Talk:Hereditarily countable set

Wrong definition?

...a set is called hereditarily countable if and only if its transitive closure is a countable set. If the axiom of countable choice holds, then a set is hereditarily countable if and only if it is a countable set of hereditarily countable sets...

Well, okay, I think I see the appeal to countable choice, but I don't understand why, in the absence of countable choice, "countable set of hereditarily countable sets" wouldn't be what we really mean by "hereditarily countable set". If we're concerned that this isn't a proper definition, then why couldn't we use some sort of "disjoint transitive closure"? There are many choices seeing as we only care about its countability. And of course the countable disjoint union of countable sets is countable. --Unzerlegbarkeit (talk) 20:41, 4 June 2008 (UTC)

You appear to be correct, if we judge by Jech. I added a link to a paper by him on this subject. I cannot remember any more where I got the idea that it was the other way. Please feel free to fix the article. JRSpriggs (talk) 04:20, 5 June 2008 (UTC)