Talk:Critical point (set theory)

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[edit] Large cardinal properties belong to their critical points

Many of the large cardinal properties assert the existence of elementary embeddings, including the largest (strongest) of them. Often these properties involve two or more cardinal (or ordinal) numbers. Nonetheless, I believe that it is the critical point of the elementary embedding which is the cardinal number which the property implies is a very strong limit, even though some of the other cardinals may be larger. Unfortunately, I do not know how to prove or even formalize this idea, so I am putting it here in the comments only rather than the text of the article. JRSpriggs 06:28, 6 May 2006 (UTC)

[edit] Decomposition of elementary embeddings

I think (but I am not positive) that any non-trivial elementary embedding of one standard inner model of ZFC into another can be decomposed into an ultrapower followed by another (possibly trivial) elementary embedding. The ultrapower having the same critical point as the original embedding; and the other embedding, if non-trivial, having a larger critical point. JRSpriggs 05:05, 8 May 2006 (UTC)