Abstract

We will prove that some so-called union theorems are equivalent in ZF0 to statements about the transitive closure of relations. The special case of “bounded” union theorems dealing with κ-hereditary sets yields equivalents to statements about the transitive closure of κ-narrow relations. The instance κ = ω1 yields an equivalent to Howard-Rubin's Form 172 of every hereditarily countable set x is countable). In particular, the countable union theorem and, a fortiori, the axiom of countable choice imply Form 172.