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The Strengths of Some Violations of Covering
Author(s) -
Mildenberger Heike
Publication year - 2001
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 - 10.1002/1521-3870(200108)47:3<291::aid-malq291>3.0.co;2-p
Subject(s) - cofinality , uncountable set , mathematics , cardinality (data modeling) , transitive relation , regular cardinal , consistency (knowledge bases) , forcing (mathematics) , countable set , cardinal number (linguistics) , order (exchange) , set (abstract data type) , combinatorics , discrete mathematics , pure mathematics , computer science , mathematical analysis , data mining , philosophy , finance , economics , programming language , linguistics
We consider two models V 1 , V 2 of ZFC such that V 1 ⊆ V 2 , the cofinality functions of V 1 and of V 2 coincide, V 1 and V 2 have that same hereditarily countable sets, and there is some uncountable set in V 2 that is not covered by any set in V 1 of the same cardinality. We show that under these assumptions there is an inner model of V 2 with a measurable cardinal κ of Mitchell order κ ++ . This technical result allows us to show that changing cardinal characteristics without changing cofinalities or ω ‐sequences (which was done for some characteristics in [13]) has consistency strength at least Mitchell order κ ++ . From this we get that the changing of cardinal characteristics without changing cardinals or ω ‐sequences has consistency strength Mitchell order ω 1 , even in the case of characteristics that do not stem from a transitive relation. Hence the known forcing constructions for such a change have lowest possible consistency strength. We consider some stronger violations of covering which have appeared as intermediate steps in forcing constructions.