Creature forcing and five cardinal characteristics in Cicho\'{n}'s diagram

We use a (countable support) creature construction to show that consistently \[ \mathfrak d=\aleph_1= \text{cov}(\text{NULL})<\text{non}(\text{MEAGER})<\text{non}(\text{NULL})<\text{cof}(\text{NULL})<2^{\aleph_0}. \] The same method shows the consistency of \[ \mathfrak d=\aleph_1= \text{cov}(\text{NULL})<\text{non}(\text{NULL})<\text{non}(\text{MEAGER})<\text{cof}(\text{NULL})<2^{\aleph_0}. \]

1. Introduction 1.1. The result and its history. Let N denote the ideal of Lebesgue null sets, and M the ideal of meager sets. We prove (see Theorem 6.2.1) that consistently, several cardinal characteristics of Cichoń's Diagram (see Figure 1) are (simultaneously) different: Since our model will satisfy d = ℵ 1 , will also have non(M) = cof(M). The desired A (by now) classical series of theorems [Bar84,BJS93,CKP85,JS90,Kam89,Kra83,Mil81,Mil84,RS83,RS99,She98] proves these (in)equalities in ZFC and shows that they are the only ones provable. More precisely, all assignments of the values ℵ 1 and ℵ 2 to the characteristics in Cichoń's Diagram are consistent, provided they do not contradict the above (in)equalities. (A complete proof can be found in [BJ95,chapter 7].) This does not answer the question whether three (or more) characteristics can be made simultaneously different. The general expectation is that this should always be possible, but may require quite complicated forcing methods. We cannot use the two best understood methods, countable support iterations of proper forcings (as it forces 2 ℵ0 ≤ ℵ 2 ) and, at least for the "right hand side" of the diagram, we cannot use finite support iterations of ccc forcings in the straightforward way (as it adds lots of Cohen reals, and thus increases cov(M) to 2 ℵ0 ).
There are ways to overcome this obstacle. One way would be to first increase the continuum in a "long" finite support iteration, resulting in cov(M) = 2 ℵ0 , and then "collapsing" cov(M) in another "short" finite support iteration. In a much more sophisticated version of this idea, Mejía [Mej13] recently constructed several models with many simultaneously different cardinal characteristics in Cichoń's Diagram (building on work of Brendle [Bre91], Blass-Shelah [BS89] and Brendle-Fischer [BF11]).
We take a different approach, completely avoiding finite support, and use something in between a countable and finite support product (or, a form of iteration with very "restricted memory").
This construction avoids Cohen reals, it is in fact ω ω -bounding, resulting in d = ℵ 1 . This way we get an independence result "orthogonal" to the ccc/finitesupport results of Mejía. The fact that our construction is ω ω -bounding is not incidental, but rather a necessary consequence of the two features which, in our construction, are needed to guarantee properness: a "compact" or "finite splitting" version of pure decision, and fusion (which together give a strong version of Baumgartner's Axiom A and in particular properness and ω ω -bounding).
We think that our construction can be used for various other independence results with d = ℵ 1 , but the construction would require considerable remodeling if we want to use it for similar results with d > ℵ 1 , even more so for b > ℵ 1 .

1.2.
A very informal overview of the construction. The obvious attempt to prove the theorem would be to find a forcing for each cardinal characteristic x that increases x but leaves the other characteristics unchanged. More specifically, find the following forcing notions.
• Q nm , adding a new meager set which will contain all old reals.
Adding many such sets will tend to make non(M) large. • Q nn , adding a new measure zero set which will contain all old reals.
Adding many such sets will tend to make non(N ) large. • Q cn , adding a new measure zero set which is not contained in any old measure zero set. Adding many such sets will tend to make cof(N ) large. • Q sk , adding a kind of Sacks real, in the sense that the generic real does not change any other cardinal characteristic; in particular, every new real is bounded by an old real, is contained in an old measure zero set, etc. Adding many such reals will tend to make the continuum large. For each t ∈ {nm, nn, cn, sk}, our Q t will be a finitely splitting tree forcing; Q nm will be "lim-inf" (think of a tree forcing where we require large splitting at every node, not just infinitely many along every branch; i.e., more like Laver or Cohen than Miller or Sacks; however note that in contrast to Laver all our forcings are finitely splitting); the other ones will be "lim-sup" (think of forcings like Sacks or Silver).
We then fix for each t a cardinal κ t , and take some kind of product (or, iteration) of κ t many copies of Q t , and hope for the best. Here we arrive at the obvious problem: which product or iteration will work? As mentioned above, neither a finite support iteration 1 nor a countable support iteration will work, and it is not clear why a product will not collapse the continuum. So we will introduce a modification of the product construction.
The paper is divided into two parts. In part 1 we describe the "general" forcing construction (let us call it the "framework"), in part 2, the "application", we use the framework to construct a specific forcing that proves the main theorem.
Part 1: In Sections 2-5 we present the "framework". Starting with building blocks (so-called "subatoms"), we define the forcing Q. This is an instance of creature forcing. (The standard reference for creature forcing is Rosłanowski and Shelah [RS99], but our presentation will be self-contained. Our framework is a continuation of [KS12,KS09], where the central requirement to get properness was "decisiveness". In this paper, decisiveness does not appear explicitly, but is implicit in the way that the subatoms are combined to form so-called atoms. ) We fix a set Ξ of indices. (For the application, we will partition Ξ into sets Ξ t of size κ t for t ∈ {nm, nn, cn, sk} as above.) The forcing Q will "live" on the product Ξ × ω, i.e., a condition p ∈ Q will contain for certain (ξ, n) a "creature" p(ξ, n), a finite object that gives some information about the generic filter.
More specifically, there is a countable subset supp(p) ⊆ Ξ, and for each ξ ∈ supp(p) the condition up to some level n 0 (ξ) consists of a so-called trunk (where a finite initial segment of the generic realỹ ξ is already completely determined), and for all n > n 0 (ξ) there is a creature p(ξ, n), an element of a fixed finite set K ξ,n , which gives several (finitely many) possibilities for the corresponding segment of the generic realỹ ξ . We assign a "norm" to the creature, a real number that measures the "number of possibilities" (or, the amount of freedom that the creature leaves for the generic). More possibilities means larger norm.
Moreover, for each m there are only finitely many ξ with n 0 (ξ) ≤ m (i.e., at each level m there live only finitely many creatures of p). We can then set the norm of p at m to be the minimum of the norms of p(ξ, n) over all ξ "active" at level m.
A requirement for a p to be a valid condition in Q is that the norms at level m diverge to infinity for m → ∞ (i.e., the lim-inf of the norms is infinite).
So far, Q seems to be a lim-inf forcing, but recall that we want to use lim-inf as well as lim-sup.
So let us redefine Q. We will "cheat" by allowing "gluing". We declare a subset of Ξ to be the set Ξ ls of "lim-sup indices" (in the application this will be Ξ nn ∪ Ξ cn ). 1 To avoid giving a wrong impression, our specific forcings Qt will not be ccc, so a finite support iteration would not work anyway.
