# Talk:Mostowski collapse lemma

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WikiProject Mathematics (Rated Start-class, Mid-priority)

If you extend Mostowski collapse to well-founded (not just well-ordered) relations, surely the result is not an isomorphism! —Preceding unsigned comment added by 192.75.48.150 (talk) 19:36, 17 April 2008 (UTC)

Well-ordered relations are not involved in any version of Mostowski lemma. The result is an isomorphism for set-like well-founded extensional relations, which is the usual statement of the lemma, and what this article says. If you try to extend the lemma to nonextensional wellfounded relations then it obviously can't be an isomorphism. So what? — EJ (talk) 10:01, 18 April 2008 (UTC)
Actually, as a consequence of the lemma one obtains that any set-like (not necessarily extensional) well-founded relation is isomorphic to membership on a (not necessarily transitive) class. Neither the class nor the isomorphism is unique then. You can count this as an extension of Mostowski lemma to nonextensional relations which does give an isomorphism. — EJ (talk) 11:53, 18 April 2008 (UTC)
My mistake, I thought "extensional relation" meant something else entirely (basically, one which respects the intended equivalence relation). Thanks for spelling it out in the article. --192.75.48.150 (talk) 15:13, 18 April 2008 (UTC)

## generalization

The recently added "homomorphism version" of the lemma, and specifically its uniqueness part, is false. Consider the relation R = {(a,b), (b,c)} on X = {a, b, c}. We can map (X, R) homomorphically onto the transitive set {∅, {∅}, {{∅}}} via f(a) = ∅, f(b) = {∅}, f(c) = {{∅}}, but we can also map it homomorphically onto the transitive set {∅, {∅}, {∅,{∅}}} via f(c) = {∅,{∅}}. The problem is even more pronounced in nonwellfounded set theory, where any relation whatsoever can be mapped homomorphically to the transitive urelement x = {x}. Let alone the fact that Mostowski collapse lemma is Mostowski collapse lemma, not any of its generalizations. — Emil J. (formerly EJ) 09:46, 14 July 2008 (UTC)

## The name

Shouldn't this name Mostowski's collapse lemma? —Preceding unsigned comment added by 83.6.96.216 (talk) 17:11, 31 August 2008 (UTC)