On Closed Unbounded Sets Consisting of Former Regulars

A method of iteration of Prikry type forcing notions as well as a forcing for adding clubs is presented. It is applied to construct a model with a measurable cardinal containing a club of former regulars, starting with o(κ) = κ+ 1. On the other hand, it is shown that the strength of above is at least o(κ) = κ. Suppose that κ is an inaccessible cardinal. We wish to find a generic extension (usually cardinal preserving) such that {α < κ | α is regular in V } contains a club. Radin [Ra] introduced a basic method to do this. Simply start with a measurable κ with o(κ) = κ and then force with the Radin forcing constructed from o(κ) = κ. If one wishes to keep κ a measurable in the extension, then a weak repeat point suffices. Both facts are proved by Mitchell [Mi1], We show how to reduce assumptions rendering the above possible. A method of iteration generalizing those of [Gi1] is presented. Then a variant of it is used to iterate forcing for shooting clubs. We think that this method of iteration can be applied to other distributive forcings as well. We like to thank the referee for pointing out that the proof of section 2 gives only o(κ) = κ and not o(κ)κ + 1 as was claimed in the previous version, for long and detailed list of corrections and for his requests on explaining certain parts of the paper. 1. A Forcing Construction We will now prove the following two theorems. Theorem 1.1. Suppose that κ is an inaccessible cardinal such that for every δ < κ the set of α’s below κ with o(α) ≥ δ is stationary. Then there is a cardinal preserving generic extension such that the set {α < κ | α is regular in V } contains a club. Theorem 1.2. Suppose that κ is a measurable cardinal with o(κ) = κ+1. Then there is a cardinal preserving extension satisfying the following: (1) κ is a measurable, (2) {α < κ | α is a regular in V contains a club}. The proofs of these theorems use an iteration Prikry type forcing notion that was introduced in [Gi1]. Basically for every α < κ with α > o(α) > o we are forcing a Prikry or Magidor sequence to α without adding new bounded subsets. The order type of the sequence is ω, where the exponentiation is the ordinal one. The forcing used for this is 〈P(α, o(α)),≤,≤〉