Forget the "norm of p at level m" and the lim-inf condition above. Instead, we partition the set of levels ω into finite intervals ω = I 0 ∪ I 1 ∪ . . . (this partition depends on the condition and can be coarsened when we go to a stronger condition). For such an interval I, we declare all creatures whose levels belong to I to constitute a "compound creature" with a "compound norm", intuitively computed as follows: • for each ξ ∈ Ξ ls we set nor(p, I, ξ) to be the maximum of the norms of p(ξ, m) with m ∈ I; • for other ξ we take the minimum rather than the maximum; and • we set nor(p, I) to be the minimum of nor(p, I, ξ) for all (finitely many) ξ active at some level in I.
The new lim-inf condition is that nor(p, I k ) diverges to infinity with k → ∞.
While this may give some basic idea about the construction, things really are more complicated. We will require the well-known "halving" property of creature forcing (to prove Axiom A). Moreover, the Sacks part, i.e., Q sk on the indices Ξ sk ⊂ Ξ, does not fit well into the framework as presented above and requires special treatment. This will not be very complicated mathematically but will unfortunately make our notation much more awkward and unpleasant.
A central requirement on our building blocks (subatoms) will be another wellknown property of creature-forcing: "bigness". This is a kind of Ramsey property connected to the statement that creatures at a level m are "much bigger" than everything that "happened below m".
Using these requirements, we will show the following.
• (Assuming CH in V ) Q is ℵ 2 -cc. (Accomplished via a standard ∆-system argument.) • We say that p "essentially decides" a nameτ of an ordinal if there is a level m such that whenever we increase the trunk of p up to m (for this there are only finitely many possibilities), we know the value ofτ . In other words, knowing the generic up to m (on some finite set of indices), we also know the value ofτ . • Pure decision and fusion. Given a nameτ of an ordinal and a condition p, we can strengthen p to a condition q essentially decidingτ . Moreover, we can do this in such a way that p and q agree below a given level h and the norms above this level do not drop below a given bound. (This is called "pure decision".) This in turn implies "fusion" in that we can iterate this strengthening for infinitely many namesτ ℓ , resulting in a common extension q ∞ which essentially decides eachτ ℓ .
(While fusion is an obvious property of the framework, pure decision is the central result of part 1, and will use the requirements on bigness and halving).
• The usual standard argument then gives continuous reading (every real is a continuous image of (countably many) generic reals), a strong version of Axiom A, and thus ω ω -bounding and properness. (Recall that we have "finite splitting", i.e., essentially deciding implies that there are only finitely many potential values.) • We also get a Lipschitz variant of continuous reading, "rapid reading", which implies that the forcing adds no random reals (and which will be essential for many of the proofs in part 2).
To simplify notation, we further assume: • if | poss(y)| = 1, then nor(y) < 1; and • for each x ∈ K and a ∈ poss(x) there is a y ≤ x with poss(y) = {a}. (Such a subatom will be called a singleton.) Notation 2.1.2. Abusing notation, we will just write K for the subatomic family (K, ≤, nor, poss). If y ≤ x we will also say that y is "stronger than x" or is "a successor of x".
Remark 2.1.3. Our subatomic families will also have the following properties (which might make the picture clearer, but will not be used in any proof).
In the usual way we often identify a natural number n with the set {0, . . . , n−1}, and write m ∈ n for m < n; for example in the following definition.
Definition 2.1.4. Fix a natural number B > 0. We say that a subatom x ∈ K has B-bigness if for each coloring c : poss(x) → B there is a y ≤ x such that c ↾ poss(y) is constant and nor(y) ≥ nor(x) − 1. 3 We say that the subatomic family K has B-bigness if each x ∈ K has B-bigness.
Given a subatom x in a fixed subatomic family K, we have the following facts. Example 2.1.5. The basic example of a subatomic family with B-bigness is the following "counting norm". For a fixed finite set POSS, a subatom x is a nonempty subset of POSS, with poss(x) := x, y ≤ x defined as y ⊆ x, and nor(x) := log B |x|.
We get a stronger variant of bigness if we divide the norm by B: Then for each F : poss(x) → B there is a y ≤ x such that F ↾ poss(y) is constant and nor ′ (y) ≥ nor ′ (x) − 1 /B. 2 The analogous statement will not be true for "compound creatures" (cf. Definition 2.5.1) because of the halving parameters. 3 As only the number of "colors" is of importance, we may consider the codomain of the coloring function to be any set of cardinality B.
Remark 2.1.6. The above example (in the version nor ′ ) is actually used for the non(M)-subatoms (cf. 7.1.1). The cof(N )-subatoms (cf. Section 10.1) still use a counting norm, i.e., nor(x) only depends on the cardinality of poss(x), but the relation between | poss(x)| and nor(x) is more complicated. The non(N )-subatoms (cf. Section 8.1) will use a different kind of norm which does not just depend on the cardinality of poss(x), but also on its structure.
Given a subatomic family with 2-bigness, it is straightforward to construct another subatomic family with arbitrary bigness by only altering the norm.
Lemma 2.1.7. If K is a subatomic family with 2-bigness, then given any b ≥ 1 replacing the norm of K with nor ′ defined by nor ′ (x) := nor(x) /b results in a subatomic family with 2 b -bigness.
Proof. Given x ∈ K, and a coloring c : poss(x) → P(b), use the 2-bigness of the original subatomic family to inductively pick Remark 2.1.8. Of course, any subatomic family K can be made to have arbitrary bigness by simply ensuring that all subatoms have norm ≤ 1. The benefit of the method presented in Lemma 2.1.7 is that the norm of each subatom decreases proportionally to the logarithm of the desired bigness. As our construction depends on the existence of subatomic families with "big" bigness and also having subatoms with "large" norm, the above Lemma gives an indication of how this can be achieved.

Atomic creatures.
We now describe how to combine subatomic families to create so-called atoms. Fix a natural number J > 0, and fix a parameter ℓ ∈ ω. We will first define the "measure" of subsets of J with respect to this parameter. 4 We will later use the following easy observation about the "measure".
Proof. Note that if for some i ≤ k we have that µ ℓ (A i ) ≤ 1, then simply picking B i := ∅ will introduce no obstructions. We may then assume that µ ℓ (A i ) > 1 (meaning that |A i | ≥ 3 ℓ +1 ) for each i ≤ k. We now inductively construct (k + 1)- (and A j+1 i = A j i for all other i ≤ k). As |A i 0 | ≥ 3 ℓ+1 it follows by the induction that |A j i 0 | ≥ 3, and similarly |A j i 1 | ≥ 3, and so it is possible to partition the intersection As each A i was modified at most k times in the inductive construction it follows that |B i | ≥ |Ai| /3 k , and so µ ℓ Suppose now that for each j ∈ J we have a subatomic family K j living on a finite set POSS j . We can now define the atoms built from the subatoms.
