Talk:Alternative Definition of Ordinal in Well-Founded Theory

Naming

New day, same problems. This definition only works if one accepts the axiom of foundation. Otherwise things like the set $a = \{ a \}$ end up being ordinals, which we do not want. --Dfeuer (talk) 21:20, 4 April 2013 (UTC)

Even so, a) it's a definition so the thing being defined needs to be in the definition namespace, b) the word "alternative" is lame, and this page should be called "equivalence of definitions ..." blah blah, c) if this page is part of a different schema of mathematical foundations from ZFC then the whole schema needs to be defined from ground up. It is inadequate to bolt bits on the side of the existing stuff by just flabbily defining it as "alternative". --prime mover (talk) 21:33, 4 April 2013 (UTC)

I have no idea what source defines it so, so I cannot put it on a definition page without risking your eagerness to delete things. This definition is acceptable under ZFC, but not under theories that exclude or negate AoF (Smullyan and Fitting, for example, explore various consequences of AoF, but don't accept it as an axiom, per se, and they note without any details that set theories with anti-foundation axioms have found application in computer science and linguistics). --Dfeuer (talk) 21:48, 4 April 2013 (UTC)