Philipp Schlicht (Bristol)

Forcing over choiceless models

Forcing over models of ZF set theory without the axiom of choice has been studied in particular for L(ℝ) in work of Steel, Van Wesep, Woodin and more recently Larson and Zapletal. However, the axiom of choice can fail in much stronger ways than in L(ℝ). For example, in Gitik’s celebrated model all uncountable cardinals are singular. Since virtually all known forcing techniques fail in this situation, it is interesting to understand what forcing does to such models. We develop a toolbox for forcing over arbitrary choiceless models. We further introduce very strong absoluteness principles and show their relation with Gitik’s model. This is joint work with Daisuke Ikegami and in part with W. Hugh Woodin.