# Hyperclass Forcing in Morse Kelley Set Theory

• Antos, C (Universitt Wien)
• Monday 24 August 2015, 14:00-14:30
• Seminar Room 1, Newton Institute.

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.