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ON THE TRANSITIVE HULL OF A κ‐NARROW RELATION
Author(s) -
Diener KarlHeinz,
Diener K.H.
Publication year - 1992
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.19920380137
Subject(s) - mathematics , corollary , transitive relation , axiom of choice , injective function , surjective function , countable set , hull , class (philosophy) , pure mathematics , relation (database) , axiom , discrete mathematics , combinatorics , set (abstract data type) , set theory , geometry , database , artificial intelligence , marine engineering , computer science , engineering , programming language
We will prove in Zermelo‐Fraenkel set theory without axiom of choice that the transitive hull R * of a relation R is not much “bigger” than R itself. As a measure for the size of a relation we introduce the notion of κ + ‐ narrowness using surjective Hartogs numbers rather than the usul injective Hartogs values. The main theorem of this paper states that the transitive hull of a κ + ‐narrow relation is κ + ‐narrow. As an immediate corollary we obtain that, for every infinite cardinal κ, the class HC κ of all κ‐hereditary sets is a set with von Neumann rank ϱ(HCκ) ≤ κ + . Moreover, ϱ(HCκ) = κ + if and only if κ is singular, otherwise ϱ(HCκ) = κ. The statements of the corollary are well known in the presence of the axiom of choice (AC). To prove them without AC ‐ as carried through here ‐ is, however, much harder. A special case of the corollary (κ = ω 1 , i.e., the class HCω 1 of all hereditarily countable sets) has been treated independently by T. JECH .