The purpose of requiring $M$ to have a well-founded $\omega$ is the Krajewski phenomenon that countable $\omega$-nonstandard models of set theory which admit ‘truth predicates’ which measure the truth of all—internal, possibly nonstandard—formulae (full satisfaction classes, to use the jargon) admit continuum many different ones. So in such case there is not a unique object to kill off (indeed, it’s not sensible to speak of the truth predicate for $M$ in the absence of an already-fixed second-order part) and the situation becomes much trickier.

But more on that later. Let’s see why this proposition is true.

Proof: Suppose that truth is strongly amenable for $M$. Otherwise, no $\mathsf{GBC}$-realization for $M$ can contain the truth predicate and there is nothing to prove. We will produce the desired $\mathcal X$ by adding a carefully constructed Cohen-generic subclass of $\mathrm{Ord}$. Let $\mathbb C$ be this class forcing, i.e. the partially ordered class consisting of set-sized partial functions $\mathrm{Ord} \to 2$, ordered by reverse inclusion. This forcing is $\kappa$-closed and $\kappa$-distributive for every cardinal $\kappa$ so it does not add new sets.

Claim: There is a sequence $\langle D_i : i \in \mathrm{Ord} \rangle$ of definable dense subclasses of $\mathbb C$ which is (1) definable from the truth predicate for $M$ and (2) meeting every $D_i$ is sufficient to guarantee a filter is generic over $M$.

It is clear that the collection of dense subclasses of $\mathbb C$ is definable from the truth predicate; the truth predicate allows one to see whether $\varphi(x,a)$ defines a dense subclass of $\mathbb C$ and so the whole collection can be indexed by $(\varphi, a)$. The trick is to extract a sequence of ordertype $\mathrm{Ord}$. This is trivial if $M$ has a definable global well-order, but in general this may not be true. In such case, use that $\mathbb C$ is $\kappa$-distributive for every $\kappa$. So given a set-sized collection of dense subclasses we can find a single dense subclass below all of them. So we get the $D_i$s as in the claim by taking $D_i$ to be below all dense subclasses definable from parameters in $V_i$.

We will now use this $\langle D_i : i \in \mathrm{Ord} \rangle$ to define a Cohen-generic subclass $C$ of $\mathrm{Ord}$. Externally to the model, fix an $\omega$-sequence cofinal in $\mathrm{Ord}^M$. Think of this sequence as an $\mathrm{Ord}^M$-length binary sequence $\langle b_i : i \in \mathrm{Ord} \rangle$, consisting mostly of zeros with the rare one. This sequence is amenable over $M$, since its initial segments have only finitely many ones. On the other hand, it is not strongly amenable since from this sequence can be defined a cofinal map $\omega \to \mathrm{Ord}$, contradicting an instance of Replacement.

We build $C$ in $\mathrm{Ord}^M$ steps. Start with $c_0 = \emptyset$. Given $c_i$ let $c_{i+1}' = c_i {}^\smallfrown \langle b_i \rangle$ and then extend $c_{i+1}'$ to $c_{i+1}$ meeting $D_i$, where we require the extension to be minimal in length to meet $D_i$. (If there is more than one way to meet $D_i$ of minimal length, choose between them arbitrarily.) At limit stages take unions of the previous stages. Finally, set $C = \bigcup_{i \in \mathrm{Ord}} c_i$. Then $C$ is generic over $M$ since it meets every $D_i$. But hidden within $C$ is this bad sequence $\langle b_i \rangle$.

I claim now that from both $C$ and the truth predicate for $M$ one can define $\langle b_i \rangle$. This is done inductively. We know $b_0$ because it is the first bit of $C$. From the truth predicate we know the minimal amount we have to extend $\langle b_0 \rangle$ in order to meet $D_0$, because the sequence $\langle D_i \rangle$ is definable from the truth predicate. So we can recover $c_1$. We now repeat this process, discovering $b_2$ as the first bit of $C$ after $c_1$ and thereby can define $c_2$ by knowing the minimal length we had to extend to meet $D_2$. And so on we define $b_i$ and $c_i$ for all $i \in \mathrm{Ord}^M$. So if we have access to both $C$ and the truth predicate then we can define the sequence $\langle b_i \rangle$.

