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 $\beta$-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 $\kappa^*$ is the height of $M^+$, $\kappa$ strongly inaccessible in $M^+$ and $X\subseteq\kappa$. 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 $\beta$-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 [BJW82] and [Fri00] to show that every $\beta$-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.