Talk:Hereditary set

In the process of copyediting hereditary set, I found myself writing the sentence

In non-well-founded set theories where such objects are allowed, a set that contains only itself is also a hereditary set.

It then occurred to me not only that this may or may not be true, but that it might not even be a meaningful statement. Consider the set E = {E}. By the definition of hereditary sets, if E is hereditary, then {E} is hereditary, which merely restates the initial premise. If E isn't hereditary, then E isn't hereditary, again restating the inital premise. I can't see how to get a better handle on this problem. Can anyone help? -- The Anome (talk) 14:59, 15 January 2010 (UTC)

EmilJ replied to this as follows on the Wikipedia:Reference desk/Mathematics:

The usual way to unambiguously phrase such definitions in non-well-founded set theories is to define that A is a hereditary xxx iff every object in the transitive closure of {A} is a xxx (note that this is equivalent to the inductive definition if the universe is well-founded). Your E is thus indeed a hereditary set. — Emil J. 15:09, 15 January 2010 (UTC)

Can anyone help update the article to reflect this? I'm afraid I'm outside my area of competence. -- The Anome (talk) 15:38, 15 January 2010 (UTC)

Proposal to merge "Hereditarily finite set" and "Hereditarily countable set" to here

Proposal to merge Hereditarily finite set and Hereditarily countable set into Hereditary set.

104.228.101.152 (talk) 13:58, 9 November 2019 (UTC)

• No: You are treating these articles as if "Hereditary" is the main point. It is not. JRSpriggs (talk) 05:12, 10 November 2019 (UTC)
  • Okay, should I remove the merge proposals? Just asking for input before I make yet another change to the articles. Also, your reply would add to the history of the talk page. 104.228.101.152 (talk) 22:41, 11 December 2019 (UTC)

    Yes, in my opinion, you should remove the {{merge to}} templates from hereditarily finite set and hereditarily countable set, with an edit summary stating "merge proposal withdrawn by nominator" or some language to that effect.
    To expand on what JR said, the notion of being a hereditarily finite/countable set is quite a different thing from the notion of being a hereditary set. The former notion is more or less a size constraint (not just at the top level, but the whole tree), whereas the second is about whether you allow urelements. These are more or less orthogonal considerations. --Trovatore (talk) 03:53, 12 December 2019 (UTC)

    Done. 104.228.101.152 (talk) 01:40, 13 December 2019 (UTC)

    Thanks! --Trovatore (talk) 03:05, 13 December 2019 (UTC)