Iterating along a Prikry sequence
In this paper we introduce a new method which combines Prikry forcing with an iteration between the Prikry points. Using our method we prove from large cardinals that it is consistent that the tree property holds at אn for n ≥ 2, אω is strong limit and 2אω = אω+2. The typical method for interleaving collapses with a Prikry sequence is to use a product of collapses. The original paper using this method is Magidor’s paper [8] where he obtains the failure of the Singular Cardinals Hypothesis (SCH) at אω. There are many generalizations and variations on this method, examples include [9, 5, 6, 3]. In this paper we introduce a method for replacing the usual product with an iteration. Broadly speaking Prikry forcing is a poset of finite approximations to a witness that some large cardinal κ (and perhaps some cardinals above κ) is singular of cofinality ω. The finite approximations are often called the stem of a condition. To avoid collapsing κ, the growth of the stem is controlled by measure one sets from some appropriate measure. The key lemma which allows us to prove that κ is preserved is called the Prikry Lemma. The result of the forcing is a singular cardinal κ which is still large in the sense that it is a limit of inaccessible cardinals. Collapses can be added to Prikry forcing in order to make κ in to a small cardinal like אω. In particular we add collapses between the ordinals in the stem. We cannot only use a finite support product of collapses, since this would collapse κ. The solution is to constrain the values of future collapse conditions. For this we define constraining functions whose domains are measure one sets and whose values are elements of collapsing posets. To extend a condition, we select an element x from a measure one set which we add to the stem and we select a collapse condition which is below the value of the constraining function at x. This allows us to recover a version of the Prikry Lemma and show that again κ is preserved and the collapses have their desired effect. The usual method of adding collapses to a Prikry forcing uses a product of posets which collapse between elements of the stem. A trivial but important observation is that in the usual scenario the values of the constraining functions do not depend on the stem. In particular the collapses are defined in a uniform way and each condition only has one constraining function. When we seek to iterate between the elements of the stem, the value of a given constraining function must depend on the stem, because the stem determines the space from which the name for a constrained collapse condition is taken. To deal with this our technique has a complex system of constraining functions rather than just a single constraining function. One could think of our forcing as 2010 Mathematics Subject Classification. Primary 03E35, Secondary 03E55.