On hereditarily small sets in ZF

We show in ZF (the usual set theory without Choice) that for any set X, the collection of sets Y such that each element of the transitive closure of {Y } is strictly smaller in size than X (the collection of sets hereditarily smaller than X) is a set. This result has been shown by Jech in the case X = ω1 (where the collection under consideration is the set of hereditarily countable sets). In [2], Thomas Jech showed that the collection of hereditarily countable sets exists in ZF by showing that it has ordinal rank ≤ ω2. In [1], Thomas Forster observes in effect that this result can be extended to show the existence of H(κ) for any aleph κ: the result he actually proves is in the context of NF and stated differently, but the reasoning is transferable. In this note, we demonstrate the existence for any set X of the set H(|X|) of all sets which are hereditarily of size less than |X|, by generalizing Jech’s technique of establishing a bound on the ranks of the elements of this collection. The generalization is not altogether trivial. We give two different proofs, one being the way we first established the result, and one emulating Jech’s original argument for the existence of H(א1) more closely. We work in ZF, in which we have all the usual axioms of set theory except Choice. Definition (transitive closure): We define TC(A) for any set A as the intersection of all transitive sets which include A as a subset (so A 6∈ TC(A); when we want to include A we consider TC({A})). Definition (ordinals, order types): Ordinals for us are the usual von Neumann ordinals. We will take well-orderings to be non-strict and define