Regressive Partition Relations, n-Subtle Cardinals, and Borel Diagonalization

Kanamori, A., Regressive partition relations, n-subtle cardinals, and Bore1 diagonalization, Annals of Pure and Applied Logic 52 (1991) 65-77. We consider natural strengthenings of H. Friedman’s Bore1 diagonahzation propositions and characterize their consistency strengths in terms of the n-subtle cardinals. After providing a systematic survey of regressive partition relations and their use in recent independence results, we characterize n-subtlety in terms of such relations requiring only a finite homogeneous set, and then apply this characterization to extend previous arguments to handle the new Bore1 diagonahzation propositions. In previous papers [6,7] we showed how regressive partition relations provide a simplifying and unifying scheme for establishing the independence of the Paris-Harrington as well as the Friedman [3] propositions. In these contexts the more informative approach of using regressive partition relations to generate indiscemibles in models can replace the abstract diagonalization technique of Cantor and Gijdel for substantiating transcendence. Friedman’s proposition correlated with the n-Mahlo cardinals. Here we show how the regressive partition formulation leads directly to an extension that correlates with the n-subtle cardinals, far stronger in consistency strength. In Section 1 we provide a systematic survey of regressive partition relations, their use in independence results, and related open questions. In Section 2 we establish a regressive partition result about n-subtle cardinals, and finally in Section 3 we use it to motivate and characterize the aforementioned extension. 1. Regressive partition relations Let X be a set of ordinals and n E o. If f is a function with domain [Xl”, we write f(ab, . . . , CY~-~) for f({LYo,. . . , CX~_~}), with the understanding that 0168~0072/91/$03.50