Using the analysis developed in our earlier paper, we show that every uncountable cardinal in Gitik's model of in which all uncountable cardinals are singular is almost Ramsey and is also a Rowbottom cardinal carrying a Rowbottom filter. We assume that the model of is constructed from a proper class of strongly compact cardinals, each of which is a limit of measurable cardinals. Our work consequently reduces the best previously known upper bound in consistency strength for the theory math formula (...) + “All uncountable cardinals are singular” + “Every uncountable cardinal is both almost Ramsey and a Rowbottom cardinal carrying a Rowbottom filter”. (shrink)

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 $$\textsf {ZFC}$$ ZFC, then $$\textsf {DC}_{<\kappa }$$ DC < κ can be preserved in the symmetric extension of V in terms of symmetric system $$\langle {\mathbb {P}},{\mathcal {G}},{\mathcal {F}}\rangle $$ ⟨ P, G, F ⟩, if $${\mathbb {P}}$$ P is $$\kappa $$ (...) κ -distributive and $${\mathcal {F}}$$ F is $$\kappa $$ κ -complete. Further we observe that if $$\delta <\kappa $$ δ < κ and V is a model of $$\textsf {ZF}+\textsf {DC}_{\delta }$$ ZF + DC δ, then $$\textsf {DC}_{\delta }$$ DC δ can be preserved in the symmetric extension of V in terms of symmetric system $$\langle {\mathbb {P}},{\mathcal {G}},{\mathcal {F}}\rangle $$ ⟨ P, G, F ⟩, if $${\mathbb {P}}$$ P is ($$\delta +1$$ δ + 1 )-strategically closed and $${\mathcal {F}}$$ F is $$\kappa $$ κ -complete. (shrink)

We improve on results and constructions by Apter, Dimitriou, Gitik, Hayut, Karagila, and Koepke concerning large cardinals, ultrafilters, and cofinalities without the axiom of choice. In particular, we show the consistency of the following statements from certain assumptions: the first supercompact cardinal can be the first uncountable regular cardinal, all successors of regular cardinals are Ramsey, every sequence of stationary sets in is mutually stationary, an infinitary Chang conjecture holds for the cardinals, and all are singular. In each of the (...) cases, our results either weaken the hypotheses or strengthen the conclusions of known proofs. (shrink)