Hyperclass Forcing in Morse-Kelley Class Theory

Abstract

In this article we introduce and study hyperclass-forcing (where the conditions of the forcing notion are themselves classes) in the context of an extension of Morse-Kelley class theory, called MK^∗∗. We define this forcing by using a symmetry between MK^∗∗ models and models of ZFC⁻ plus there exists a strongly inaccessible cardinal (called SetMK^∗∗). We develop a coding between β-models $\mathcal {M}$ of MK^∗∗ and transitive models M ⁺ of SetMK^∗∗ which will allow us to go from $\mathcal {M}$ to M ⁺ and vice versa. So instead of forcing with a hyperclass in MK^∗∗ we can force over the corresponding SetMK^∗∗ model with a class of conditions. For class-forcing to work in the context of ZFC⁻ we show that the SetMK^∗∗ model M ⁺ can be forced to look like $L_{\kappa ^*}[X]$, where κ ^∗ is the height of M ⁺, κ strongly inaccessible in M ⁺ and X ⊆ κ. Over such a model we can apply definable class forcing and we arrive at an extension of M ⁺ from which we can go back to the corresponding β-model of MK^∗∗, which will in turn be an extension of the original $\mathcal {M}$. Our main result combines hyperclass forcing with coding methods of Beller et al. (Coding the universe. Lecture note series. Cambridge University Press, Cambridge, 1982) and Friedman (Fine structure and class forcing. de Gruyter series in logic and its applications, vol 3, Walter de Gruyter, New York, 2000) to show that every β-model of MK^∗∗ can be extended to a minimal such model of MK^∗∗ with the same ordinals. A simpler version of the proof also provides a new and analogous minimality result for models of second-order arithmetic.

Originally published in C. Antos, S.D. Friedman, Hyperclass Forcing in Morse-Kelley Class Theory. J. Symb. Logic 82(2), 549–575 (2017).