Arch. Math. Logic

DOI 10.1007/s00153-014-0375-x Mathematical Logic

Some considerations on amoeba forcing notions

Giorgio Laguzzi

Received: 19 June 2013 / Accepted: 14 February 2014 © Springer-Verlag Berlin Heidelberg 2014

Abstract In this paper we analyse some notions of amoeba for tree forcings. In particular we introduce an amoeba-Silver and prove that it satisfies quasi pure decision but not pure decision. Further we define an amoeba-Sacks and prove that it satisfies the Laver property. We also show some application to regularity properties. We finally present a generalized version of amoeba and discuss some interesting associated questions.

Keywords Amoeba · Sacks · Silver

Mathematics Subject Classification 03E15 1 Introduction

The amoeba forcings play an important role when dealing with questions concerning the real line, such as cardinal invariants and regularity properties. As an intriguing example, one may consider the difference between the amoeba for measure and category in Shelah’s proof regarding the use of the inaccessible cardinal to build models for regularity properties, presented in [7] and [8]; in fact, since the amoeba for category is sweet (a strengthening of σ -centeredness), one can construct, via amalgamation, a

Boolean algebra as limit of length ω1 (without any need of the inaccessible), in order to get an extension where all projective sets have the Baire property. On the contrary, for

Lebesgue measurability, Shelah proved that if we assume all 13 sets to be Lebesgue measurable, we obtain, for all x ∈ ωω, L[x] | “ωV1 is inaccessible”. If one then goes deeply into Shelah’s construction of the model satisfying projective Baire property just mentioned, one can realize that the unique difference with Lebesgue measurabilG. Laguzzi (B)

Universität Hamburg, Hamburg, Germany e-mail: giorgio.laguzzi@libero.it 123

G. Laguzzi ity consists of the associated amoeba forcing, which is not sweet for measure. Such an example is probably one of the oldest and most significant ones to underline the importance of the amoeba forcing notions in set theory of the real line. In other cases, it is interesting to define amoeba forcings satisfying certain particular features, like not adding specific types of generic reals, not collapsing ω1 and so on; these kinds of constructions are particularly important when one tries to separate regularity properties of projective sets, or when one tries to blow up certain cardinal invariants without affecting other ones. For a general and detailed approach to regularity properties, one may see [4]. The main aim of the present paper is precisely to study two versions of amoeba, for Sacks and Silver forcing, respectively.

Definition 1 Let P be either Sacks or Silver forcing. We say that AP is an amoeba-P iff for any ZFC-model M ⊇ NAP, we have

M | ∀T ∈ P ∩ N ∃T ′ ∈ M ∩ P (T ′ ⊆ T ∧ ∀x ∈ [T ′](x is P-generic over N)).

Note that this definition works even when P is any other tree forcing notions (Laver,

Miller, Mathias, and so on). We would like to mention that a similar work for Laver and Miller forcing is developed in detail by Spinas in [10] and [11].

Let us now recall some basic notions and standard notation. Given t, t ′ ∈ 2<ω, we write t ′t iff t ′ is an initial segment of t . A tree T is a subset of 2<ω closed under initial segments, i.e., for every t ∈ T , for every k < |t |, tk ∈ T , where |t | represents the length of t . Given s, t ∈ T , we say that s and t are incompatible (and we write s ∦ t) iff neither st nor t s; otherwise one says that s and t are compatible (s ‖ t). We denote with Stem(T ) the longest element t ∈ T compatible with every node of T . For every t ∈ T , we say that t is a splitting node whenever both t0 ∈ T and t1 ∈ T , and we denote with Split(T ) the set of all splitting nodes. Moreover, for n ≥ 1, we say t ∈ T is an nth splitting node iff t ∈ Split(T ) and there exists n ∈ ω maximal such that there are natural numbers k0 < · · · < kn−1 with tk j ∈ Split(T ), for every j ≤ n − 1. We denote with Splitn(T ) the set consisting of the nth splitting nodes. For a finite tree

T , the height of T is defined by ht(T ) := max{n : ∃t ∈ T, |t | = n}, while Term(T ) denotes the set of terminal nodes of T , i.e, those nodes having no proper extensions in T . Finally, for every t ∈ T , the set {s ∈ T : s ‖ t} is denoted by Tt , the body of T is defined as [T ] := {x ∈ 2ω : ∀n ∈ ω(xn ∈ T )}, and T |n := {t ∈ T : |t | ≤ n}.

Further, given a tree T and a finite subtree p ⊂ T , we define: • T↓p := {t ∈ T : ∃s ∈ Term(p)(s ‖ t)}; • p T ⇔ ∀t ∈ T \p ∃s ∈ Term(p)(s t), and we will say that p is an initial segment of T , or equivalently T end-extends p.

Our attention is particularly focused on the following two types of infinite trees of 2<ω: • T ⊆ 2<ω is a perfect (or Sacks) tree iff each node can be extended to a splitting node. • T ⊆ 2<ω is a Silver tree (or uniform tree) iff T is perfect and for every s, t ∈ T , such that |s| = |t |, one has s0 ∈ T ⇔ t0 ∈ T and s1 ∈ T ⇔ t1 ∈ T . 123

Amoeba forcing notions

Sacks forcing S is defined as the poset consisting of Sacks trees, ordered by inclusion, and Silver forcing V is analogously defined by using Silver trees. Further, if G is the S-generic filter over N, we call the generic branch zG = ⋃{Stem(T ) : T ∈ G} a Sacks real (and analogously for Silver). Other common posets that will appear in the paper will be the Cohen forcing C, consisting of finite sequences of 0’s and 1’s, ordered by extension, and the random forcing B, consisting of perfect trees T with strictly positive measure, ordered by inclusion.

We recall the notion of axiom A, which is a strengthening of properness.

Definition 2 A forcing P satisfies Axiom A if and only if there exists a sequence {≤n : n ∈ ω} of orders of P such that: 1. for every a, b ∈ P , for every n ∈ ω, b ≤n+1 a implies both b ≤n a and b ≤ a; 2. for every sequence 〈an : n ∈ ω〉 of conditions in P such that for every n ∈ ω, an+1 ≤n an , there exists b ∈ P such that for every n ∈ ω, b ≤n an ; 3. for every maximal antichain A ⊆ P , b ∈ P , n ∈ ω, there exists b′ ≤n b such that {a ∈ A : a is compatible with b′} is countable.