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On coding uncountable sets by reals
Author(s) -
Bagaria Joan,
Kanovei Vladimir
Publication year - 2010
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.200910056
Subject(s) - uncountable set , mathematics , forcing (mathematics) , extension (predicate logic) , set (abstract data type) , discrete mathematics , coding (social sciences) , combinatorics , computer science , countable set , mathematical analysis , programming language , statistics
If A ⊆ ω 1 , then there exists a cardinal preserving generic extension [ A ][ x ] of [ A ] by a real x such that 1) A ∈ [ x ] and A is Δ 1 HC ( x ) in [ x ]; 2) x is minimal over [ A ], that is, if a set Y belongs to [ x ], then either x ∈ [ A , Y ] or Y ∈ [ A ]. The forcing we use implicitly provides reshaping of the given set A (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)