I shall make two points, but unfortunately neither of them fully answers your question.

**First**, let me elaborate on the applicability of Usuba's results to this question, as pointed out also by Gabe in the comments.

Every omniscient theory has some large cardinal strength, since we will get truth predicates for $L$ and indeed $L[y]$ for every set $y$, and this requires strictly greater consistency strength than ZFC. But omniscience, meanwhile, will be incompatible with certain very large cardinals, such as an extendible cardinal.

**Theorem.** Every omniscient theory refutes the existence of an extendible cardinal.

This is because a theorem of Toshimichi Usuba (Extendible cardinals and the mantle, ZBL07006127) shows that if there
is an extendible cardinal, then the universe has a bedrock, a
ground model of the universe by set forcing that is least among all
such grounds. This ground model is the mantle $M$, which is a
parameter-free definable class, defined as the intersection of all
ground models. (For background on bedrocks, the mantle, and set-theoretic geology generally, see *Fuchs, Gunter; Hamkins, Joel David; Reitz, Jonas*, **Set-theoretic geology**, Ann. Pure Appl. Logic 166, No. 4, 464-501 (2015). ZBL1348.03051.)

The relevance of this is that if the universe has a bedrock model $M$, then $V=M[G]$ for some $M$-generic set forcing $G\subset\mathbb{P}\in M$, and since $M$ is definable as the mantle in a forcing-absolute manner, it would follow that $V$ itself has a forcing-invariant parametric definition—it is "the forcing extension of the mantle $M$ using $G$ and $\mathbb{P}$."

In particular, if $V$ has truth predicates for every forcing-invariant definable class, then this would have to include $V$ itself. But we cannot have a truth predicate for $V$ definable in $V$, since this contradicts Tarski's theorem.

**Second**, let me offer a positive answer to a weakened version of your question. I would like to weaken the notion of omniscience by modifying the notion of forcing-invariance in the following ways:

- You had wanted the class defined by $\varphi(\cdot,y)$ to be forcing invariant from $V$ to its forcing extensions; but I shall consider such classes that are invariant both upward and downward, also from $V$ to its ground models $W$, provided they contain the parameter $y\in W$. (In fact, I shall need
*only* the downward part of absoluteness.)
- I would like to consider not just set forcing, but also certain very-well-behaved class forcing notions, the progressively closed Easton iterations. (In fact, I shall be able to handle every $\varphi$ that is invariant merely for a very specific type of Easton support progressively closed iteration.)

This kind of class forcing is always very nice: it is tame; it preserves ZFC, GBC, and KM; it has definable forcing relations; and so on.

To achieve weakened omniscience, let me begin with any model $V$ of Kelley-Morse KM set theory, although it is sufficient to have GBC+ETR${}_\omega$ and even less than this. I shall construct a class-forcing extension $V[G]$ by progressively closed Easton-support forcing iteration of length Ord, such that for any first-order formula $\varphi$, ordinal $\alpha$ and parameter $y\in V[G_\alpha]$, there is a definable truth predicate for the class $\{x\mid\varphi(x,y)\}^{V[G_\alpha]}$ defined in that model.

We can easily obtain this simply by coding into the GCH pattern. We had started with KM, which has plenty of truth-predicate classes for first-order truth over any class. So we can reserve a definable class of coding points (which will be sufficiently absolutely definable as in the usual geology arguments) for $\langle\varphi,\alpha,y\rangle$, and then gradually force the GCH to hold or fail at those coding points in such a way so as to code a truth predicate for that class. It seems to me that it suffices for us to code a truth predicate for each $V[G_\alpha]$ itself, since truth about $\varphi$ in this model reduces to truth in this model. We can interleave all this coding forcing together into one big progressively closed iteration.

Consider the final model $V[G]$. Consider any formula $\varphi$ and parameter $y$, which must have been added by some stage $V[G_\alpha]$. If $\varphi(\cdot,y)$ defines an invariant class in $V[G]$, then by the downward aspect of this, the class defined in $V[G]$ will be the same as defined in $V[G_\alpha]$. And we specifically coded a truth predicate for this into the GCH pattern of $V[G]$.

So we have constructed a model of KM in which every first-order definable invariant class admits a definable truth predicate.

One can produce a theory $T$ that simply describes what we have done: the theory $T$ asserts that the universe was obtained by forcing over a model $W$ so as to code truth predicates into the GCH patter for every model $W[G_\alpha]$ that was used along the way.

This theory is entirely first-order expressible, and so we can throw away the classes after doing the forcing. The theory $T$ expresses:

- There is a definable inner model $W$ such that the GCH pattern at the first class of coding points codes a truth predicate for $W$.
- and there is a class $G$ that is defined by the GCH pattern at the second class of coding points, which is a sequence of sets $G_\alpha$
- the GCH pattern at the $\alpha$th class of coding points codes a truth predicate for $W[G_\alpha]$
- The class $G$ is $W$-generic for the class forcing that would perform the forcing creating this very situation.

This all seems to involve only first-order matters.

So we've got a first-order theory $T$ extending ZFC, which is consistent relative to KM, which satisfies the omniscience property for definable forcing-invariant classes, when this is understand to refer to downward-invariance by progressively closed class forcing.

9more comments