• An atomic creature, or atom, a consists of a sequence (x j ) j∈J where x j is a K j -subatom for all j ∈ J.
• The norm of an atom a = (x j ) j∈J , nor(a), is the maximal r for which there is a set A ⊆ J with µ ℓ (A) ≥ r and nor(x j ) ≥ r for all j ∈ A. We say that such an A "witnesses the norm" of a.
So the norm of an atom is large if there is a "large" subset A of J such that all subatoms in A are "large".
The following easy fact will be useful later.
Fact 2.2.4. Suppose A ⊆ J witnesses the norm of an atom a = (x j ) j∈J , and let b = (y j ) j∈J be any atom which agrees with a on all indices in A. Then nor(b) ≥ nor(a). In particular, if nor(y j ) ≤ nor(x j ) for all j / ∈ A, then nor(b) = nor(a).

Sacks columns.
Given a (finite) tree T , its splitting-size, nor split (T ), is defined as the maximal ℓ ∈ ω such that there is a subset S ⊆ T (with the induced order) which is order isomorphic to the complete binary tree 2 ≤ℓ (of height ℓ with 2 ℓ many leaves). Equivalently, 2 ≤ℓ order-embeds into T . Given a finite subset I of ω and F ⊆ 2 I , we can identify F with the tree of its restrictions T F = F ∪ {η ↾ n : η ∈ F, n ∈ I} (a tree of partial functions from I to 2, ordered by inclusion). We write nor split (F ) for nor split (T F ).
The following establishes a basic combinatorial fact about this norm.
Definition and Lemma 2.3.1. There exists a function f with the following properties.
• For each j, n, c, whenever (2 f (j,n,c) ) j is colored with c colors there are subsets A 1 , . . . , A j of 2 f (j,n,c) such that the set A 1 ×· · ·×A j is homogeneous, and nor split (A i ) ≥ n for all i. 5,6 • f is monotone in each argument.
Proof. We define f (j, n, c) recursively on j by f (1, n, c) = n · c, and f (j + 1, n, c) = f (1, n, c 2 j·f (j,n,c) ) = n · c 2 j·f (j,n, c) . Note that f (j, n, 1) = n, and clearly any coloring 5 As in the case of the bigness of subatoms, only the number of "colors" of our coloring functions is of importance. Moreover, by the definition of the splitting norm it follows that T 1 , . . . , T j are trees each of splitting size at least f (j, n, c) and π : T 1 × · · · × T j → c is a coloring, then there are A i ⊆ T i (i ≤ j) such that nor split (A i ) ≥ n for each i and π ↾ A 1 × · · · × A j is constant. 6 If j = 1 this condition becomes whenever 2 f (1,n,c) is colored with c colors there is a homogeneous subset A of 2 f (1,n,c) such that nor split (A) ≥ n.
Then as π * is a coloring of T j by at most c colors, and as nor split (T ) = p = f (j, n, c) by hypothesis for each i ≤ j there are A i ⊆ T ⊆ 2 n·q with nor split (A i ) ≥ n (for i ≤ j) such that A 1 × · · · × A j is homogeneous for π * . It then follows that A 1 × · · · × A j × A j+1 is homogeneous for π. Definition 2.3.2. Suppose that I is a nonempty (finite) interval in ω. By a Sacks column on I we mean a nonempty s ⊆ 2 I . We say that another Sacks column s ′ on I is stronger than s, and write s ′ ≤ s, if s ′ ⊆ s.
We can naturally take products of columns that are stacked above each other. Definition 2.3.3. Let s 1 be a Sacks column on an interval I 1 and let s 2 be a Sacks column on an interval I 2 . If min(I 2 ) = max(I 1 ) + 1, then the product s ′ = s 1 ⊗ s 2 is the Sacks column on I 1 ∪ I 2 defined by f ∈ s ′ iff f ↾ I 1 ∈ s 1 and f ↾ I 2 ∈ s 2 . Iterating this, we can take products of finitely many properly stacked 7 Sacks columns.
We now define the norm of a Sacks column s on an interval I. Actually, we define a family of norms, using two parameters B and m. Later, we will virtually always use values of B and m determined by min(I); more details will come in Subsection 2.5 and Section 4. In other words, . The exact definition of this norm will not be important in the rest of the paper; we will only require the following properties.  Sacks (s i ) ≥ n + 1. Then for any "coloring" function π : i<m s i → B there are Sacks columns (5), just prune all unnecessary branches. In more detail, note that nor split (2 I ) = |I|, and that nor B,m Sacks is determined by the splitting-size nor split . So we have to finds ⊆ s with splitting size r := min(nor split (s), |I|). Obviously we can find the binary tree 2 ≤r inside s (as a suborder). Extend each of its maximal elements (uniquely), and take the downwards closure. This givess.
Setting the stage. We fix for the rest of this paper a nonempty (index) set Ξ. We furthermore assume that Ξ is partitioned into subsets Ξ ls , Ξ li , Ξ sk (Ξ li is nonempty, but Ξ ls and Ξ sk could be empty). For each ξ ∈ Ξ, we say that ξ is of type lim-sup, lim-inf or Sacks if ξ is an element of Ξ ls , Ξ li , or Ξ sk , respectively. We set Ξ non-sk := Ξ ls ∪ Ξ li = Ξ \ Ξ sk . Our forcing will "live" on Ξ × ω. For (ξ, ℓ) ∈ Ξ × ω we call ξ the index and ℓ the level.
The "frame" of the forcing will be as follows.
To be able to use this frame to construct a reasonable (in particular, proper) forcing, we will have to add several additional requirements of the following form. The Sacks intervals I sk,ℓ (that "appear" at sublevel ℓ) are "large" with respect to everything that was constructed in sublevels v below ℓ; and the subatoms at a subatomic sublevel u have "large" bigness with respect to everything that was constructed at sublevels v < u. The complete construction with all requirements will be given in Section 4.

Compound creatures.
We can now define compound creatures, which are made up from subatomic creatures and Sacks columns. (2) a nonempty, finite 9 subset supp of Ξ (3) for each ξ ∈ supp ∩ Ξ sk a Sacks column c(ξ) between m dn and m up ; (4) for each ξ ∈ supp ∩ Ξ non-sk and each subatomic sublevel u = (ℓ, j) with m dn ≤ ℓ < m up a subatom c(ξ, u) ∈ K ξ,u ; and (5) for each m dn ≤ ℓ < m up a real number d(ℓ) ≥ 0, called the "halving parameter" of c at level ℓ). 10 We additionally require "modesty": 11 (6) for each subatomic sublevel u with m dn < u < m up there is at most one ξ ∈ supp ∩ Ξ non-sk such that the subatom c(ξ, u) is not a singleton. Note that by (4) for each level ℓ with m dn ≤ ℓ < m up and each ξ ∈ supp ∩ Ξ non-sk there is a naturally defined atom c(ξ, ℓ) := (c(ξ, (ℓ, j))) j∈J ℓ .
