The exact strength of the class forcing theorem

Journal of Symbolic Logic 85 (3):869-905 (2020)
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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 $\mathcal {L}_{\text {Ord},\omega }$, allowing any class parameter A; to the existence of truth predicates for the language $\mathcal {L}_{\text {Ord},\text {Ord}}$ ; to the existence of $\text {Ord}$ -iterated truth predicates for first-order set theory $\mathcal {L}_{\omega,\omega }$ ; 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 $\text {KM}$.



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Forcing and the Universe of Sets: Must We Lose Insight?Neil Barton - 2020 - Journal of Philosophical Logic 49 (4):575-612.
Indeterminateness and `The' Universe of Sets: Multiversism, Potentialism, and Pluralism.Neil Barton - 2021 - In Melvin Fitting (ed.), Research Trends in Contemporary Logic (Series: Landscapes in Logic). College Publications. pp. 105-182.
Varieties of Class-Theoretic Potentialism.Neil Barton & Kameryn J. Williams - forthcoming - Review of Symbolic Logic:1-45.

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Classes and truths in set theory.Kentaro Fujimoto - 2012 - Annals of Pure and Applied Logic 163 (11):1484-1523.
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Algebraicity and Implicit Definability in Set Theory.Joel David Hamkins & Cole Leahy - 2016 - Notre Dame Journal of Formal Logic 57 (3):431-439.

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