Logic and Set Theory seminar: Forcing minimal β-models in an extension of MK

26 April 2016, 3.00 PM - 26 April 2016, 4.00 PM

Carolin Antos-Kuby, University of Vienna

Howard House 4th floor seminar room

A minimal β-model M(S) of Morse-Kelley class theory (MK) is the least
β-model of MK containing a real S. We show that in an extension of MK,
called MK**, every β-model of MK** can be extended to a minimal β-model
of MK** with the same ordinals. This requires a hyperclass forcing taking
place in MK**, a forcing where the conditions are themselves classes. 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 β-models M of MK** and transitive models
M+ of SetMK** which will allow us to go from M to M+ and vice versa. So
to force a minimal β-model of MK**, we start with a β-model of MK**, go
to the corresponding SetMK** model and force to obtain a minimal model
M(S) of SetMK**. This is done by eliminating all possible smaller models by
a series of forcings involving reshaping, almost disjoint forcing, club shooting
and Jensen Coding. Then we show that by going back to MK**, we arrive
at a minimal β-model of MK**.

Contact information

Organisers: Andrew Brooke-Taylor, Philip Welch

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