Premium
On hereditarily small sets in ZF
Author(s) -
Holmes M. Randall
Publication year - 2014
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201300089
Subject(s) - mathematics , countable set , transitive relation , set (abstract data type) , transitive closure , closure (psychology) , combinatorics , discrete mathematics , computer science , economics , market economy , programming language
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 = ω 1(where the collection under consideration is the set of hereditarily countable sets).