Artículos con la etiqueta ‘models of ZFC’

On ground model definability

Por • 6 dic, 2013 • Category: Opinion

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.