Canonical functions, non-regular ultrafilters and Ulam's problem on ω

Journal of Symbolic Logic 68 (3): 713- 739 (2003)
Abstract
Our main results are: Theorem 1. Con(ZFC + "every function $f : \omega_{1} \rightarrow \omega_1$ is dominated by a canonical function") implies Con(ZFC + "there exists an inaccessible limit of measurable cardinals"). [In fact equiconsistency holds.] Theorem 3. Con(ZFC + "there exists a non-regular uniform ultrafilter on ω1") implies Con(ZFC + "there exists an inaccessible stationary limit of measurable cardinals"). Theorem 5. Con (ZFC + "there exists an $\omega_{1}-sequence$ T of $\omega_{1}-complete$ uniform filters on ω1 s.t. every $A \subseteq \omega_1$ is measurable w.r.t. a filter in T (Ulam property)") implies Con(ZFC + "there exists an inaccessible stationary limit of measurable cardinals"). We start with a discussion of the canonical functions and look at some combinatorial principles. Assuming the domination property of Theorem 1, we use the Ketonen diagram to show that $\omega_{2}^V$ is a limit of measurable cardinals in Jensen's core model $K_{MO}$ for measures of order zero. Using related arguments we show that $\omega_{2}^V$ is a stationary limit of measurable cardinals in $K_{MO}$ , if there exists a weakly normal ultrafilter on ω1. The proof yields some other results, e.g., on the consistency strength of weak*-saturated fiters on ω1, which are of interest in view of the classical Ulam problem
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