ON THE TRANSITIVE HULL OF A κ‐NARROW RELATION

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