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Supercompact cardinals and trees of normal ultrafilters1

Published online by Cambridge University Press:  12 March 2014

Julius B. Barbanel
Julius B. Barbanel*
Affiliation:
Union College, Schenectady, New York, 12308
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Extract

Supercompact cardinals are usually defined in terms of the existence of certain normal ultrafilters. It is well known that there is a natural partial ordering on the collection of all normal ultrafilters associated with a super-compact cardinal, that of normal ultrafilter restriction. Using this notion, we define a tree structure T on the collection of normal ultrafilters associated with a fixed supercompact cardinal. Many results already appearing in the literature can be conveniently phrased in terms of structural properties of T (see, e.g. [4] or [6]). In this paper, we establish additional structural facts concerning T.

In §1 we standardize our notation and review some of the basic facts and methods that will be used throughout. §2 begins with a presentation of an important technique, due to Solovay, which will be an important tool for us. Also in §2, we begin a detailed study of the structure of T in terms of branching and the existence of many successors to branches at limit levels. §3 contains results proving the existence of many nodes of T which do not have successors above certain levels of T. This complements work of Magidor [6] who established the existence of many nodes which have successors at all higher levels of T.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 47 , Issue 1 , March 1982 , pp. 89 - 109
Copyright
Copyright © Association for Symbolic Logic 1982

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Footnotes

1

This paper is part of the author's Ph.D. thesis [1] written at the State University of New York at Buffalo under the supervision of Professor Nicolas Goodman to whom the author is grateful.

References

REFERENCES

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