Von Rimscha's Transitivity Conditions

In Zermelo-Fraenkel set theory with the axiom of choice every set has the same cardinal number as some ordinal. Von Rimscha has weakened this condition to “Every set has the same cardinal number as some transitive set.” In set theory without the axiom of choice, we study the deductive strength of this and similar statements introduced by von Rimscha. We shall use the standard notation and terminology of set theory. In particular we recall that a set x is transitive if for every y ∈ x and every t ∈ y, t ∈ x. The transitive closure, TC(x), of a set x is the smallest transitive set z such that x ⊆ z. In addition, for any set x, we will use TC′(x) to stand for TC(x) ∪ {x}. Transitive sets play an important role in set theory. The ordinal numbers, for example, are transitive sets and under the assumption of the axiom of choice (which we shall denote by AC) given any set x, there is a bijection from x to some ordinal. In [R] von Rimscha introduces a similar statement, Tr, weakened by replacing the word “ordinal” with the words “transitive set”. Tr. ∀x∃u∃f such that u is transitive and f is a bijection from x onto u. Two strengthenings of Tr are also considered in [R]: Tr′. ∀x∃u∃f such that u is transitive, u ⊆ TC(x), and f is a bijection from x onto u. Tr′′. ∀x∃u∃f such that u is transitive, f is a bijection from x onto u, and ∀s ∈ x, f(s) ∈ TC′. Working in Zermelo-Fraenkel set theory (ZF), without AC, von Rimscha shows that AC implies Tr′′, Tr′′ implies Tr′, Tr′ implies Tr, and Tr implies that every Dedekind finite set is finite. We prefer to work in ZF, where ZF is modified to allow the existence of atoms. The same proofs hold with minor modifications. Von 1991 Mathematics Subject Classification. 03E25, 03E35, 03E50, 04A25, 04A30.