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Isols and maximal intersecting classes
Author(s) -
Dekker Jacob C. E.
Publication year - 1993
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.19930390110
Subject(s) - mathematics , transfinite number , cardinality (data modeling) , combinatorics , finite set , class (philosophy) , discrete mathematics , family of sets , equivalence (formal languages) , set (abstract data type) , mathematical analysis , artificial intelligence , computer science , data mining , programming language
In transfinite arithmetic 2 n is defined as the cardinality of the family of all subsets of some set v with cardinality n. However, in the arithmetic of recursive equivalence types (RETs) 2 N is defined as the RET of the family of all finite subsets of some set v of nonnegative integers with RET N. Suppose v is a nonempty set. S is a class over v , if S consists of finite subsets of v and has v as its union. Such a class is an intersecting class (IC) over v , if every two members of S have a nonempty intersection. An IC over v is called a maximal IC (MIC), if it is not properly included in any IC over v. It is known and readily proved that every MIC over a finite set v of cardinality n ≥ 1 has cardinality 2 n ‐1 . In order to generalize this result we introduce the notion of an ω‐MIC over v. This is an effective analogue ot the notion of an MIC over v such that a class over a finite set v is an ω‐MIC iff it is an MIC. We then prove that every ω‐MIC over an isolated set v of RET N ≥ 1 has RET 2 N ‐1 . This is a generalization, for while there only are χ 0 finite sets, there are ϰ isolated sets, where c denotes the cardinality of the continuum, namely all the finite sets and the c immune sets. MSC: 03D50.