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Hyperclass Forcing in Morse Kelley Set Theory

Monday 24th August 2015 - 14:00 to 14:30
INI Seminar Room 2
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 $\beta$-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 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 $\mathcal{M}$. We conclude by giving an application of this forcing in sho wing that every $\beta$-model of MK$^*$ can be extended to a minimal $\beta$-model of MK$^*$ with the same ordinals.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons