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University of Cambridge > Talks.cam > Isaac Newton Institute Seminar Series > Hyperclass Forcing in Morse Kelley Set Theory

## Hyperclass Forcing in Morse Kelley Set TheoryAdd to your list(s) Download to your calendar using vCal - Antos, C (Universitt Wien)
- Monday 24 August 2015, 14:00-14:30
- Seminar Room 1, Newton Institute.
If you have a question about this talk, please contact webseminars. 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$ ^{$ which will allow us to go from $mathcal{M}$ to $M}$ and vice versa. So instead of forcing with a hyperclass in MK$$ we can force over the corresponding SetMK$^{}$ model with a class of conditions. For class-forcing to work in the context of ZFC $$ model $M^{$ we show that the SetMK$}^{$ 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. ## This talk is included in these lists:- All CMS events
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