We also write m dn (c), m up (c), supp(c), d(c, h). We will use the following assumptions (later there will be more; a complete list will be given in Section 4). 9 We could assume without loss of generality that the size of supp is at most m dn . This will be shown in Lemma 3.4.3.
10 One could (without loss of generality, in some sense) restrict the halving parameter to a finite subset of the reals; then for fixed supp, m dn , m up there are only finitely many compound creatures. 11 Again, without this requirement, the resulting forcing poset would be equivalent.
Using these assumptions, we can now define the norm of a compound creature.
Definition 2.5.3. The norm of a compound creature c, nor (c), is defined to be the minimum of the following values.
(1) The "width norm": (2) For each ξ ∈ supp ∩ Ξ sk the "Sacks norm" at index ξ:  where N := min{nor(c(ξ, h)) : ξ ∈ supp ∩ Ξ li }. 12,13 (So for both nor limsup and nor liminf we use the norms of atoms c(ξ, h); recall that the level h of this atom is used in Definition 2.2.3 of nor(c(ξ, h)), more specifically, µ h is used to measure the size of subsets of J h .) Remark 2.5.4. As supp(c) is nonempty, the width norm (and thus nor(c) as well) is at most m dn (c).
The assumptions imply the following.  . 13 The reason for the logarithm, and the use of the halving parameters, will become clear only in Section 5.2.
Proof. We can first use for all subatoms and Sacks columns the "large" ones guaranteed by the assumptions. However, this will in general not satisfy modesty. So we just apply Lemma 2.2.2 at each m dn ≤ ℓ < m up , resulting (for each ℓ) in disjoint sets A ℓ ξ ⊆ J ℓ for ξ ∈ supp ∩ Ξ non-sk . We keep the large subatoms at the sublevels in A ℓ ξ , and choose arbitrary singleton subatoms at other sublevels. Now we have a compound creature, whose norm is the minimum of the following: • the width norm; • the (unchanged) Sacks norms, which are ≥ m dn > nor width (supp); • the lim-sup norms, noting that all atoms at level ℓ have norm ≥ 2 ℓ·maxposs(<ℓ) − 1 ≥ 2 m dn ·maxposs(<m dn ) − 1 > nor width (supp), so all lim-sup norms drop by at most 1; and • the lim-inf norms, which drop by an even smaller amount, due to the logarithm.
Fact 2.5.6. Let c be a compound creature and u ⊆ supp(c) such that u∩Ξ sk , u∩Ξ li , u ∩ Ξ ls are all nonempty. Then the naturally defined c ↾ u is again a compound creature with norm at least nor(c).
Definition 2.5.7. A compound creature d is "purely stronger" than c, if c and d have the same m dn , m up , the same halving parameters, the same supp; and if for each ξ ∈ supp ∩ Ξ sk the Sacks column d(ξ) is stronger than c(ξ) and for each subatomic sublevel u that appears in c and ξ ∈ supp ∩ Ξ non-sk the subatom d(ξ, u) is stronger than c(ξ, u). (In other words, the only difference between c and d occurs at the Sacks columns and the subatoms, where they become stronger.) For r ≥ 0 we say that d is "r-purely stronger" than c, if additionally nor(d) ≥ nor(c) − r.
To show that our forcing has the ℵ 2 -cc, we will use the following property.
More generally, the same is true if c 1 and c 2 are not necessarily disjoint, but identical on the intersection u : Proof. Let d ′ be the "union" of c 1 and c 2 , which is defined in the obvious way. 14 As d ′ may not satisfy the modesty requirement (6) or Definition 2.5.1, we apply the procedure from the first part of the proof of Lemma 2.5.5 to ensure that the resulting object d does. Then d is a compound creature with norm ≥ x 2 − 1. (The factor 1 2 comes from doubling the size of the support, which decreases the width norm.) 2.6. The elements (conditions) of the forcing poset Q. Definition 2.6.1. ∅ is the weakest condition. Any other condition p consists of w p , (p(h)) h∈w p and t p such that the following are satisfied.
is a compound creature whose m dn is h, and whose m up is the w P -successor of h.
Note that Assumption 2.5.2 guarantees that Q is nonempty (cf. Lemma 2.5.5).
Notation 2.6.2. Given p ∈ Q, h ∈ w p and ℓ which is ≥ h and less than the w p -successor of h, and a sublevel u = (ℓ, j) we use the following notations.
, the Sacks column at index ξ starting at level h (note that we require h ∈ w p ).
2.7. The set of possibilities. We will now define the "possibilities" of a condition p, which give information about the possible value of the generic objectsỹ ξ and which we will use to define the order of the forcing. The possibilities of a condition p come from three sources, informally described below.
• The trunk t p , where there is a unique possibility.
• The Sacks columns p(ξ, h) (which we interpret as a set of possible branches) which "live" between h ∈ w p and the w p -successor h + of h. The possibilities of the whole Sacks column have to be counted as belonging to the sublevel (h, −1), i.e., we have to list them before the subatomic sublevel (h, 0), even though their domain reaches up to just below h + . This property of the Sacks columns will make our notation quite awkward. As a consequence, the following section has the worst ratio of mathematical contents to notational awkwardness. Things will improve later on. We promise.
We first (in 2.7.1) describe a way to define the set of possibilities separately for each ξ ∈ supp(p); all possibilities then are the product over the ξ-possibilities.
Then (in 2.7.2) we will describe a variant in which possibilities at a sublevel u are defined, and all possibilities are a product over the u-possibilities.
Both versions result in the same set of possibilities (up to an awkward but canonical bijectionl; see Fact 2.7.3). The first version is more useful in formulating things such as "a stronger condition has as smaller set of possibilities", whereas the second is the notion that will actually be used later in proofs.
Definition 2.7.1. Fix a condition p and an index ξ ∈ supp(p).
• If ξ ∈ Ξ non-sk , then for each subatomic sublevel u = (ℓ, j) we define the set poss(p, ξ, =u) to be either the singleton We then define poss(p, ξ, <u) to be the set of all functions η ∈ 2 [0,min(I sk,ℓ )) compatible 15 with the trunk and the Sacks columns of p at ξ. • We set poss(p, <u) to be ξ∈supp(p) poss(p, ξ, <u).
Note that each possibility below u restricted to the non-Sacks part can be seen as a "rectangle" with width supp(p) ∩ Ξ non-sk and height u; whereas the restriction to the Sacks part is a rectangle with height in w p (which is generally above u). So together this gives an "L-shaped" domain. Only in case u = (ℓ, −1) for ℓ ∈ w p do we get a more pleasant overall rectangular shape.
