Combinatorial properties and dependent choice in symmetric extensions based on Lévy collapse
Combinatorial properties and dependent choice in symmetric extensions based on Lévy collapse
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基于Lévy塌陷的对称扩展的组合性质和相关选择
DOI：
10.1007/s00153-022-00845-3
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发表时间：
2019
影响因子：
0.3
通讯作者：
A. Banerjee
中科院分区：
文献类型：
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作者：
A. Banerjee
我们研究了基于L折叠的对称扩张，推广了Apter，Cody和Koepke的一些结果。我们从迪米特里欧的博士论文中证明了她的一个猜想。我们还观察到，如果V是$$\Textsf{ZFC}$$ZFC的模型，则$$\Textsf{dc}_{<；\kappa}$$dc<；κ可以按对称系统$$\⟨{P}，{\Mathcal{G}}，{\Mathcal{F}}保持在V的对称扩张中，如果$$\Mathbb{P}$$P是$$\kappa$$⟩，G，Fκ，如果$$\kappa$$κ-分配且$${\Mathcal{F}$$F是$$\kappa$$κ-完全的.此外，我们还观察到，如果$$\δ<；κ和V是$$\Textsf{ZF}+\Textsf{DC}_{\Delta}$$ZF+DCδ的模型，则$$\Textsf{DC}_{\Dc}$$DCδ可以关于对称系统$$\Lange{\Mathbb{P}，{\Mathcal{G}}，{\Mathcal{F}}\Ranging$$⟨P，G，F⟩，如果$${\mathbb{P}}$$P为($$\Delta+1$$δ+1)-战略闭合，且$${\mathcal{F}}$$F为$$\kappa$$κ-完全。
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.