On coding uncountable sets by reals

✍ Scribed by Joan Bagaria; Vladimir Kanovei

• Book ID: 102487117
• Publisher: John Wiley and Sons
• Year: 2010
• Tongue: English
• Weight: 273 KB
• Volume: 56
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.200910056

✦ Synopsis

Abstract

If A ⊆ ω~1~, then there exists a cardinal preserving generic extension 𝕃[A ][x ] of 𝕃[A ] by a real x such that

1. A ∈ 𝕃[x ] and A is Δ~1~^HC^ (x) in 𝕃[x ];

2. x is minimal over 𝕃[A ], that is, if a set Y belongs to 𝕃[x ], then either x ∈ 𝕃[A, Y ] or Y ∈ 𝕃[A ].

The forcing we use implicitly provides reshaping of the given set A (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)