In the following alternative definition we ignore a part of p which is "trivial" because we have no freedom/choice left. More specifically, we ignore the trunk and singleton subatoms (but not singleton Sacks columns). Also, we do not first concentrate on some fixed index ξ, but directly define poss ′ (p, =u) for certain sublevels u. Definition 2.7.2. We define the set sblvls(p) of "active" sublevels of p by case distinction, and then for each u ∈ sblvls(p) we define the object poss ′ (p, =u).
if there is a non-singleton subatom at sublevel u, say at index ξ. In this case according to the modesty condition (6) of Definition 2.5.1 this is the only non-singleton subatom at u. We call ξ the "active index" at u, set p(u) := p(ξ, u) (the "active subatom") and define poss ′ (p, =u) := poss(p(u)). So sblvls(p) is a subset (and thus suborder) of the set of all sublevels, also of order type ω. We set poss ′ (p, <u) The definition of the following bijection ι is easy to see/understand, but very awkward to formulate precisely, and hence left as an exercise.
Fact 2.7.3. There is a natural/canonical correspondence ι : poss(p, <u) → poss ′ (p, <u). Given an η ∈ poss(p, <u), we first omit from η all the "trivial" information contained in the trunk and in the singleton subatoms; and then "relabel" the resulting sequence (instead of a sequence indexed by elements of ξ we wish to have one indexed by elements of sblvls(p)).
Later in this paper we will not distinguish between poss and poss ′ ; actually, we will mostly use poss ′ , and often use the following trivial observation. 15 In more detail, for each h < ℓ an element of {0, . . . , where we set poss ′ (p, >v) := v ′ ∈sblvls(p),v<v ′ <u poss(p, =v ′ ). poss ′ (p, =v) is a product of Sacks columns if v is Sacks, otherwise it is poss(x) for the active subatom at v.
2.8. The order of the forcing. Definition 2.8.1. A condition q is stronger than p, written q ≤ p, iff the following conditions hold.
The trunk t q of q extends the trunk t p of p and is "compatible" with p in the sense that for each ξ ∈ supp(p) the singleton poss(q, ξ, <trklg q (ξ)) is a subset of poss(p, ξ, <trklg q (ξ)). 17 (I.e., the subatoms and Sacks columns of p that disappeared have become part of the trunk of q which is compatible with the respective possibilities of p.) (4) If ξ ∈ supp(p) ∩ Ξ non-sk and u is a subatomic sublevel above trklg q (ξ), then the subatom q(ξ, u) is stronger than p(ξ, u).
, then the Sacks column q(ξ, h) is stronger than (i.e., a subset of) the product of the Sacks The halving parameters do not decrease; i.e., d(q, ℓ) ≥ d(p, ℓ) for all ℓ ∈ ω with ℓ ≥ min(w q ).
3. Some simple properties of Q 3.1. Increasing the trunk. We now introduce an obvious way to strengthen a condition: increase the trunk.
The definition of the order yields the following simple consequences.
• In particular, assuming that p and q are conditions that above some ℓ 1 have the same w and the same compound creatures, 18 and that poss(q, <ℓ 1 ) ⊆ poss(p, <ℓ 1 ), then q ≤ * p. 19 We can define a variant of ∧, which works for any sublevel (not only those Sacks sublevels u = (ℓ, −1) with ℓ ∈ w p ).
Definition 3.1.3. Given η ∈ poss(p, <u), we define p η as the condition q obtained by replacing the according parts of p with the singleton subatoms (or singleton Sacks columns) given by η. More formally, the only possible differences between p and q are that for each subatomic sublevel v < u and each ξ ∈ supp(p, v)∩Ξ non-sk the subatomic creature q(ξ, v) is the singleston subatom {η(ξ, v)}, and for each ℓ ∈ w p strictly below u and each ξ ∈ supp(p, ℓ) ∩ Ξ sk the Sacks column p(ξ, ℓ) is the We can now define the generic sequence added by the forcing. (Note that the generic filter will generally not be determined by this sequence, due to additional information given by w and the halving parameters.) For ξ ∈ Ξ sk , we setỹ ξ to be • For η ∈ poss(p, <u), p η ≤ p.
In particular, p forces thatȳ extends t p . • η ∈ poss(p, <u) iff 20 p does not force that η is incompatible with the generic realsȳ. • For η ∈ poss(p, <u), p "ȳ extends η ⇔ p η ∈ G." • Q forces thatȳ is "defined everywhere"; i.e.,ỹ ξ ∈ 2 ω for all ξ ∈ Ξ sk , and y η (u) ∈ POSS ξ,u is defined for all ξ ∈ Ξ non-sk and every subatomic sublevel u. Proof of the last item. Given a condition p and ξ ∈ Ξ, we have to show that we can find a q ≤ p with ξ ∈ supp(q). This is shown just as Lemma 2.5.8, using at ξ the large Sacks columns/subatoms guaranteed by 2.5.2. Then "increasing the trunk" shows thatỹ ξ (n) is defined for all n.
18 More formally, ℓ 1 ∈ w p , w p \ ℓ 1 = w q \ ℓ 1 , and p(h) = q(h) for all h ∈ w p \ ℓ 1 . Note that this implies supp(p) = supp(q). 19 Here, q ≤ * p means that q forces that p belongs to the generic filter. Equivalently, every r ≤ q is compatible with p. 20 For the direction "right to left", which we will not need in this paper, we of course have to assume that η has the right "format", i.e., η = ξ∈supp(p) η(ξ) and each η(ξ) has the appropriate length/domain.
Note that we can use the equivalent poss ′ (defined in 2.7.2) instead of poss. Formally, we could use the bijection ι of 2.7.3 and set p ∧ η ′ : <u)). But what we really mean is that for some η ′ ∈ poss ′ we can define p ∧ η ′ (p η ′ ) in the obvious and natural way; and this results in the same object as when using p ∧ η (p η) for the η ∈ poss that corresponds to η ′ (i.e., for η = ι −1 (η ′ )).

3.2.
The set of possibilities of stronger conditions. If q ≤ p, then poss(q, <u) is "morally" a subset of poss(p, <u) for any u.
If we just consider a sublevel (ℓ, −1) for ℓ ∈ w q then this is literally true: Assume that q ≤ p, ξ ∈ supp(p) and ℓ ∈ w q . Then poss(q, ξ, <ℓ) ⊆ poss(p, ξ, <ℓ). In the general case it is more cumbersome to make this explicit for the Sacks part. However, we will only need the following.
Remark 3.2.2. Note that q ≤ p does not imply sblvls(q) ⊆ sblvls(p), as a previously "inactive" sublevel of p can become active in q (with active index outside of supp(p), of course). Also, u can be an active subatomic sublevel in both p and q, but with different active indices. The "old" active subatom at ξ can shrink to a singleton in q, while q gains a new index with an active subatom (outside of supp(p)). Because of this, it is even more cumbersome to formulate an exact version of "stronger conditions have fewer possibilities" for poss ′ than it is for poss.
