Boolean-valued class forcing

This is a talk at the CUNY Logic Workshop in New York, May 4, 2018.

There are two standard ways to do forcing: using partial orders or using complete Boolean algebras. Since every partial order densely embeds into a complete Boolean algebra and whenever two partial orders densely embed, they produce the same forcing extensions, the two approaches to forcing are essentially interchangeable. But meanwhile each approach offers distinct advantages.

The kind of new object we would like to have in the forcing extension dictates what the conditions in the partial order should be. If you would like to add a new real, make the partial order consist of finite binary sequences. If you would like to add a tree, make the partial order consist of small trees. It is usually clear how different conditions in the partial order alter the properties of the new object. The Boolean completion of the partial order breaks the direct connection between the properties of the object we would like to add and the conditions in the forcing, which is precisely why pretty much no one uses the Boolean algebras approach in forcing practice. The Boolean algebras approach however offers both philosophical and practical advantages.

We often speak of the forcing relation for a partial order being definable. But what does the forcing relation really mean when we are talking about forcing over the entire universe $V$ and not just some toy countable model? In this context it does not make sense to say that $p\Vdash\varphi$ means that whenever $p$ is an element of a generic filter $G$, then $V[G]\models\varphi$. On the other hand, the Boolean-valued model construction with a complete Boolean algebra makes sense without reference to any transitive set models. Indeed, using any ultrafilter $U$ on the Boolean algebra, we can turn the Boolean-valued model into a definable model $M\models{\rm ZFC}$ of the form $\bar V[G]$, a forcing extension of submodel $\bar V$, where $V$ elementarily embeds into $\bar V$. So that by passing from $V$ to $\bar V$, a model with the same theory, we get access to both the universe and its forcing extension.

There are theorems about forcing which have no known proofs without the use of complete Boolean algebras. A fundamental result about forcing is the Intermediate Model Theorem, which states that every intermediate universe $W\models{\rm ZFC}$ between a universe $V$ and its forcing extension $V[G]$ is itself a forcing extension of $V$. Indeed, if $V[G]$ is a forcing extension by $\mathbb P$ and $\mathbb B_{\mathbb P}$ is a Boolean completion of $\mathbb P$, then $\mathbb B_{\mathbb P}$ has a complete subalgebra $\mathbb D$ such that $W=V[\mathbb D\cap G]$ is a forcing extension by $\mathbb D$. Grigorief in [1] attributes the Intermediate Model Theorem to Solovay. The theorem makes a fundamental use of the Axiom of Choice, but is true in a weaker form for models of ${\rm ZF}$. If $V[a]\models{\rm ZF}$ with $a\subseteq V$ is an intermediate universe between $V\models{\rm ZF}$ and its forcing extension $V[G]$, then $V[a]$ is a forcing extension of $V$.

Everything that has been said so far holds true for set partial orders. What about class partial orders? Analyzing the properties of class partial orders is best done in a second-order setting where we have objects for classes as well as sets. One of the weakest second-order theories is the Gödel-Bernays set theory ${\rm GBC}$, so let's take it for the moment as our second-order foundation. We will think of models of second-order set theory as triples $\mathscr V=\langle V,\in,\mathcal C\rangle$, where $V$ is the collection of sets and $\mathcal C$ is the collection of classes. If $\mathbb P\in\mathcal C$ is partial order and $G\subseteq \mathbb P$ is $\mathscr V$-generic, then the forcing extension $\mathscr V[G]=\langle V[G],\in,\mathcal C[G]\rangle$, where $V[G]$ is all interpretations of $\mathbb P$-names and $\mathcal C[G]$ is all interpretations of class $\mathbb P$-names, which are collections of pairs $\langle \sigma,p\rangle$ where $\sigma$ is a $\mathbb P$-name and $p\in\mathbb P$. Now we can talk about the properties of class partial orders.

From the perspective of forcing, class partial orders don't behave anywhere as nicely as set partial orders. Class partial orders may not preserve ${\rm GBC}$. Think for example, of the partial order to add bijection from $\omega$ onto ${\rm ORD}$. The forcing relation for a class partial order may not be definable because it can, for instance, code in a truth predicate for $\mathscr V$. Two class partial orders can densely embed but produce different forcing extensions. Class partial orders may not have Boolean completions. (The results are from [2]) To understand this last point, let's be more precise about the kind of Boolean completion we would like a class partial order to have.

The completeness of a Boolean algebra is used in defining the Boolean values of formulas in the forcing language. Suppose $\mathbb B$ is a Boolean algebra. For example, the Boolean value of an atomic assertion $$[[\sigma\in \tau]]=\bigwedge_{\langle \mu,b\rangle\in \tau}[[\sigma=\mu]]\cdot b$$ and the Boolean value of an existential assertion $$[[\exists x\varphi(x,\tau)]]=\bigvee_{\sigma\in V^{\mathbb B}}[[\varphi(\sigma,\tau)]].$$ So if $\mathbb B$ is class, then the definition of Boolean values for atomic assertions requires the existence suprema for sets (set-completeness), but the existential assertions require the existence of suprema for classes as well (class-completeness).

