Skip to main content
×
Home
    • Aa
    • Aa

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

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

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[4] H.-D. Donder , Families of almost disjoint functions, Contemporary Mathematics, vol. 31 (1984), pp. 7178.

[7] H.-D. Donder and P. Koepke , On the consistency strength of ‘accessible’ Jonsson cardinals and of the weak Chang conjecture, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 233261.

[9] M. Gitik and S. Shelah , Forcing with ideals and simple forcing notions, Israel Journal of Mathematics, vol. 68 (1989), pp. 129160.

[12] A. Kanamori , Weakly normal filters and irregular ultrafilters, Transactions of the American Mathematical Society, vol. 220 (1976), pp. 393399.

[14] J. Ketonen , Non-regular ultrafilters and large cardinals. Transactions of the American Mathematical Society, vol. 224 (1976), pp. 6173.

[16] S. Shelah , Proper and improper forcing, 2nd ed., Perspectives in Mathematical Logic, Springer, Berlin, 1998.

[17] A.D. Taylor , Regularity properties of ideals and ultrafilters. Annals of Mathematical Logic, vol. 16 (1979), pp. 3355.

[20] W.H. Woodin , The axiom of determinacy, forcing axioms, and the nonstationary ideal, de Gruyter Series in Logic and Its Applications, vol. 1, Walter de Gruyter, Berlin, 1999.

[21] M. Zeman , Inner models and large cardinals, de Gruyter Series in Logic and Its Applications, vol. 5, Walter de Gruyter, Berlin, 2002.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 2 *
Loading metrics...

Abstract views

Total abstract views: 72 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 26th July 2017. This data will be updated every 24 hours.