  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.