A class partial order can be densely embedded into a set-complete Boolean algebra if and only if its forcing relation is definable [3]. So in ${\rm GBC}$, it is not the case that every class partial order can be embedded into a set-complete Boolean algebra. To define the forcing relation however requires only a bit more comprehension, already in ${\rm GBC}$ together with $\Pi^1_1$-comprehension (for assertions with a single class quantifier) every class partial order has a definable forcing relation (indeed this is already true for a much weaker theory ${\rm GBC}+{\rm ETR}_{\rm {ORD}}$, see [4]). So in the much stronger theory Kelley-Morse ${\rm KM}$ (which consists of ${\rm GBC}$ together with comprehension for all second-order assertions) every class partial order can be embedded into a set-complete Boolean algebra. But indeed, it will never be possible to embed every class partial order into a class-complete Boolean algebra. A Boolean algebra with a proper class antichain can never be class complete, and therefore no class partial order with a proper class antichain can be embedded into such a Boolean algebra [3]. But without class completeness, we cannot define the Boolean-valued model!

What happens when we try to carry out the standard Boolean completion construction with a class partial order? The elements of the completion are subsets of $\mathbb P$ that are regular cuts (a cut is subset of $\mathbb P$ that is closed downwards and a cut $U$ is regular if whenever $p\notin U$, then there is $q\leq p$ such that $U_q\cap U=\emptyset$, where $U_q$ is the cut consisting of all conditions in the cone below $p$). If $\mathbb P$ is a class, then regular cuts are definable sub-classes of $\mathbb P$, and so the collection of regular cuts of $\mathbb P$ is a hyperclass - a (second-order) definable collection of classes. What properties does this hyperclass, call it $\mathbb B_{\mathbb P}$, have? We can definably impose a Boolean operations structure on it in the usual way, so it is a Boolean algebra. How complete is it? First, let's explain how some hyperclasses can actually be quite small, namely class-sized. We will say that a hyperclass, given a formula $\varphi(X,A)$, is coded by a class $S$ if the classes satisfying $\varphi(X,A)$ are precisely the slices of $S$, where a class $T$ is a slice of $S$ if there is a set $a$ such that $T=\{y\mid \langle y,a\rangle\in S\}$. We should think of coded hyperclasses as being class-sized. The hyperclass Boolean algebra $\mathbb B_{\mathbb P}$ is class-complete in the sense that it is complete for all coded hyperclasses. Given a hyperclass coded by $S$, its join is the least regular cut containing the union of all slices $S_a$. But because the elements of $\mathbb B_{\mathbb P}$ are now classes, it should be intuitively clear that to have any hope of using it to define a Boolean-valued model, we need $\mathbb B_{\mathbb P}$ to be complete for all hyperclasses, not just the coded ones. Remarkably it turns out that given any partial order $\mathbb P$ with a proper class antichain, if $\mathbb B_{\mathbb P}$ is hyperclass complete, then we have comprehension for all second-order assertions, and therefore ${\rm KM}$ holds [5]. Thus, the completeness of the hyperclass object is directly connected to the amount of comprehension a model satisfies.

In models of ${\rm KM}$, the hyperclass object $\mathbb B_{\mathbb P}$ has all the desired properties. It is complete for all hyperclasses and indeed, as aught to be the case because it has a class-sized dense sub-partial-order, all its hyperclass antichains are coded. But it is still very difficult if not impossible to define the Boolean-valued model with a Boolean algebra whose elements are classes. A natural solution to this problem is to move a slightly stronger second-order set theory Kelley-Morse together with the Choice Scheme.

