Talk:Critical point (set theory)

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)

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)

I now think that this is incorrect, because if an elementary embedding comes from an ultrapower, then its range cannot contain the ultrafilter generating the ultrapower and that would contradict many of the large cardinal properties such as huge cardinal and rank-into-rank cardinal. JRSpriggs (talk) 00:45, 15 March 2018 (UTC)

I think it's almost correct.

You can do it (I think — it's been a long time since I've touched this stuff) with an "ultrapower" in the appropriate sense, but not an ultrapower by a single ultrafilter or measure. You need an ultrapower by an extender. That works out to a directed system of ultrafilters, and your final ultrapower is the direct limit of ultrapowers along the way.

Then, analogously to the way that the ultrapower by a particular ultrafilter cannot contain that ultrafilter, an ultrapower by an extender cannot contain that extender (I think). But I am pretty sure the extender notion is flexible enough to capture all elementary embeddings that can be defined from a set parameter, and that (with some quibbles) all known (consistent) large-cardinal axioms can be expressed in terms of the existence of extenders.

However the version in our article may not be that flexible, as it says somewhere that λ is at most j(κ); I think that excludes so-called "long extenders". The analysis gets more complicated when you allow long extenders, but the basic definition is the same. You probably need long extenders to talk about superstrong and up. But again, that's fairly off the cuff, and it's been many many years since I've looked at it. --Trovatore (talk) 08:38, 15 March 2018 (UTC)