Our main results are:
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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.]
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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”).
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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”).
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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.