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.
Categories: math.LO