Abstract

There are mainly two different types: set-forcing and class-forcing, where the forcing notion is a set or class respectively. Here, we want to introduce and study the next step in this classification by size, namely 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 $eta$-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 $X ubseteqkappa$. Over such a model we can apply class-forcing and we arrive at an extension of $M}$ from which we can go back to the corresponding $eta$-model of MK$^{$, which will in turn be an extension of the original $mathcal{M}$. We conclude by giving an application of this forcing in sho wing that every $eta$-model of MK$}$ can be extended to a minimal $eta$-model of MK$^*$ with the same ordinals.