Talk:Hereditary set

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Mathematics rating:
Start Class
Mid Importance
 Field: Foundations, logic, and set theory

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.[citation needed]

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)