On ground model definability

Por • 6 dic, 2013 • Sección: Opinion

Victoria Gitman, Thomas A. Johnstone

Abstract: Laver, and Woodin independently, showed that models of ZFC  are uniformly definable in their set-forcing extensions, using a ground model parameter. We investigate ground model definability for models of fragments of ZFC  , particularly of ZF+DC δ   and of ZFC −   , and we obtain both positive and negative results. Generalizing the results of Laver and Woodin, we show that models of ZF+DC δ   are uniformly definable in their set-forcing extensions by posets admitting a gap at δ  , using a ground model parameter. In particular, this means that models of ZF+DC δ   are uniformly definable in their forcing extensions by posets of size less than δ  . We also show that it is consistent for ground model definability to fail for models of ZFC −   of the form H κ +    . Using forcing, we produce a ZFC  universe in which there is a cardinal κ>>ω  such that H κ +    is not definable in its Cohen forcing extension. As a corollary, we show that there is always a countable transitive model of ZFC −   violating ground model definability. These results turn out to have a bearing on ground model definability for models of ZFC  . It follows from our proof methods that the hereditary size of the parameter that Woodin used to define a ZFC  model in its set-forcing extension is best possible.

Xiv:1311.6789v1 [math.LO]

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