Talk:Supercompact cardinal

Do supercompact cardinals exist?
The article says "M contains all of its &lambda;-sequences". If that were true, then it must contain their ranges (images) as well. So in particular, any ordinal less than the next regular ordinal after &lambda; would have to be in M. Thus &kappa; would have to be greater than &lambda;. But no ordinal is greater than all ordinals, so no ordinal could be supercompact. JRSpriggs 09:08, 7 March 2007 (UTC)
 * I am not sure why you conclude that &kappa;>&lambda;. Perhaps you meant j(&kappa;)>&lambda;? --Aleph4 14:58, 7 March 2007 (UTC)


 * Perhaps I am misunderstanding part of the definition, maybe what &lambda;M is?
 * I am assuming that it means the class of all functions (in V) from &lambda; to M. Suppose &kappa; is less than the least regular ordinal greater than &lambda;. Then the cofinality of &kappa; (which is a regular ordinal less than or equal to &kappa;) must be less than or equal to &lambda; itself. Choose a cofinal sequence of ordinals approaching &kappa; from below. Then extend (if necessary), that sequence with zeros to make it of length &lambda;. The range of this sequence must be in M and is a set of ordinals whose supremum (in V) is &kappa;. The union of that set must be in M, but it is also equal to &kappa; which is not in M because it is the critical point of j &mdash; a contradiction.
 * Since the supposition leads to a contradiction, it must be false. That is, &kappa; is greater than or equal to the least regular ordinal greater than &lambda;. Consequently, &kappa; is greater than &lambda;. But for a supercompact cardinal, any &lambda; must work. So &kappa; is greater than any ordinal &mdash; another contradiction. So supercompact cardinals do not exist.
 * A possible way out is to restrict the definition to all &lambda; less than &kappa;.
 * I do not see what j(&kappa;) has to do with the issue. JRSpriggs 05:31, 8 March 2007 (UTC)


 * It is nowhere claimed that j is surjective, and indeed it is not. Kope 10:49, 8 March 2007 (UTC)


 * I agree that j is not onto M (and certainly not onto V). But I do not see what that has to do with what I was saying. JRSpriggs 05:42, 9 March 2007 (UTC)


 * You wrote:"...&kappa; which is not in M because it is the critical point of j". Kope 15:22, 9 March 2007 (UTC)


 * &kappa; is not in j[V], but &kappa; is in M. In the context of elementary embeddings, M usually denotes a transitive models containing all the ordinals. --Aleph4 15:53, 9 March 2007 (UTC)


 * Sorry, I was very confused (I thought I knew this stuff). My thanks are due to Aleph4 for reminding me that these models M are transitive and thus contain all the ordinals. Even though I wrote the article on critical point (set theory), I forgot that the critical point is simply the least ordinal for which &kappa; < j(&kappa;) [rather than an ordinal outside the model]. JRSpriggs 07:35, 10 March 2007 (UTC)