Proof. Assume that A = {p i : i ∈ ℵ 2 } is a set of conditions. By thinning out A (only using CH and the ∆-system lemma for families of countable sets), we may assume that there is a countable set ∆ ⊆ Ξ such that for p = q in A the following hold: , and, moreover, supp(p, ℓ) ∩ ∆ = supp(q, ℓ) ∩ ∆ for all ℓ ∈ w p ; and • p and q are identical on ∆, i.e., for each ℓ ∈ w p the compound creatures p(ℓ) and q(ℓ) are identical on the intersection, as in Lemma 2.5.8; and the trunks agree on ∆, i.e., t p (ξ, ℓ) is the same as t q (ξ, ℓ) for each ξ ∈ ∆ ∩ Ξ sk and ℓ < h(ξ), and analogously for the subatomic sublevels. As in Lemma 2.5.8 we can (for each p, q ∈ A and ℓ ∈ w p ) find a compound creature d(ℓ) "stronger than" both p(ℓ) and q(ℓ). These creatures (together with the union of the trunks) form a condition stronger than both p and q. Hence A is not an antichain.
3.4. Pruned conditions. Let p be a condition. All compound creatures p(ℓ) above some ℓ 0 will have norm at least 1. Note that by the definition of nor width this implies that | supp(p, ℓ)| ≤ ℓ.
The norm of a compound creature c is at most m dn (where we set m dn := m dn (c)).
We assumed that nor Furthermore, if we replace all Sacks columns in c with appropriate stengthenings, the resulting compound creature d will be 0-purely stronger than c. 21 This leads us to the following definitions.
Definition 3.4.1. We call a Sacks column s between ℓ and n Sacks-pruned if Sacks-pruned and all compound creatures p(h) have norm bigger than 1.

Definition 3.4.2.
A condition q is purely stronger (r-purely stronger ) than p, if w q = w p , t q = t p , and q(ℓ) is purely stronger (r-purely stronger) than p(ℓ) for all ℓ ∈ w q . (Note that this implies q ≤ p.) For every condition p there is a Sacks-pruned condition q which is 0-purely stronger than p. Given p ∈ Q Sacks-pruned, ℓ ∈ w p sufficiently large, and η ∈ poss(p, <ℓ), the condition q = p ∧ η < p is pruned.
In particular, we have the following. • If p is pruned, then | supp(p(h))| < h for all h ∈ w p .
• The set of pruned conditions in Q is dense.

3.5.
Gluing. So far we have increased trunks to strengthen conditions, as well as taking disjoint unions and pure strengthenings. This subsection introduces two more methods of strengthening conditions. • m dn (d) = m dn (c 1 ) and m up (d) = m up (c n ) (i.e., vertically the creature lives on the union of the levels of the old creatures). • supp(d) = supp(c 1 ) (i.e., the rectangle-shape of the new creature is the result of taking the union of the old rectangles and cutting off the stuff that sticks out horizontally beyond the base). • For ξ ∈ supp(d) ∩ Ξ non-sk and subatomic sublevels u between m dn (d) and By the definition of the norm (see 2.5.3), the monotonicity of B and maxposs (Assumption 2.5.2) and Lemma 2.3.6(2),(3) we get nor(glue(c 1 , . . . , c n )) ≥ min(nor(c 1 ), . . . , nor(c n )).
This gives another way to strengthen a condition p: shrinking the set w.
Definition 3.5.3. Given a condition p and an infinite subset U of w p such that min(U ) = min(w p ), we say that q results from gluing p along U if h n enumerates the elements of w p that are ≥ h and less than the w q -successor of h, then the compound creature q(h) is glue(p(h 1 ), . . . , p(h n )); and • the new parts of the trunk are compatible with p.
Note that q is not uniquely determined by p and U , as in general there are many choices to increase the trunk (in the last item). Of course, any such resulting q is stronger than p.
We have now seen five specific ways to strengthen a condition. Actually, every q ≤ p can be obtained from p by a combination of these methods. (We will not use the following fact, nor the subsequent remark, in the rest of the paper.) Fact 3.5.4. For p, q ∈ Q, q ≤ p iff there are p 1 , p 2 , p 3 and p 4 such that: (1) p 1 results from increasing the trunk in p, i.e., p 1 = p ∧ η for some η ∈ poss(p, < min(w q )) (in fact, for the unique η which is extended by t q ); (2) p 2 ≤ p 1 results from gluing p 1 along w q , as above.
(3) p 3 is purely stronger than p 2 ; (4) p 4 ≤ p 3 results from increasing halving parameters; and (5) q is the naturally defined "disjoint union" of p 4 and some condition p ′ which has the same w and halving parameters as p 4 , supp(p ′ ) is disjoint from supp(p 4 ), and which jointly satisfies "modesty" with p 4 .
• Every q obtained by the above construction is stronger than p, provided it is a condition. Note that constructions (1), (2) and (5) always result in conditions (for (5), this is the same argument as in 2.5.8), whereas constructions (3) and (4) will generally decrease the norms of the compound creatures in an uncontrolled fashion. So to get a condition, we have to make sure that the norms of the new compound creatures still converge to infinity. Also, to be able to find a suitable p ′ in (5), we should make enough room for modesty in (3).  • The order is not entirely irrelevant, as gluing (2) has to be done before pure strengthening (3), since glued Sacks columns always have the form of products, whereas generally the Sacks columns in q will not be of this form.
We will later use the following specific gluing construction. Proof. The lim-sup norm and the Sacks-norms will be large because nor(d n ) = nor(c n ) ≥ M . The lim-inf norm will be large because we did not change anything on the lim-inf part.
Then Q Ξ ′ is a complete subforcing of Q, and the restriction map is a projection on an open dense subset. Of course, Q Ξ ′ will satisfy all the properties that we will prove generally for Q (as Q Ξ ′ is defined just like Q, only with a different underlying index set).
Proof. The dense set D is the set of all conditions p with supp(p) ∩ Ξ li = ∅. Fix p ∈ D, set p ′ := p ↾ Ξ ′ , and assume that q ′ ≤ p ′ is in Q ′ . It is enough to show that q ′ is compatible with p. To do this we will construct q ≤ p such that q ′ = q ↾ Ξ ′ . Set p 1 := p ↾ (Ξ \ Ξ ′ ). Increase the trunk of p 1 to min(w q ′ ), glue along w q ′ , and increase the halving parameters to match those of q ′ to get a condition q 1 ≤ p 1 with w q1 = w q ′ . 23 Letting q be the disjoint union of q 1 and q ′ , it follows that q is a condition of Q, and clearly q ↾ Ξ ′ = q ′ .

An inductive construction of Q
We will now review the "framework" from Definition 2.4.1, finally giving all the assumptions (including the previous Assumption 2. 5.2) that are required to make the forcing proper.
In the following construction, we have the freedom to choose the following (as long as the assumptions are satisfied).
• Natural numbers H(<u) (for each sublevel u) such that H is increasing.
Remark. The function H gives us the possibility to impose additional demands on the bigness B (as given in (4.0.2), below). It is not needed to get properness and ω ω -bounding, but will be used later 24 in our specific constructions. • For each ξ ∈ Ξ non-sk and each subatomic sublevel u the subatomic family K ξ,u living on some finite set POSS ξ,u . The other parameters are determined by the construction.