But if we can define the bad sequence then our model must fail to satisfy Class Replacement. Thus, there can be no $\mathsf{GBC}$-realization for $M$ which contains both $C$ and the truth predicate for $M$. So $\mathrm{Def}(M;C)$ is a $\mathsf{GBC}$-realization for $M$ which cannot be extended to a $\mathsf{GBC}$-realization which contains the truth predicate, as desired. (To see $(M,\mathrm{Def}(M;C)) \models \mathsf{GBC}$ observe that every Cohen-generic subclass of $\mathrm{Ord}$ codes a global well-order by comparing where each set is first coded into the generic.) ∎

As mentioned above, Kelley–Morse proves the existence of the truth predicate for the first-order part, as do many much weaker theories. As a consequence we get $\mathsf{GBC}$ models which cannot be expanded by adding classes to get a model of $\mathsf{KM}$, even if the first-order part has some other $\mathsf{KM}$-realization.

Corollary: Let $M \models \mathsf{ZFC}$ be any countable $\omega$-model. Then $M$ has a $\mathsf{GBC}$-realization which cannot be extended to a $\mathsf{KM}$-realization. The same is true if $\mathsf{KM}$ is replaced by $\mathsf{GBC}$ plus Elementary Transfinite Recursion, or even $\mathsf{GBC} + \mathsf{ETR}_\omega$.

What this result suggest is that it matters how one builds up one’s classes. It is not difficult to see that, say, $\mathsf{KM}$ proves there exist classes of ordinals which are Cohen-generic over the definable classes. (The truth predicate lets us list the definable dense subclasses of the forcing, so they can be met one at a time.) So if we want to build up a model of $\mathsf{KM}$ from a fixed universe of sets we must add Cohen-generics at some point. The above shows that it matters which generics we pick. One wrong step along the way and we stumble off the narrow path leading to paradise.

Observe that the proposition relativizes. Fixing a possible class $X \subseteq M$ if $X$ can be put in any $\mathsf{GBC}$-realization for $M$ then there is a $\mathsf{GBC}$-realization for $M$ containing $X$ which cannot be extended to one containing the truth predicate relative to the parameter $X$. As a consequence, one can find $\mathsf{GBC}$-realizations for any strong enough countable $\omega$-model $M$ which contain an iterated truth predicate of length $\Gamma$ but which cannot be extended to a $\mathsf{GBC}$-realization containing an iterated truth prediacte of length $\Gamma + 1$.

As a coda, let me return $\omega$-nonstandard models. Here, things go badly. As mentioned earlier, Krajewski showed that if a countable $\omega$-nonstandard model of set theory admits a ‘truth predicate’ (full satisfaction class) then it admits continuum many different ones. So killing off one possible truth predicate will not immediately ensure that the classes cannot be expanded to include truth; perhaps some other full satisfaction class can be added to the classes to be truth. But a version of the above argument does show that we can kill off individual full satisfaction classes. (A strongly amenable full satisfaction class will tell us about dense subclasses of the Cohen-forcing which are ‘defined’ using a nonstandard formula, but it will also tell us about all the really definable dense subclasses, so it is good enough.) Can we kill off all the full satisfaction classes? I don’t (yet?) know!

Question: If $M \models \mathsf{ZFC}$ is countable and $\omega$-nonstandard must it be the case that there is a $\mathsf{GBC}$-realization $\mathcal X$ for $M$ so that no matter how we expand $\mathcal X$ we never get $\mathcal Y$ so that $(M,\mathcal Y)$ satisfies $\mathsf{GBC}$ plus “there is a truth predicate for first-order truth”?