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Minimal extensions of Π 0 1 classes
Author(s) -
Cenzer Douglas,
Riazati Farzan
Publication year - 2005
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.200410021
Subject(s) - mathematics , extension (predicate logic) , class (philosophy) , decidability , combinatorics , limit (mathematics) , degree (music) , discrete mathematics , mathematical analysis , physics , artificial intelligence , computer science , acoustics , programming language
A minimal extension of a Π 0 1 class P is a Π 0 1 class Q such that P ⊂ Q , Q – P is infinite, and for any Π 0 1 class R , if P ⊂ R ⊂ Q , then either R – P is finite or Q – R is finite; Q is a nontrivial minimal extension of P if in addition P and Q ′ have the same Cantor‐Bendixson derivative. We show that for any class P which has a single limit point A , and that point of degree ≤ 0 , P admits a nontrivial minimal extension. We also show that as long as P is infinite, then P does not admit any decidable nontrivial minimal extension Q . (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)