Some results about (+) proved by iterated forcing

We shall show that the consistency of CH+¬(+) and CH+(+)+there are no club guessing sequences on ω1. We shall also prove that ♢ does not imply the existence of a strong club guessing sequence on ω1. §0. Introduction. The principle (+) and its variations were first considered by the second author in [2]. They are very weak club guessing principles. The properties of the principles were largely unknown until recently. While J. Moore proved that MRP implies the negation of (+), it was not known whether the negation of (+) has any large cardinal strength, or CH implies (+). The first main result is to show that just from ZFC, we can build a model of CH+¬(+). Hence, it answers both questions in the previous paragraph. We also build a model of CH+(+) in which there is no club guessing sequence on ω1. This is the first model satisfying these properties. The last part of this paper is devoted to construct a model of ♢ in which there is no strong club guessing sequence on ω1. It answers the question asked by the first author in [1]. The proof in fact builds a model of ♢ in which the “strong” version of (+) fails. This demonstrates how effective the use of variations of (+) is in the investigation of guessing principles. The structure of this paper is as follows. In Section 1, we shall give the definitions of (+)k, (+)<ω, and related notions. In Section 2, the results by S. Shelah and J. Moore about the iteration adding no new reals are described. It will be used repeatedly in the later sections. Then, we shall build a model of CH+¬(+) in Section 3. In Section 4, we shall prove some lemmas about the internalization. They are slightly improved from the ones in [1]. By using these lemmas, we shall prove the consistency of CH+(+)<ω + there is no club guessing sequence on ω1 in Section 5, and ♢ + ‘there is no strong club guessing sequence on ω1’ in Section 6. Received by the editors June 3, 2010. 1991 Mathematics Subject Classification. 03E35.