It is easy to prove in ZF− that a relation R satisfies the maximal condition if and only if its transitive hull R* does; equivalently: R is well-founded if and only if R* is. We will show in the following that, if the maximal condition is replaced by the chain condition, as is often the case in Algebra, the resulting statement is not provable in ZF− anymore . More precisely, we will prove that this statement is equivalent in ZF− to (...) the countable axiom of choice ACω. Moreover, applying this result we will prove that the axiom of dependent choices, restricted to partial orders as used in Algebra, already implies the general form for arbitrary relations as formulated first by Teichmüller and, independently, some time later by Bernays and Tarski. (shrink)

We show the existence of a high r. e. degree bounding only joins of minimal pairs and of a high2 nonbounding r. e. degree. MSC: 03D25.

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. (shrink)