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Remarks on Gitik's model and symmetric extensions on products of the Lévy collapse
Author(s) -
Banerjee Amitayu
Publication year - 2020
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201800018
Subject(s) - uncountable set , mathematics , mathematical proof , conjecture , consistency (knowledge bases) , regular cardinal , axiom , axiom of choice , combinatorics , sequence (biology) , pure mathematics , discrete mathematics , mathematical economics , countable set , set theory , computer science , geometry , set (abstract data type) , biology , genetics , programming language
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 ℵ n is mutually stationary, an infinitary Chang conjecture holds for the cardinals ℵ 2 n , and all ℵ n are singular. In each of the cases, our results either weaken the hypotheses or strengthen the conclusions of known proofs.