Subtle cardinals and linear orderings

✍ Scribed by Harvey M. Friedman

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 228 KB
• Volume: 107
• Category: Article
• ISSN: 0168-0072
• DOI: 10.1016/s0168-0072(00)00019-1

✦ Synopsis

The subtle, almost ine able, and ine able cardinals were introduced in an unpublished 1971 manuscript of R. Jensen and K. Kunen. The concepts were extended to that of k-subtle, k-almost ine able, and k-ine able cardinals in 1975 by J. Baumgartner. In this paper we give a self contained treatment of the basic facts about this level of the large cardinal hierarchy, which were established by J. Baumgartner. In particular, we give a proof that the k-subtle, k-almost ine able, and k-ine able cardinals deÿne three properly intertwined hierarchies with the same limit, lying strictly above "total indescribability" and strictly below "arrowing !".

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 (k-critical).