Sacks forcing, Laver forcing, and Martin's axiom

In this paper we study the question assuming MA+¬CH does Sacks forcing or Laver forcing collapse cardinals? We show that this question is equivalant to question of what is the additivity of Marczewski’s ideal s. We give a proof that it is consistent that Sacks forcing collapses cardinals. On the other hand we show that Laver forcing does not collapse cardinals. Introduction Let S be Sacks perfect set forcing [32]; p ∈ S iff p ⊆ 2 is a nonempty subtree and for every s ∈ p there exists t ⊇ s such that tˆ0 ∈ p and tˆ1 ∈ p. The ordering is defined by p ≤ q iff p ⊆ q. Define [p] = {x ∈ 2 : ∀n x n ∈ p}. The s ideal of subsets of 2 is defined by X ∈ s iff for every p ∈ S there exists q ≤ p with X∩ [q] = ∅. Define add(s) = min{|F | : F ⊆ s, ⋃ F / ∈ s}. Marczewski’s ideal s, which first appeared in [24], has been studied by a number of authors, Aniszczyk, Frankiewicz, Plewik [4], Brown [11][12][13], Brown, Cox [14], Brown, Prikry [15], Corazza [16], Morgan [29], and Pawlikowski [31]. Aniszczyk [5] has asked if MA implies that the ideal s is c-additive, i.e., is it true that the union of fewer than continuum many s sets is an s set, i.e., add(s) = c. It is a folklore result that assuming the proper forcing axiom the ideal s is c-additive (see Abraham [1]). It is also an easy exercise to show the consistency of add(s) = ω1 plus the continuum is large. This happens in the Cohen real model. Theorem 1.1. (MA+¬CH) add(s) is the minimum κ such that for some p ∈ S we have p |`S cof(c) = κ. This means that the question of the additivity of the s ideal is the same as the question of whether Sacks forcing collapses cardinals. In the proof in Archive for Mathematical Logic, 31(1992), 145-161. Research partially supported by NSF grant 8801139.