# Randall Holmes: On hereditarily small sets

Tuesday, January 27 from 2 to 3pm
Room: Mathematics 136
Speaker: Randall Holmes (BSU)
Title: On hereditarily small sets

Abstract: Jech showed in 1982 that $H(\omega_1)$, the collection of hereditarily countable sets, is a set in ZF. In the absence of choice, the existence of $H(\omega_1)$ is not obvious; but he showed that the rank of any hereditarily countable set is less than $\omega_2$. Forster has remarked that Jech’s proof generalizes to show the existence of $H(\kappa)$ when $\kappa$ is an $\aleph$ cardinal (a cardinal of well-orderable sets). We have shown the existence of $H(\kappa)$, the collection of all sets hereditarily of size less than $\kappa$, in ZF without Choice, for any cardinal $\kappa$ at all, by a not altogether trivial generalization of Jech’s technique.