Talk:Supercompact cardinal

From Wikipedia, the free encyclopedia

[edit] Do supercompact cardinals exist?

The article says "M contains all of its λ-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 would have to be in M. Thus κ would have to be greater than λ. 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 κ>λ. Perhaps you meant j(κ)>λ? --Aleph4 14:58, 7 March 2007 (UTC)

Perhaps I am misunderstanding part of the definition, maybe what ^λM is?

I am assuming that it means the class of all functions (in V) from λ to M. Suppose κ is less than the least regular ordinal greater than λ. Then the cofinality of κ (which is a regular ordinal less than or equal to κ) must be less than or equal to λ itself. Choose a cofinal sequence of ordinals approaching κ from below. Then extend (if necessary), that sequence with zeros to make it of length λ. The range of this sequence must be in M and is a set of ordinals whose supremum (in V) is κ. The union of that set must be in M, but it is also equal to κ 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 is greater than or equal to the least regular ordinal greater than λ. Consequently, κ is greater than λ. But for a supercompact cardinal, any λ must work. So κ 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 less than κ.

I do not see what j(κ) 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:"...κ which is not in M because it is the critical point of j". Kope 15:22, 9 March 2007 (UTC)

κ is not in j[V], but κ 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 κ < j(κ) [rather than an ordinal outside the model]. JRSpriggs 07:35, 10 March 2007 (UTC)