## SCH above a compact cardinal

January 20, 2011 at 20:41 | Posted in Uncategorized | 1 CommentTags: compact cardinal, SCH

This is really just an experiment to see if I can actually get latex2wp to work for me, but maybe it has mathematical interest as well. Anyway, it is well-known that Shelah’s pcf theory can be used to give a proof of Solovay’s result that the Singular Cardinals Hypothesis holds above a compact cardinal. What follows is a fairly short proof of a more general result; the argument consists of chaining together some ideas of Shelah taken from his *Cardinal Arithmetic*.

See “proof of 3.3 in Case gamma” and Remark 3.3B on pages 150-151 of his book.

Theorem 1Suppose is singular of countable cofinality and . Then there is no uniform countably complete ultrafilter on .

*Proof:* We actually show more: we prove that there is no weakly -saturated uniform countably complete filter on . Said another way, if is a uniform countably complete filter on for singular of countable cofinality and , then we can find disjoint -positive subsets of . In particular, cannot be an ultrafilter.

To understand the proof, we don’t really need to know what means. What we really need is one of the combinatorial consequences of this statement, namely that there is a family of countable subsets of (not !) such that for every , the collection is essentially disjoint, in the sense that there is a function with domain such that

- is a finite subset of , and
- the collection is pairwise disjoint.

Thus, for each the collection can be “disjointified” by removing a finite subset from each .

Note that since the are subsets of , no -sized subcollection of can be disjointified, as this would give us disjoint subsets of . One should think of as a failure of “compactness” at which necessarily arises from the failure of the singular cardinals hypothesis at .

Let us assume now that is as above, and is a uniform countably complete filter on our cardinal ; we will show contains disjoint -positive subsets. Let us also assume that we have fixed some way of enumerating each , so that expressions such as “the first 5 elements of ” make sense.

Given and , define to be those greater than for which is contained in the first elements of . For each , the sequence is increasing with union . Since is uniform and countably complete, it follows that there is an such that is -positive.

For each , let be the “th element” of . Then clearly the set

is -positive for each , as includes .

We know for each , and so there exists an such that

Given , we define

We are now done, as given in , we see that , and therefore the collection is a collection of disjoint -positive subsets of .

Corollary 2If where is strongly compact and is a singular cardinal of countable cofinality, then .

*Proof:* The co-bounded filter on is a uniform -complete filter. Since is strongly compact, this filter can be extended to a -complete ultrafilter (necessarily uniform) on . The conclusion now follows immediately from the theorem.

Corollary 3If is strongly compact, then the Singular Cardinals Hypothesis holds above .

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In fact the construction of the sets gives that each one is a cofinal subset of with order type , so that “the first five elements of ” already makes

perfect sense.

Comment by James Cummings— April 12, 2011 #