The Calculus of Partition Sequences, Changing Cofinalities, and a Question of Woodin

We study in this paper polarized infinite exponent partition relations. We apply our results to constructing a model for the theory “ZF+DC+ω1 is the only regular, uncountable cardinal ≤ ωω1+1.” This gives a partial answer to a question of Woodin. In 1994, J. Steel proved what had long been suspected: that assuming AD, the regular cardinals of L(R) below Θ are all measurable, where Θ is the least ordinal onto which the real numbers R cannot be mapped [St3]. This remarkable theorem motivated the work of [A], which showed essentially that the cofinality of any regular cardinal below Θ can be changed to ω without perturbing cardinal structure. Specifically, the following theorem was proved. Theorem 0.1. Assume V′ AD, and let V = L(R)V . Then for any subset A of the regular, uncountable cardinals below Θ, there is a partial ordering P ∈ V and a symmetric inner model N of ZF such that (1) V ⊆ N ⊆ VP , (2) N and V contain the same cardinals, (3) Θ = Θ, (4) for all κ, cof(κ) = ω if cof(κ) ∈ A and cof(κ) = cof(κ) otherwise, and (5) the measurable cardinals not in A remain measurable in N. What was left unanswered is whether similar facts can be proven if the cofinality of κ < Θ is to be made uncountable. Consideration of this question was the genesis of this paper. The principal tool, infinite–exponent partition sequences, proved as interesting as its application. We begin, in the first three sections, with background material on infinite– exponent partition properties of cardinals, their abundance in models of AD, and their use in “Magidor–like” forcing to change cofinalities. The second three sections deal with infinite–exponent partition properties of cardinal sequences. With appropriate notation, the basic theory of partition sequences and the associated Magidor–like forcing exactly parallels the cardinal theory. In “Finite Support,” we apply the theory to show the extent to which possibly uncountable cofinal sequences can be added if we sacrifice DC. In “Countable Support,” we preserve DC. Received by the editors September 30, 1997. 1991 Mathematics Subject Classification. Primary 03E15, 03E35, 03E60. The first author’s research was partially supported by PSC–CUNY grant 665337. c ©1999 American Mathematical Society