On the transitive Hull of a κ-narrow relation

Mathematical Logic Quarterly 38 (1):387-398 (1992)
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Abstract

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 HCκ 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 = ω1, i.e., the class HCω1 of all hereditarily countable sets) has been treated independently by T. JECH.

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Model theory for infinitary logic.H. Jerome Keisler - 1971 - Amsterdam,: North-Holland Pub. Co..
Cylindric Algebras. Part I.Leon Henkin, J. Donald Monk, Alfred Tarski, L. Henkin, J. D. Monk & A. Tarski - 1985 - Journal of Symbolic Logic 50 (1):234-237.
Basic Set Theory.William Mitchell - 1981 - Journal of Symbolic Logic 46 (2):417-419.
On hereditarily countable sets.Thomas Jech - 1982 - Journal of Symbolic Logic 47 (1):43-47.
Book Review. Basic Set Theory. Azriel Levy. [REVIEW]Harold T. Hodes - 1981 - Philosophical Review 90 (2):298-300.

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