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Canonical functions, non-regular ultrafilters and Ulam's problem on ω 1

  • Oliver Deiser (a1) and Dieter Donder (a2)
Abstract

Our main results are:

Theorem 1. Con(ZFC + “every function f: ω 1ω 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 ω 1”) implies Con(ZFC + “there exists an inaccessible stationary limit of measurable cardinals”).

Theorem 5. Con (ZFC + “there exists an ω 1-sequence of ω 1-complete uniform filters on ω 1 s.t. every A ⊆ ω 1 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 ω 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 ω 2 V is a stationary limit of measurable cardinals in KMO , if there exists a weakly normal ultrafilter on ω 1. The proof yields some other results, e.g., on the consistency strength of weak*-saturated filters on ω 1, which are of interest in view of the classical Ulam problem.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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