SubtleLinear.040998 - 1 SUBTLE CARDINALS AND LINEAR...

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1 SUBTLE CARDINALS AND LINEAR ORDERINGS by Harvey M. Friedman Department of Mathematics Ohio State University April 9, 1998 INTRODUCTION The subtle, almost ineffable, and ineffable cardinals were introduced in an unpublished 1971 manuscript of R. Jensen and K. Kunen, and a number of basic facts were proved there.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 by means of a striking universal second order property of linear orderings.
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2 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 ) Æ {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 Æ X, there exists b 1 < . .. < b k+1 such that f(b 1 ,...,b k ) = f(b 2 ,...,b k+1 ). We prove that for k ≥ 0, the least k-subtle cardinal is the least cardinality of a (k+1)-critical linear ordering.
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SubtleLinear.040998 - 1 SUBTLE CARDINALS AND LINEAR...

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