Another c.c.c. forcing that destroys presaturation

Paul B. Larson  Teruyuki Yorioka

January 10, 2010

An ideal I on ω_{1} is said to be strong if I is precipitous and in any forcing extension by P(ω_{1})/I, j(ω_{1}^{V}) = ω_{2}^{V} holds, where j is the corresponding embedding [1, Definition 5.6.], and presaturated if for any countable collection D of predense subsets of P(ω_{1})/I there are densely many A ∈ I^{+} such that for each D ∈ D, A intersects at most ℵ_{1} many members of D positively. It is a standard fact, and not hard to see, that if I is presaturated, then the forcing P(ω_{1})/I does not collapse ω_{2} [1, §4]. In their study of ideals on ω_{1}, Baumgarter and Taylor [1, Theorem 5.7.] show that strong ideals are preserved by c.c.c. forcing, and mention that they do not know whether the same holds for presaturated ideals. They also mention that they do not know whether these two notions are equivalent. These questions were answered by Veličković [5, Theorem 4.6.], who showed (from the consistency of ZFC+SPFA) that consistently the nonstationary ideal on ω_{1} (NS_{ω_{1}}) is saturated and there is a c.c.c. forcing which adds a Kurepa tree, and thus destroys the presaturation of NS_{ω_{1}}. Here we show that the main argument from [4] gives another example.

Theorem 0.1. If ZF is consistent with the Axiom of Determinacy, then it is consistent that NS_{ω_{1}} is saturated and there exists a Suslin tree S such that the forcing to specialize S with finite conditions destroys the presaturation of NS_{ω_{1}}.

By a theorem of Baumgartner (see [2, Theorem 3]), the forcing to specialize S (i.e., to cover it with countably many antichains) with finite conditions is c.c.c. in this model, as S has no cofinal branches.

Proof of Theorem 0.1. In [4], a model is constructed (assuming ZF + AD) in which NS_{ω_{1}} is saturated and there is a Suslin tree S such that, when one forces with P(ω_{1})/NS_{ω_{1}} and takes the corresponding generic embedding j: V → M, j(S) contains a cofinal branch in V [G] (though of course not in M, where it is a Suslin tree). This means that in this model there are functions f_{α}: ω_{1} → S (α < ω_{2}) such that for all α < β < ω_{2}, the set {γ < ω_{1} | f_{α}(γ) <_{S} f_{β}(γ)} contains a club. Let V [H] be an extension of V by this specializing forcing for S, and consider a V [H]-generic filter G for P(ω_{1})/NS_{ω_{1}} with corresponding embedding j: V [H] → M. Then ⟨j(f_{α})(ω_{1}^{V}) : α < ω_{2}^{V} ⟩ forms a (possibly noncofinal) path through j(S), which is special in M. Therefore ω_{2}^{V} is countable in V [H][G] (which means that NS_{ω_{1}} is not presaturated in V [H]).

References

[1] A.D. Taylor, J. Baumgartner, Saturation properties of ideals in generic extensions. II. Trans. Amer. Math. Soc. 271 (1982), no. 2, 587–609.

[2] J. Baumgartner, J. Malitz, W. Reinhardt, Embedding trees in the rationals. Proc. Nat. Acad. Sci. U.S.A. 67 (1970), 1748–1753.

[3] P.B. Larson, An 𝕊_{max} variation for one Souslin tree. J. Symbolic Logic 64 (1999), no. 1, 81–98.

[4] P.B. Larson, Saturation, Suslin trees and meager sets. Arch. Math. Logic 44 (2005), no. 5, 581–595.

[5] B. Veličković, Forcing axioms and stationary sets. Adv. Math. 94 (1992), no. 2, 256–284.

Department of Mathematics Miami University Oxford, Ohio 45056 USA larsonpb@muohio.edu

Department of Mathematics Faculty of Science, Shizuoka University Ohya 836, Shizuoka, 422-8529 JAPAN styorio@ipc.shizuoka.ac.jp