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

Abstract

Our main results are: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.]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.

Bounding by Canonical Functions, With Ch

“…• for all countable X ≺ (H(χ), ∈, < χ ) with λ,Q ∈ X, and for all β < ω 1 there exists a countable z ⊂ λ such that [z] <ω is a subset of…”

“…Let X be a countable elementary substructure of (M, ∈, < * ). Let η be an ordinal in X and let z 0 , z 1 |b| → ω, defined by letting f a (y) = f (a ∪ y), is a function in X = Sk (M,∈,< * ) (X ∪ z 0 ), so f (a ∪ b) ∈ X ∩ ω 1 = X ∩ ω 1 .…”

Function Chains From Uniform Sigma-1-1 Well Orders

“…A characterization using schemes is given in [3]. Here, a fourth characterization from [1] will be given.…”

“…If α = β + n where β is a limit ordinal and n is an integer, let α (i) = β + 2n + i for i = 0, 1. Using obvious notation, Fld(C) = Fld(A) (0) ∪ Fld(B) (1) and α (i) , β (j) ∈ C iff i < j or i = j = 0 and α, β ∈ A or i = j = 1 and α, β ∈ B.…”

Covering theorems for the core model, and an application to stationary set reflection

“…3 (5) The property of being a premouse without a top extender is a Q-property. • n > 0, or • n = 0 and cr(E M top ) + M ≥ ht(Q ) then ult n ( M, σ ) is a premouse.…”