Combinatorial properties and dependent choice in symmetric extensions based on Lévy collapse

Abstract

We work with symmetric extensions based on Lévy collapse and extend a few results of Apter, Cody, and Koepke. We prove a conjecture of Dimitriou from her Ph.D. thesis. We also observe that if V is a model of \(\textsf {ZFC}\), then \(\textsf {DC}_{<\kappa }\) can be preserved in the symmetric extension of V in terms of symmetric system \(\langle {\mathbb {P}},{\mathcal {G}},{\mathcal {F}}\rangle \), if \({\mathbb {P}}\) is \(\kappa \)-distributive and \({\mathcal {F}}\) is \(\kappa \)-complete. Further we observe that if \(\delta <\kappa \) and V is a model of \(\textsf {ZF}+\textsf {DC}_{\delta }\), then \(\textsf {DC}_{\delta }\) can be preserved in the symmetric extension of V in terms of symmetric system \(\langle {\mathbb {P}},{\mathcal {G}},{\mathcal {F}}\rangle \), if \({\mathbb {P}}\) is (\(\delta +1\))-strategically closed and \({\mathcal {F}}\) is \(\kappa \)-complete.