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

Published online by Cambridge University Press:  12 March 2014

Oliver Deiser
Affiliation:
Mathematisches Institut der Lmu-München, Theresienstr, 33, D-80333 München, Germany, E-mail: deiser@rz.mathematik.uni-muenchen.de
Dieter Donder
Affiliation:
Mathematisches Institut der Lmu-München, Theresienstr, 33, D-80333 München, Germany, E-mail: donder@rz.mathematik.uni-muenchen.de

Abstract

Our main results are:

Theorem 1. Con(ZFC + “every function f: ω1ω1 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 of ω1-complete uniform filters on ω1 s.t. every A ⊆ ω1 is measurable w.r.t. a filter in (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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2003

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