Second-order Replacement and Collection for H(X) in ZF

Ascertain whether, for an arbitrary set X, the class H(X) (defined as the union of all transitive sets M that are surjective images of some S ∈ X) satisfies Second-order Replacement and Second-order Collection in ZF; establish general conditions under which these axioms hold or fail for H(X).

Background

The paper discusses several choiceless definitions of H(X) and proves that H(X) is always a set in ZF, with basic closure properties yielding a model of Second-order Zermelo set theory under mild hypotheses. However, the status of Second-order Replacement and Collection for H(X) is left unresolved in general, and examples indicate that Collection may fail even when Replacement holds in choiceless contexts.