Canonical functions, non-regular ultrafilters and Ulam’s problem on ω1

Abstract

Our main results are:

Theorem 1. Con(ZFC + “every function f : ω₁ → ω₁ 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 ω₁-sequence ℱ of ω₁-complete uniform filters on ω₁ s.t. every A ⊆ ω₁ is measurable w.r.t. a filter in ℱ (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 ω₂^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 ω₂^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 filters on ω₁, which are of interest in view of the classical Ulam problem.