THE EXACT STRENGTH OF THE CLASS FORCING THEOREM. (29th September 2020)

• Record Type: Journal Article
• Authors: GITMAN, VICTORIA; HAMKINS, JOEL DAVID; HOLY, PETER; SCHLICHT, PHILIPP; WILLIAMS, KAMERYN J.
• Is Part Of: Journal of symbolic logic. Volume 85:Number 3(2020)
• Page Start: 869
• Page End: 905
• Publication Date: 2020-09-29
• Subjects: 03E40, -- 03E70; class forcing, -- forcing theorem, -- truth predicates, -- infinitary logic; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1017/jsl.2019.89
• Languages: English
• ISSNs: 0022-4812

Abstract

Abstract: The class forcing theorem, which asserts that every class forcing notion ${\mathbb {P}}$ admits a forcing relation $\Vdash _{\mathbb {P}}$, that is, a relation satisfying the forcing relation recursion—it follows that statements true in the corresponding forcing extensions are forced and forced statements are true—is equivalent over Gödel–Bernays set theory $\text {GBC}$ to the principle of elementary transfinite recursion $\text {ETR}_{\text {Ord}}$ for class recursions of length $\text {Ord}$ . It is also equivalent to the existence of truth predicates for the infinitary languages $\mathcal {L}_{\text {Ord}, \omega }(\in, A)$, allowing any class parameter A ; to the existence of truth predicates for the language $\mathcal {L}_{\text {Ord}, \text {Ord}}(\in, A)$ ; to the existence of $\text {Ord}$ -iterated truth predicates for first-order set theory $\mathcal {L}_{\omega, \omega }(\in, A)$ ; to the assertion that every separative class partial order ${\mathbb {P}}$ has a set-complete class Boolean completion; to a class-join separation principle; and to the principle of determinacy for clopen class games of rank at most $\text {Ord}+1$ . Unlike set forcing, if every class forcing notion ${\mathbb {P}}$ has a forcing relation merely for atomic formulas, then every such ${\mathbb {P}}$ has a uniform forcing relation applicable simultaneously to all formulas. Our results situate the class forcing theorem in the rich hierarchy of theories between $\text {GBC}$ and Kelley–Morse set theory $\text {KM}$ .