Pmax VARIATIONS FOR SEPARATING CLUB GUESSING PRINCIPLES

In his book on Pmax [6], Woodin presents a collection of partial orders whose extensions satisfy strong club guessing principles on ω1. In this paper we employ one of the techniques from this book to produce Pmax variations which separate various club guessing principles. The principle (+) and its variants are weak guessing principles which were first considered by the second author [3] while studying games of length ω1. It was shown in [1] that the Continuum Hypothesis does not imply (+) and that (+) does not imply the existence of a club guessing sequence on ω1. In this paper we give an alternate proof of the second of these results, using Woodin’s Pmax technology. We also present a variation which produces a model with a ladder system which weakly guesses each club subset of ω1 club often. We show that this model does not satisfy the Interval Hitting Principle, thus separating these statements. The main technique in this paper, in addition to the standard Pmax machinery, is the use of condensation principles to build suitable iterations. In Chapter 8 of his book on Pmax [6], Woodin presents a collection of Pmax variations whose extensions satisfy strong club guessing principles on ω1, along with the statement that the nonstationary ideal on ω1 (NSω1) is saturated (see pages 499-500, for instance). In this paper we employ one of the techniques from that chapter to produce Pmax variations which separate various club guessing principles. The arguments and results in this paper are significantly simpler than the ones used there. The separation of club guessing principles is carried out via iterations; no local forcing arguments are used. We present these iteration arguments in full and outline the way in which they are incorporated in the standard Pmax machinery. The principle (+) and its variants are weak guessing principles which were first considered by the second author [3] while studying games of length ω1. It was shown in [1] that the Continuum Hypothesis does not imply (+) and that (+) does not imply the existence of a club guessing sequence on ω1. In this paper we give an alternate proof of the second of these results, using Woodin’s Pmax technology. With the Pmax approach it is more natural to produce sequences which weakly guess clubs at club many points, so our model for (+) satisfies a strengthening of (+) for which the guessing happens club often. As always with Pmax variations, the continuum has cardinality א2 in our models. This research was done at the same time as [1], though the results in that paper were proved first. As a warm-up we present a variation which produces a model with a ladder system which weakly guesses each club subset of ω1 club often. We show that this model does not satisfy the Interval Hitting Principle, thus separating these statements. Date: June 15, 2010. The first author is supported in part by NSF grant DMS-0700983. The second author is supported in part by NSF grants DMS-0401603 and DMS-0801009.