subtlecardinals

subtlecardinals - 1 SUBTLE CARDINALS AND LINEAR ORDERINGS...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 SUBTLE CARDINALS AND LINEAR ORDERINGS by Harvey M. Friedman Department of Mathematics Ohio State University friedman@math.ohio-state.edu www.math.ohio-state.edu/~friedman/ April 9, 1998 INTRODUCTION The subtle, almost ineffable, and ineffable cardinals were introduced in an unpublished 1971 manuscript of R. Jensen and K. Kunen, and a number of basic facts were proved there. These concepts were extended to that of k-subtle, k-almost ineffable, and k-ineffable cardinals in [Ba75], where a highly developed theory is presented. This important level of the large cardinal hierarchy was discussed in some detail in the survey article [KM78], section 20. However, discussion was omitted in the subsequent [Ka94]. This level is associated with the discrete/finite combinatorial independence results in [Fr98] and [Fr ]. In section 1 we give a self contained treatment of the basic facts about this level of the large cardinal hierarchy, which were established in [Ba75]. In particular, we give a proof that the k-subtle, k-almost ineffable, and k-ineffable cardinals define three properly intertwined hierarchies with the same limit, lying strictly above &total indescribability and strictly below &arrowing w . The innovation here is presented in section 2, where we take a distinctly minimalist approach. Here the subtle cardinal hierarchy is characterized by very elementary properties that do not mention closed unbounded or stationary sets. This development culminates in a characterization of the hierarchy 2 by means of a striking universal second order property of linear orderings. As is usual in set theory, we treat cardinals and ordinals as von Neumann ordinals. We use w for the first limit ordinal, which is also N. For sets X and integers k 1 , we let S k (X) be the set of all k element subsets of X. To orient the reader, we mention two results proved in this paper. The first is an important result from [Ba75] which is perhaps the simplest way of defining the k-ineffable cardinals from the point of view of set theoretic Ramsey theory. We say that l is k-SRP if and only if l is a limit ordinal, and for every f:S k ( l ) f {0,1}, there exists a stationary E l such that f is constant on S k (E). Here SRP stands for "stationary Ramsey property." In [Ba75] it is proved that for all k 0 and ordinals l , l is k-ineffable if and only if l is a regular cardinal with (k+1)-SRP. The second is a new result that is the most elementary way we know of defining the least k-subtle cardinal. We say that a linear ordering (X,<) is k-critical if and only if it has no endpoints, and: for all regressive f:X k f X, there exists b 1 < ... < b k+1 such that f(b 1 ,...,b k ) = f(b 2 ,...,b k+1 )....
View Full Document

Page1 / 52

subtlecardinals - 1 SUBTLE CARDINALS AND LINEAR ORDERINGS...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online