## Journal of Symbolic Logic

### 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 ω2V is a limit of measurable cardinals in Jensen’s core model KMO for measures of order zero. Using related arguments we show that ω2V is a stationary limit of measurable cardinals in KMO, 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.

#### Article information

Source
J. Symbolic Logic, Volume 68, Issue 3 (2003), 713- 739.

Dates
First available in Project Euclid: 17 July 2003

https://projecteuclid.org/euclid.jsl/1058448434

Digital Object Identifier
doi:10.2178/jsl/1058448434

Mathematical Reviews number (MathSciNet)
MR2000073

Zentralblatt MATH identifier
1057.03040

#### Citation

Deiser, Oliver; Donder, Dieter. Canonical functions, non-regular ultrafilters and Ulam’s problem on ω 1. J. Symbolic Logic 68 (2003), no. 3, 713-- 739. doi:10.2178/jsl/1058448434. https://projecteuclid.org/euclid.jsl/1058448434

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