Talk:Supercompact cardinal

Do supercompact cardinals exist?[edit]

The article says "M contains all of its λらむだ-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 λらむだ would have to be in M. Thus κかっぱ would have to be greater than λらむだ. But no ordinal is greater than all ordinals, so no ordinal could be supercompact. JRSpriggs 09:08, 7 March 2007 (UTC)[reply]

I am not sure why you conclude that κかっぱ>λらむだ. Perhaps you meant j(κかっぱ)>λらむだ? --Aleph4 14:58, 7 March 2007 (UTC)[reply]

Perhaps I am misunderstanding part of the definition, maybe what ^λらむだM is?

I am assuming that it means the class of all functions (in V) from λらむだ to M. Suppose κかっぱ is less than the least regular ordinal greater than λらむだ. Then the cofinality of κかっぱ (which is a regular ordinal less than or equal to κかっぱ) must be less than or equal to λらむだ itself. Choose a cofinal sequence of ordinals approaching κかっぱ from below. Then extend (if necessary), that sequence with zeros to make it of length λらむだ. The range of this sequence must be in M and is a set of ordinals whose supremum (in V) is κかっぱ. The union of that set must be in M, but it is also equal to κかっぱ which is not in M because it is the critical point of j — a contradiction.

Since the supposition leads to a contradiction, it must be false. That is, κかっぱ is greater than or equal to the least regular ordinal greater than λらむだ. Consequently, κかっぱ is greater than λらむだ. But for a supercompact cardinal, any λらむだ must work. So κかっぱ is greater than any ordinal — another contradiction. So supercompact cardinals do not exist.

A possible way out is to restrict the definition to all λらむだ less than κかっぱ.

I do not see what j(κかっぱ) has to do with the issue. JRSpriggs 05:31, 8 March 2007 (UTC)[reply]

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

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)[reply]

You wrote:"...κかっぱ which is not in M because it is the critical point of j". Kope 15:22, 9 March 2007 (UTC)[reply]

κかっぱ is not in j[V], but κかっぱ 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)[reply]

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 κかっぱ < j(κかっぱ) [rather than an ordinal outside the model]. JRSpriggs 07:35, 10 March 2007 (UTC)[reply]