SubtleLinear.040998

# 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 http://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April 9, 1998 INTRODUCTION The subtle, almost ineffable, and ineffable cardinals were introduced in an unpublished 1971 manuscript of R. Jensen and K. Kunen, and a number of basic facts were proved there.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 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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