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On Possible Restrictions of the Null Ideal

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On Possible Restrictions of the Null Ideal On possible restrictions of the null ideal Ashutosh Kumar,∗ Saharon Shelahy Abstract We prove that the null ideal restricted to a non-null set of reals could be isomorphic to a variety of sigma ideals. Using this, we show that the following are consistent: (1) There is a non-null subset of plane each of whose non-null subsets contains three collinear points. (2) There is a partition of a non-null set of reals into null sets, each of size @1, such that every transversal of this partition is null. 1 Introduction In [9], starting with a measurable cardinal, Shelah constructed a model in which the null ideal restricted to a non-null set of reals is @1-saturated. We would like to prove the following generalization of this. Theorem 1.1. Suppose κ is a cardinal of uncountable cofinality and I is a sigma ideal on κ that contains all bounded subsets of κ. Then there is a ccc forcing P such that in V P, the null ideal restricted to some non-null set is isomorphic to the ideal generated by I. An immediate corollary of Theorem 1.1 (see Lemma 5.1 below) is that the null ideal restricted to some non-null set of reals can be isomorphic to the non-stationary ideal on !1. This answers a question of Fremlin (Problem 9 in [2] and Problem DX in [3]). We also give the following applications. Theorem 1.2. The following is consistent: There is a non-null X ⊆ R2 such that for every Y ⊆ X if Y is non-null, then Y contains three collinear points. Theorem 1.3. The following is consistent: There exists a non-null X ⊆ R and a partition hXi : i < !1i of X into null sets of size @1 such that for every Y ⊆ X, if jY \ Xij = 1 for every i < !1, then Y is null. ∗Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J Safra Campus, Givat Ram, Jerusalem 91904, Israel; email: [email protected]; Supported by a Postdoctoral Fellowship at the Einstein Insititute of Mathematics funded by European Research Council grant 338821 and by NSF grant no. 136974 yEinstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J Safra Campus, Givat Ram, Jerusalem 91904, Israel and Department of Mathematics, Rutgers, The State University of New Jersey, Hill Center-Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA; email: [email protected]; Partially supported by European Research Council grant 338821 and by NSF grant no. 136974; Publication no. 1102 1 Recently, we were able to combine the methods of this paper with that of Komjath's in [4] to construct a model where the null and the meager ideals are both somewhere @1-saturated. This will appear in [7]. On notation: µ denotes the standard Lebesgue measure on 2!. A subset W ⊆ 2! is fat if for every clopen set C, either W \ C = φ or µ(W \ C) > 0. A subtree T ⊆ <!2 is fat if ! ! [T ] = fx 2 2 :(8n < !)(x n 2 T )g is fat. For a clopen subset C ⊆ 2 , define supp(C) to be ! the smallest (finite) set F such that (8x; y 2 2 )((x F = y F ) =) (x 2 C () y 2 C)). Random denotes the random real forcing. Note that f[T ]: T ⊆ <!2 is a fat treeg is dense in Random. Cohenκ denotes the forcing for adding κ Cohen reals. In forcing we use the convention that a larger condition is the stronger one - p ≥ q means p extends q. If P; Q are forcing notions and Q ⊆ P, we write Q l P if every maximal antichain in Q is also a maximal antichain in P. 2 On category The category analogue of Theorem 1.1 follows from the argument in [4] where Komj´ath proved the following: Let κ be measurable. Then there is a ccc forcing P such that in V P, the meager ideal restricted to some non-meager set of reals is @1-saturated. Let us sketch this argument. Suppose κ ≥ !1 and I is a sigma ideal on κ that contains ! P all singletons. Let P = Cohenκ adding κ Cohen reals hci : i < κi where each ci 2 2 . In V , Q let Q be the finite support product fQA : A 2 Ig where QA is defined as follows: p 2 QA iff p = (Fp;Np; n¯p; σ¯p) = (F; N; n;¯ σ¯) where • F is a finite subset of A • N < ! • n¯ = hnk : k ≤ Ni is a strictly increasing sequence of integers with n0 = 0 [nk;nk+1) • σ¯ = hσk : k < Ni where each σk 2 2 For p; q 2 QA, define p ≤ q iff Fp ⊆ Fq, Np ≤ Nq,n ¯p n¯q,σ ¯p σ¯q and for every Np ≤ k < Nq, for every i 2 Fp, σq;k 6= ci [nq;k; nq;k+1). Note that QA is a sigma centered forcing making fci : i 2 Ag meager. The set of conditions (p; q) 2 P ? Q where p 2 P and for each A 2 dom(q), p forces an actual value to q(A) is dense in P ? Q. Put S = P ? Q. S We claim that, in V the meager ideal restricted to W = fci : i < κg is isomorphic to J = fX ⊆ κ :(9A 2 I)(X ⊆ A)g - the ideal generated by I. It is clear that the for each S X 2 J , fci : i 2 Xg is meager. Also, in V , for every X ⊆ κ, if X2 = J , then fci : i 2 Xg ! S is non-meager. To see this, let B ⊆ 2 be a meager Fσ-set coded in V . Since S is ccc, we S Q can find a countable F ⊆ I such that B is coded in V [hci : i 2 Fi][ fG : A 2 Fg]. S QA Choose i? 2 X n ( F). Note that for each A 2 F, QA 2 V [hci : i 2 Ai]. It follows that ci? S Q is Cohen over V [hci : i 2 Fi][ fGQA : A 2 Fg]. Hence ci? 2= B and therefore fci : i 2 Xg is non-meager. 2 This proof does not have an obvious analogue in the case of the null ideal since, e.g., if we start by adding a set X = fri : i < κg of κ random reals, and do a finite support iteration (for ccc) to make certain subsets of X null, then we'd inevitably add Cohen reals at stages of countable cofinality making X null. To get around this difficulty, Shelah came up with the following idea in [8, 9]. κ Let hXα : α < λi be a list of members of I, each occurring λ = 2 times. Perform a finite support iteration hPα; Qα : α < λ + κi with limit P where for α < λ, Qα is V [hτα:α2Aλ+ξi] Cohen forcing with generic real τα and for ξ < κ, Qλ+ξ = (Random) , where Aλ+ξ = fα < λ : ξ2 = Xαg [ [λ, λ + ξ), with generic partial random real τλ+ξ. Using the fact that P is ccc, it is easy to show that if A ⊆ κ is not in the ideal generated by I, then P fτλ+ξ : ξ 2 Ag is not null in V . The other direction is supposed to follow from the fact that for α < λ, τα codes a null Gδ-set that should cover fτλ+ξ : ξ 2 Xαg because for ξ 2 Xα, the memory Aλ+ξ of the partial random τλ+ξ does not contain α. But Aλ+ξ contains [λ, λ + ξ) and the partial random added at these stages can use τα so this is not a simple product forcing argument as in the category case. Some remarks on the structure of the proof of Theorem 1.1 follow. The next section introduces the model witnessing the theorem and proves some preliminary facts about the iteration used in defining it. The main difficulty in the verification appears in Claim 3.8 whose proof is reduced to constructing a condition p? satisfying the hypothesis of Lemma 3.9. The concluding remarks in Section 3 explain the difficulty in an inductive construction of p? due to the use of partial memories for the random reals. Section 4 introduces the main tools to get around this difficulty via Definition 4.6 and Claim 4.7 where the existence of p? is reduced to the existence of certain finitely additive measures on P(!) \ V P. Lemma 4.8 completes the proof by showing that such measures exist. The use of blueprints is to allow a sufficiently general statement in this lemma so that an inductive proof using automorphisms of higher iterations can be given. 3 Forcing κ +! Let κ, I be as in the hypothesis of Theorem 1.1. Put λ0 = 2 . For λ0 ≤ λ < λ0 , define the following. +! (1) hXα : α < λ0 i is a sequence of members of I. +n +n (2) For every n < ! and X 2 I, jfα < λ0 : Xα = Xgj = λ0 . λ (3) For ξ < κ, Aλ+ξ = Aλ+ξ = fα < λ : ξ2 = Xαg [ [λ, λ + ξ). ¯ (4) Pλ = hPα; Qα : α < λ + κi is a finite support iteration with limit Pλ+κ such that ! (a) For α < λ, Qα is Cohen forcing with generic real τα 2 ! . V [hτi:i2Aλ+ξi] ! (b) For ξ < κ, Qλ+ξ = (Random) with generic partial random τλ+ξ 2 2 . 3 ¯ The reason for considering iterations Pλ for λ > λ0 will become clear during the proof of Lemma 4.8 where we use automorphisms of Pλ+κ for λ > λ0 to construct certain finitely additive measures on P(!) \ V Pλ0+κ .
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