Modern class forcing

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C. Antos and V. Gitman, “Modern Class Forcing,” Submitted.

Set theorists started forcing with class partial orders to modify global properties of the universe very soon after general forcing techniques were developed. Once Cohen showed that ${\rm CH}$ could fail, and that indeed the continuum could assume any reasonable value, it was natural to ask whether, for instance, the ${\rm GCH}$ can fail unboundedly often, or more generally what global patterns were possible for the continuum function. Since a set partial order can only affect a set-sized chunk of the continuum function, a class partial order was required to modify the ${\rm GCH}$ pattern unboundedly. Easton used a class product of set-forcing notions, with what became known as Easton support, to show that the ${\rm GCH}$ can fail at every regular cardinal, and that, in fact, any reasonable pattern on the continuum function was consistent with ${\rm ZFC}$ [1]. Since then class products and iterations, usually with Easton support, have been used, for example, to globally kill large cardinals, to make all supercompact cardinals indestructible, or to code the universe into the continuum function.

There are two approaches to handling class forcing in first-order set theory. We can use a generic filter $G$ for a class forcing notion $\mathbb P$ to interpret the $\mathbb P$-names and obtain the forcing extension $V[G]$, and then throw $G$ away. We can also work in the structure $(V[G],\in,G)$ with a predicate for $G$. While both approaches were used by set theorists to handle specific instances of class forcing, neither approach could provide a robust framework for the study of general properties of class forcing because it relegated properties of classes to the meta-theory.

Important early results on class forcing, due to Friedman and Stanley, showed that class partial orders satisfying `niceness' conditions, such as pretameness, behave like set partial orders. But despite the pervasive use of class forcing in set theoretic constructions, a general theory of class forcing was not fully developed until recently. These recent results demonstrated that properties of class forcing depend on what other classes exist around them, establishing that the theory of class forcing can only be properly investigated in a second-order set theoretic setting. The mathematical framework of second-order set theory has objects for both sets and classes, and allows us to move the study of classes out of the meta-theory.

Class forcing becomes even more important in the context of second-order set theory, where it can be used to modify the structure of classes. With class forcing, we can, for instances, add a global well-order or shoot class clubs with desirable properties. Both these forcing notions leave the first-order part of the model intact because they do not add sets. Such forcing constructions over models of second-order set theory can actually be used to prove theorems about models of ${\rm ZFC}$. One of the most important results in the model theory of set theory is the Keisler-Morley extension theorem asserting that every countable model of ${\rm ZFC}$ has an elementary top-extension (adding only sets of rank above the ordinals of the model) [2]. Given a countable model $M$ of ${\rm ZFC}$, the top-extension is built as an ultrapower of $M$ by a special ultrafilter, which we can construct provided that $M$ can be expanded to a model of ${\rm GBC}$. If $M$ doesn't already have a definable global well-order, we use class forcing to add it without altering $M$ itself.

In many ways class forcing does not behave as nicely as set forcing. Class forcing can destroy ${\rm ZFC}$; even the atomic forcing relation may fail to be definable for a class forcing notion; densely embedded class forcing notions need not be forcing equivalent; most class forcing notions don't have Boolean completions; the intermediate model theorem fails badly for class forcing, etc. At the same time, conditions on class forcing, such as pretameness, guarantee that these pathologies are avoided, and these conditions hold for most familiar class forcing notions used by set theorists, such as progressively closed Easton support products and iterations.

In this article, we survey most of the recent results developing a general theory of class forcing in the second-order setting and establishing surprising connections between properties of class forcing and the structure of second-order set theories.

References

  1. W. B. Easton, “Powers of regular cardinals,” Ann. Math. Logic, vol. 1, pp. 139–178, 1970.
  2. H. J. Keisler and M. Morley, “Elementary extensions of models of set theory,” Israel J. Math., vol. 6, pp. 49–65, 1968.