Combinatorial Properties and Dependent choice in symmetric extensions based on L\'{e}vy Collapse

DOI10.1007/S00153-022-00845-3arXiv1903.05945WikidataQ114231460 ScholiaQ114231460MaRDI QIDQ6315592

Amitayu Banerjee

Publication date: 14 March 2019

Abstract: We work with symmetric extensions based on L'{e}vy 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 $langlemathbbP,mathcalG,mathcalFangle$, if $mathbbP$ is $kappa$-distributive and $mathcalF$ 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 $langlemathbbP,mathcalG,mathcalFangle$, if $mathbbP$ is $kappa$-strategically closed and $mathcalF$ is $kappa$-complete.

Mathematics Subject Classification ID

Consistency and independence results (03E35) Large cardinals (03E55) Axiom of choice and related propositions (03E25)

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