The Choice Scheme ${\rm CC}$ is a choice principle (or a collection principle depending on the point of view) for classes which states for every second-order assertion $\varphi(x,X,A)$ that if for every set $x$, there is a witnessing class $X$ such that $\varphi(x,X,A)$ holds, then there is a single class $Y$ collecting witnesses for every set $x$, in the sense that $\varphi(x,Y_x,A)$ holds for every set $x$ (where $Y_x$ is the $x$-th slice of $Y$ defined as above). Kelley-Morse does not prove even the weakest instances of the Choice Scheme, those for first-order assertions and making just $\omega$-many choices [6]. But nevertheless the two theories ${\rm KM}$ and ${\rm KM}+{\rm CC}$ are equiconsistent, basically because the "constructible universe" of a model of ${\rm KM}$ satisfies ${\rm KM}+{\rm CC}$. The theory ${\rm KM}+{\rm CC}$ has one remarkable feature. It is bi-interpretable with a very well-understood first-order set theory. Given a model $\mathscr V\models{\rm KM}+{\rm CC}$ we can consider its class well-founded relations. For instance, $\mathscr V$ has relations coding ${\rm ORD}+{\rm ORD}$, ${\rm ORD}\times\omega$, $V\cup{\{V\}}$, etc. We can view such relations as coding transitive sets that sit "above" the sets of $V$. Modulo isomorphism, we can define a membership relation on the equivalence classes of these relations resulting in a first-order structure extending $V$. We will call this structure the companion model $M_{\mathscr V}$ of $\mathscr V$. The model $M_{\mathscr V}$ satisfies the theory ${\rm ZFC}^-_I$, consisting of the axioms of ${\rm ZFC}$ without powerset $(-)$, with the assertion that there is a largest cardinal $\kappa$ which is inaccessible ($I$). Natural models of ${\rm ZFC}^-_I$ are $H_{\kappa^+}$ for an inaccessible cardinal $\kappa$. The $V_\kappa$ of $M_{\mathscr V}$ consists of the sets $V$ of $\mathscr V$ and the subsets of $V_\kappa$ are the classes of $\mathscr V$. In the other direction if $M\models{\rm ZFC}^-_I$, then $\mathscr V=\langle V_\kappa^M,\in,\mathcal C\rangle$, where $\mathcal C$ consists of the subsets of $V_\kappa$ in $M$, is a model of ${\rm KM}+{\rm CC}$ whose companion model $M_{\mathscr V}$ is isomorphic to $M$.

Thus, whenever we work in a model $\mathscr V\models{\rm KM}+{\rm CC}$, we might as well be working in its companion model $M_{\mathscr V}$. But now the trick is that in $M_{\mathscr V}$, the class partial order $\mathbb P$ is a set (subset of $V_\kappa^{M_{\mathscr V}}$) and the hyperclass $\mathbb B_{\mathbb P}$ is a class-complete Boolean algebra with set-sized antichains. In $M_{\mathscr V}$, it is now easy to define the collection $M_{\mathscr V}^{\mathbb B_{\mathbb P}}$ of $\mathbb B_{\mathbb P}$-names as well the Boolean values of all assertions in the forcing language, using the completeness of $\mathbb B_{\mathbb P}$. In this sense, we have the Boolean algebras approach to forcing in models of ${\rm KM}+{\rm CC}$.

One consequence of this is that we get the ${\rm ZF}$-analogue of the Intermediate Model Theorem for models of ${\rm KM}+{\rm CC}$. Let us say that $\mathscr W$ is a simple extension of $\mathscr V\models{\rm KM}+{\rm CC}$ if $\mathscr W$ is generated by the classes of $\mathscr V$ together with a single class. In particular, forcing extensions of $\mathscr V$ are simple. Using, the Boolean-valued model, we can show that every intermediate simple extension $\mathscr W\models{\rm KM}+{\rm CC}$ between a model $\mathscr V\models{\rm KM}+{\rm CC}$ and its forcing extension $\mathscr V[G]\models{\rm KM}+{\rm CC}$ is itself a forcing extension of $\mathscr V$. The result is optimal because it is possible to have intermediate models between a model $\mathscr V\models{\rm KM}+{\rm CC}$ and its forcing extension $\mathscr V[G]\models\rm KM+{\rm CC}$ that are not simple, and therefore cannot be forcing extensions themselves. [5]

This is joint work with Carolin Antos and Sy-David Friedman.

References

1. S. Grigorieff, “Intermediate submodels and generic extensions in set theory,” Ann. Math. (2), vol. 101, pp. 447–490, 1975.
2. P. Holy, R. Krapf, and P. Schlicht, “Characterizations of pretameness and the Ord-cc,” Ann. Pure Appl. Logic, vol. 169, no. 8, pp. 775–802, 2018.
3. P. Holy, R. Krapf, P. Lücke, A. Njegomir, and P. Schlicht, “Class forcing, the forcing theorem and Boolean completions,” J. Symb. Log., vol. 81, no. 4, pp. 1500–1530, 2016.
4. V. Gitman, J. D. Hamkins, P. Holy, P. Schlicht, and K. J. Williams, “The exact strength of the class forcing theorem,” J. Symb. Log., vol. 85, no. 3, pp. 869–905, 2020. Available at: https://doi.org/10.1017/jsl.2019.89
5. C. Antos, S.-D. Friedman, and V. Gitman, “Boolean-valued class forcing,” Fund. Math., vol. 255, no. 3, pp. 231–254, 2021.
6. V. Gitman and J. D. Hamkins, “Kelley-Morse set theory and choice principles for classes.”