Two Results in Combinatorial Set Theory

In this note we sketch the proofs of two results in combinatorial set theory. The common theme of the results is singular cardinal combinatorics: they involve the interaction between forcing, large cardinals, PCF theory and versions of Jensen’s weak square principle. 1. Changing cofinality Our first result involves an old question about changes of cofinality. Suppose that V,W are inner models of set theory with V ⊆ W , that κ is a V -cardinal and that (κ) = (κ) . The question is whether it is necessarily true that W |= cf(κ) = cf(|κ|). Let ∗(κ) denote the assertion “In every outer model in which κ is still a cardinal, cf(κ) = cf(|κ|)”. We summarise some known results: (1) (Shelah [14]): If κ is regular, or κ is singular and ∗κ holds, then ∗(κ). (2) (Cummings [4]): If κ is singular and there is a good scale of length κ at κ then ∗(κ). Shelah [14] used the result that κ regular implies ∗(κ) to show that for every cardinal λ, any saturated ideal on λ must concentrate on points of cofinality cf(λ). The results of [14] were used by Burke and Matsubara [2] and Foreman and Magidor [7] to constrain saturated ideals on Pκλ. The property ∗(κ) is also relevant to Chang’s Conjecture and problems in PCF theory. For the rest of this discussion we focus on the case κ = אω. It is easy to see that forcing with the Levy collapse Coll(ω,אω) will produce a generic extension in which אω+1 = א V [G] 1 , so the question is whether there can be an outer model W such that אω+1 = א W n for 1 < n < ω. Fact 1.1 (Cummings [4]). Let V ⊆ W with אVω+1 = א W n for some n with 1 < n < ω. Then 2000 Mathematics Subject Classification. 03E55.