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 Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her P.h.d. thesis. We also observe that if $V$ is a model of ZFC, then $D{C}_{<\kappa }$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $⟨\mathbb{P},\mathcal{G},\mathcal{F}⟩$, if $\mathbb{P}$ is $\kappa$-distributive and $\mathcal{F}$ is $\kappa$-complete. Further we observe that if $V$ is a model of ZF + $D{C}_{\kappa }$, then $D{C}_{<\kappa }$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $⟨\mathbb{P},\mathcal{G},\mathcal{F}⟩$, if $\mathbb{P}$ is $\kappa$-strategically closed and $\mathcal{F}$ is $\kappa$-complete.

Revised version

References in corpus (3)

• Iterating Symmetric Extensions
• Preserving Dependent Choice
• Spectra of uniformity

Cited by in corpus (3)

• Sequential and distributive forcings without choice
• On Erdos--Dushnik--Miller theorem without AC
• Iterated failures of choice