Class Forcing, the Forcing Theorem and Boolean Completions

Abstract

The forcing theorem is the most fundamental result about set forcing, stating that the forcing relation for any set forcing is definable and that the truth lemma holds, that is everything that holds in a generic extension is forced by a condition in the relevant generic filter. We show that both the definability (and, in fact, even the amenability) of the forcing relation and the truth lemma can fail for class forcing.In addition to these negative results, we show that the forcing theorem is equivalent to the existence of a (certain kind of) Boolean completion, and we introduce a weak combinatorial property (approachability by projections) that implies the forcing theorem to hold. Finally, we show that unlike for set forcing, Boolean completions need not be unique for class forcing.2010 Mathematics Subject Classification. 03E40, 03E70, 03E99. Key words and phrases. Class forcing, Forcing theorem, Boolean completions. The first, third and fifth author were partially supported by DFG-grant LU2020/1-1. The authors would like to thank Joel David Hamkins for the careful reading of the paper and numerous useful comments.1 Note that in the absence of the power set axiom, Collection does not follow from Replacement and many important set-theoretical results can consistently fail in the weaker theory. For further details, consult [GHJ11]. 2 A partial order (or, more generally, a preorder) P is separative if for all p, q ∈ P , if p ≤ q then there exists r ≤ p such that r ⊥ q.

Forcing and the Universe of Sets: Must We Lose Insight?

“…Hence, whenever G is P-generic over M, they contain exactly the same first-order objects. 13 For details, see [Holy et al, 2016]. 14 We shall argue later that for the purposes of talking about forcings over V , there is no reason why ZFC preservation is especially desirable.…”

“…We are rather trying to code in V the effects of viewing V as part of a forcing multiversist framework despite the fact that the relevant extensions do not exist (strictly speaking). Thus, there seems to be no objection to considering models where, say, there is a bijection between ω and V (as is the case when forcing 41 For the details of the proof, and further discussion of the Truth and Definability lemmas in context of class forcing, see [Holy et al, 2016]. 42 Pretameness implies the preservation of Replacement in a class forcing, and tameness additionally requires that the forcing preserve the Power Set Axiom.…”

Hyperclass Forcing in Morse-Kelley Class Theory

“…6 Also, κ will still be inaccessible in the extension because Jensen coding preserves inaccessibles. 7 Note that the result from Step 1 still holds: In L κ * [R] we have that ifκ < κ * is a cardinal then there is no transitive model of SetMK * * containing R in whichκ is inaccessible as otherwise there would have been such a model containing X ∩κ as the latter is coded by R in Lκ[R].…”

The Exact Strength of the Class Forcing Theorem

“…Unlike set forcing, for which one may prove in ZFC that every set forcing notion has corresponding forcing relations, with class forcing it is consistent with Gödel-Bernays set theory GBC that there is a proper class forcing notion lacking a corresponding forcing relation, even merely for the atomic formulas. For certain forcing notions (see [6,9], also Theorem 17), the existence of an atomic forcing relation implies Con(ZFC) and much more, and so the consistency strength of the class forcing theorem strictly exceeds GBC, if this theory is consistent. Nevertheless, the class forcing theorem is provable in stronger theories, such as Kelley-Morse set theory.…”

“…Specifically, extending the analysis of [2,6,7,9], we show that the class forcing theorem is equivalent over GBC to the principle of elementary transfinite recursion ETR Ord for transfinite class recursions of length Ord; to the existence of various kinds of truth predicates and iterated truth-predicates; to the existence of Boolean completions for any separative class partial order; to a class-join separation principle; and to the principle of determinacy for clopen class games of rank at most Ord + 1. In addition, by separating the class forcing theorem from the nearby theories of Figure 1, placing it strictly between the theory with ETR α simultaneously for all ordinals α and the theory ETR Ord• , we locate it finely in the hierarchy of secondorder set theories.…”