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SUMMARY:Hyperclass Forcing in Morse Kelley Set Theory - Antos\, C (Univers
 itt Wien)
DTSTART:20150824T130000Z
DTEND:20150824T133000Z
UID:TALK60436@talks.cam.ac.uk
CONTACT:42080
DESCRIPTION:There are mainly two different types: set-forcing and class-fo
 rcing\, 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 a
 re themselves classes) in the context of an extension of Morse-Kelley clas
 s theory\, called MK$^*$.\nWe define this forcing by using a symmetry betw
 een MK$^*$ models and models  of ZFC$^-$ plus there exists a strongly inac
 cessible cardinal (called SetMK$^*$). We develop a coding between $eta$-m
 odels $mathcal{M}$ of MK$^*$ and transitive models $M^+$ of SetMK$^*$ whic
 h will allow us to go from $mathcal{M}$ to $M^+$ and vice versa. So instea
 d of forcing with a hyperclass in MK$^*$ we can force over the correspondi
 ng SetMK$^*$ model with a class of conditions. For class-forcing to work i
 n the context of ZFC$^-$ we show that the SetMK$^*$ model $M^+$ can be for
 ced 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$^*$\, wh
 ich will in turn be an extension of the original $mathcal{M}$. We conclude
  by giving an application of this forcing in sho\nwing that every $eta$-m
 odel of MK$^*$ can be extended to a minimal $eta$-model of MK$^*$ with th
 e same ordinals.\n
LOCATION:Seminar Room 1\, Newton Institute
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