This will turn out to be an upper bound to the cardinality of poss(p, <u) for any pruned condition p. 22 If we do not assume Ξ ′ ⊇ Ξli we get problems with the lim-inf norm when we combine the increased halving parameters of q ′ with the lim-inf creatures in p 1 . 23 As supp(p 1 ) ∩ Ξli = ∅ it follows that increasing the halving parameters does not affect the norms of the compound creatures, and therefore q 1 is a condition of Q. 24 Here is a very informal description of how H will be used. The basic requirement is that at each sublevel u we have bigness (namely B(u)) which is large with respect to everything that happened below. However, the notion of "large with respect to" will slightly depend on the actual construction that increases the relevant cardinal characteristic. The parameter H will allow us to accommodate these different interpretations. The function H will be used as a parameter when defining "rapid reading" in Definition 5.1.1.
• For each sublevel u, we set Note that, as usual, for a Sacks sublevel u = (ℓ, −1) we may write B(ℓ) for B(u).
(a) For each ξ ∈ Ξ non-sk , we require that K ξ,u is a subatomic family living on some finite set POSS ξ,u . (b) For each ξ ∈ Ξ non-sk , we require that there is a subatom x ∈ K ξ,u with norm at least 2 ℓ·maxposs(<ℓ) . (c) For each ξ ∈ Ξ non-sk , we require that K ξ,u is B(u)-big. (In particular this defines maxposs(<(ℓ + 1, −1)) if u = (ℓ, J ℓ − 1).) The assumptions guarantee that the previous Assumption 2.5.2 is satisfied (in particular that there are compound creatures with norm m dn , and that Q = ∅).
By induction, we immediately get the following (which is the reason for the name "maxposs"). The following shows that each p(u) is B(u)-big. Then whenever F : poss ′ (p, =u) → B(u) is a coloring, there is a strengthening q(u) of p(u) (i.e., either q(u) is a subatom stronger than p(u), or q(u) is a sequence of Sacks columns such that each one is stronger than the according column in p(u)) such that the subatomic norm (or, each Sacks norm) decreases by at most 1 and such that F ↾ poss ′ (q(u)) 26 is constant.
As B(u) is much larger than maxposs(<u), we also get a version of "compound bigness" (we will not directly use the following version, but we will use similar constructions). First note that a function G : poss ′ (p, ≤u) → H(<u) may be interpreted as F : poss ′ (p, =u) → H(<u) Y for Y := poss ′ (p, <u) (cf. 2.7.4). As | poss ′ (p, <u)| ≤ maxposs(<u), and B(u) is big with respect to maxposs(<u) and H(<u), we can use the previous item and strengthen p(u) to make G independent of the possibilities at u.
Iterating this downwards we get the following. • If G : poss ′ (p, <u) → H(<v) is a coloring, then we can strengthen the p(u ′ ) to q(u ′ ) for v ≤ u ′ < u, decreasing all subatomic/Sacks norms (and therefore also all compound norms) by at most 1, such that G restricted to poss ′ (q, <u) only depends on poss ′ (q, <v). • In particular, if G : poss ′ (p, <u) → 2, then we can strengthen p to q as above such that G ↾ poss ′ (q, <u) is constant.

5.
Properness, ω ω -bounding and rapid reading • Letτ be the name of an ordinal. We say thatτ is decided below the sublevel u (with respect to the condition p), if p η decides the value ofτ for all η ∈ poss(p, <u); in other words, there is a function R : poss(p, <u) → Ord such that p η τ = R(η) for all η ∈ poss(p, <u).
• We also write "τ is decided < u"; and we write "τ is decided ≤ u" for the obvious concept (i.e., "τ is decided < v", where v is the successor sublevel of u). • p essentially decidesτ , if there is some sublevel u such that τ is decided below u. • Letr be the name of an ω-sequence of ordinals. We say that a condition p continuously readsr, if allr(m) are essentially decided by p. • p rapidly readsr ∈ 2 ω , if, for each sublevel u,r ↾ H(<u) is decided below u.
• Let Ξ 0 ⊆ Ξ. We say that p "readsr continuously only using indices in Ξ 0 " if p readsr continuously and moreover (using the relevant functions R mentioned above) the value of R(η) depends only on η ↾ Ξ 0 . In other words: For every n there exists a sublevel u such that p η decides the value ofr(n) for all η ∈ poss(p, <u), and whenever η ↾ Ξ 0 = η ′ ↾ Ξ 0 , then p η and p η ′ agree on the value ofr(n).
• We define the notion "readsr rapidly only using indices in Ξ 0 " similarly.
• Instead of "only using indices in Ξ \ Ξ 1 " we also write "not using indices in Ξ 1 ". 26 Here poss ′ (q(u)) is either poss(q(u)) if u is a subatomic sublevel, or the product of the Sacks columns from q(u) if u is a Sacks sublevel.
Note that for X ⊇ Ξ li , a realr is read continuously from X iff it exists in the Q X -extension (cf. 3.6.1).
Remark 5.1.2. For a fixed condition p, the possibilities (at all sublevels) form an infinite tree in the obvious way. The set of branches T p of this tree carries a natural topology. p continuously reads τ iff there is a continuous function F on T p in the ground model such that p forcesτ =F (ȳ), whereF is the canonical extension of F .
In our case, the tree is finitely splitting, so T p is compact, and continuous is the same as uniformly continuous. (Note that the definition above really uses a uniform notion of continuity.) Rapid reading corresponds to a form of Lipschitz continuity. (1) If p continuously (or: rapidly) readsr and q ≤ p with supp(q) ⊇ supp(p), then q continuously (or: rapidly) readsr. The same holds if we add "only using Ξ 0 " or: "not using Ξ 1 ".
(2) If q ≤ * p, andτ is a name of an ordinal essentially decided by p, then also q essentially decides τ . Proof.
The formal proof uses Lemma 3.2.1.
(2) p forces thatτ is decided by a finite case distinction; so q forces the same. Proof. This is the usual "nice names" argument: Given p continuously readingr. We can define the obvious namer ′ continuously read by p ′ = p ↾ Ξ 0 , such that p forcesr =r ′ . There are at most κ many countable subsets of Ξ 0 , and therefore only κ many conditions p ′ with supp(p ′ ) ⊆ Ξ 0 . Given such a condition p ′ , there are only 2 ℵ0 many ways to continuously read a real (with respect to p ′ ).
We will first show that we can "densely" get from continuous reading to rapid reading. Later we will show that "densely" we can continuously read reals. Both proofs are the obvious modifications of the corresponding proofs in [KS12].
Lemma 5.1.5. Assume that p continuously readsr ∈ 2 ω , then there is a q ≤ p rapidly readingr.
The same is true if we add "only using Ξ 0 ".
