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

Journal of Symbolic Logic 68 (3): 713- 739 (2003)
Our main results are:Theorem 1. Con implies Con. [In fact equiconsistency holds.]Theorem 3. Con implies Con.Theorem 5. Con ”) implies Con.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
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DOI 10.2178/jsl/1058448434
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