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Hyperclass Forcing in Morse Kelley Set Theory

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Mathematical, Foundational and Computational Aspects of the Higher Infinite

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.

This talk is part of the Isaac Newton Institute Seminar Series series.

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