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Totally non‐immune sets
Author(s) -
Tzouvaras Athanassios
Publication year - 2015
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.201400006
Subject(s) - countable set , mathematics , converse , subclass , discrete mathematics , class (philosophy) , set (abstract data type) , generalization , combinatorics , infinite set , computer science , artificial intelligence , mathematical analysis , geometry , antibody , immunology , biology , programming language
Let L be a countable first‐order language and M = ( M , ... ) be an L ‐structure. “Definable set” means a subset of M which is L ‐definable in M with parameters. A set X ⊆ M is said to be immune if it is infinite and does not contain any infinite definable subset. X is said to be partially immune if for some definable A , A ∩ X is immune. X is said to be totally non‐immune if for every definable A , A ∩ X and A ∩ ( M ∖ X ) are not immune. Clearly every definable set is totally non‐immune. Here we ask whether the converse is true and prove that it is false for every countable structure M whose class of definable sets satisfies a mild condition. We investigate further the possibility of an alternative construction of totally non‐immune non‐definable sets with the help of a subclass of immune sets, the class of cohesive sets, as well as with the help of a generalization of definable sets, the semi‐definable ones (the latter being naturally defined in models of arithmetic). Finally connections are found between totally non‐immune sets and generic classes in nonstandard models of arithmetic.