Proof. Without loss of generality we can assume that p is pruned (use Lemmas 3.4.3 and 5.1.3). For a sublevel u, we set (5.1.6) v dec (u) is the maximal sublevel such thatr ↾ H(<v dec (u)) is decided below u, The function v dec is nondecreasing; and continuous reading implies that v dec is an unbounded function on the sublevels; but v dec can generally grow very slowly. (p "rapidly readsr" would mean that v dec (u) ≥ u for all u.) 27 More formally: reals r such that there is a p ∈ G and a namer such that p continuously readsr only using Ξ 0 and such that G evaluatesr to r.
For all sublevels v ≤ u we set (5.1.7) x u v :=r ↾ (H(< min(v, v dec (u)))) (which is by definition decided below u). There are at most many possibilities forx u v , as H((< min(v, v dec (u)))) ≤ H(<v). 1: For now, fix a Sacks sublevel u = (ℓ, −1) with ℓ ∈ w p . We will define (or rather: pick) by downwards induction on u ′ ∈ sblvls(p), u ′ ≤ u, objects d u u ′ , which are either a sequence of Sacks columns (if u ′ is Sacks) or a subatom; and functions ψ u u ′ . 1a: For u ′ = u, we set d u u := p(u), i.e., the sequence of Sacks columns of level ℓ. We let ψ u u be the function with domain poss(p, <u) which assigns to each η ∈ poss(p, <u) the corresponding value ofx u u . In other words: p η forces thatx u u = ψ u u (η) for each η ∈ poss(p, <u). 1b: We continue the induction on u ′ . For now, we write d ′ := d u u ′ , ψ ′ := ψ u u ′ , and , each η ∈ poss(p, <u ′ ) decidesx ′ to be ψ ′ (η), by which we mean: p η forces the following: If the genericȳ is compatible with d u v for each sublevel v ∈ sblvls(p) with u ′ ≤ v < u, thenx ′ = ψ ′ (η).
We can write 28 ψ ′′ 0 as a function A × B → C, for A := poss(p, <u ′ ), B = poss(p, =u ′ ) and C is the set of possible values ofx ′ , which has, according to (5.1.8), size ≤ 2 H(<u ′ ) . This defines a function from B to C A , a set of cardinality ≤ 2 maxposs(<u ′ )·H(<u ′ ) ; so according to (4.0.2) and Fact 4.0.4 we can use bigness at sublevel u ′ to find d ′ such that ψ ′′ 0 does not depend on sublevel u ′ . This naturally defines ψ ′ . 2: We perform this downwards induction from each Sacks sublevel u of p. So this defines for each v < u in sblvls(p) the objects d u v and ψ u v , satisfying (which is just 5.1.9): Also, the norms of each Sacks column and subatom drop by at most 1.

3:
Note that for a given v, there are only finitely many possibilities for d u v and ψ u v . So by König's Lemma there is a sequence ( : We now construct q by replacing the subatoms and Sacks columns in p at sublevel v with d * v (for each v ∈ sblvls(p)). So q has the same w as p, the same supports, the same halving parameters and the same trunk; and all norms decrease by at most 1. We claim that q rapidly readsr, i.e., we claim that each η ∈ poss(q, <v) decidesr ↾ H(<v).
According to the definition (5.1.6), this means thatr ↾ H(<v) is decided below v ′ . Then pick u > v ′ as in (5.1.11). Recall (from (5.1.10)) thatx u v is decided below v by ψ u v modulo the sequence x u v is decided already (by the original condition p) below v ′ . So we can omit the assumption that the generic is compatible with d u u ′′ for any v ′ ≤ u ′′ < u and still correctly computex Halving and unhalving. We will now, for the first and only time in this paper, make use of the halving parameter. We will show how to "halve" a condition q to half(q), and then "unhalve" any r ≤ half(q) with "positive norms" to some s ≤ * q with "large norms". This fact will only be used in the next section, to show pure decision.
If we increase d := d(c, h) to then the resulting lim-inf norm (hence also the compound norm) decreases by at most 1 /maxposs(<m dn ).
Definition 5.2.2. Given a compound creature c, we set half(c) to be the same compound creature as c, except that we replace each halving parameter d(h) by the d ′ (h) described above. So nor(half(c)) ≥ nor(c) − 1 /maxposs(<m dn ). Similarly, given a condition p and a level h ∈ w p , we set half(p, ≥h) to be the same as p, except that all compound creatures p(ℓ) for ℓ ≥ h are halved (and nothing changes below h).
The point of halving is the following: Assume that the norms in q are "large" and that r ≤ half(q) has norms that are just > 0. Then there is an "unhalved version" of r, an s ≤ q, such that the norms in s are "large" and still s ≤ * r.
In more detail: • a condition q, • h ∈ w q such that nor(q(ℓ)) ≥ M for all ℓ ≥ h in w q , • a condition r ≤ half(q, ≥h) such that min(w r ) = h and nor(r(ℓ)) > 0 for all ℓ in w r .
Then there is an s such that (1) s ≤ q.
• The trunks agree above h 1 .
• So in particular, supp(s) = supp(r), and the norms do not change above h 1 (hence are ≥ M ). Note that (5)  Proof. First fix h 0 ∈ w r bigger than h such that nor(r(ℓ)) > M for all ℓ ≥ h 0 . Let h 1 be the w r -successor of h 0 .
We set w s := {h} ∪ w r \ h 1 . The trunk t s will extend t r (and will contain some additional in the "area" [h, h 1 ) × (supp(r, h 0 ) \ supp(q, h))).
For ℓ ≥ h 1 in w s , we set s(ℓ) := r(ℓ). We set d 0 := glue(r(h), . . . , r(h 0 )), and choose arbitrary r-compatible elements for the new parts of the trunk t s . We then let d 1 be the restriction of d 0 to supp(q, h) (again, choosing r-compatible elements for the new parts of the trunk t s ). Now we construct d from d 1 by replacing each halving parameter d d1 (k) by d q (k) (for all h ≤ k < h 1 ). We set s(h) = d. This completes the construction of the condition s.
It is straightforward to check that the requirements are satisfied. We will show nor(s(h)) = nor(d) ≥ M − 1 /maxposs(<h): The norm of d is the minimum of several subnorms: • The width norm, which is ≥ M , as supp(d) = supp(q, h) and nor(q(h)) ≥ M . • The Sacks norms of the Sacks columns d(ξ) = r(ξ, h) ⊗ · · · ⊗ r(ξ, h 0 ) for ξ ∈ supp(d) ∩ Ξ sk : nor Sacks (d(ξ)) = nor  So we have to show that for h ≤ ℓ < h 1 , (5.2.1). These are the halving parameters used in half(q), and since r ≤ half(q) we know that d r (ℓ) ≥ d ′ (ℓ) (where d r are the halving parameters used in r).
Let m ∈ w r correspond to ℓ (i.e., m ≤ ℓ and ℓ less than the w r -successor of m). As nor(r(m)) > 0, we know that for N d ℓ as above. 29 Fix any ξ ∈ supp(q, h) ∩ Ξ li . Let k ∈ w q correspond to ℓ (as above), and set c = q(k). The inequality above gives 0 < log 2 (nor(d(ℓ, ξ)) − d r (ℓ)), which implies for all ξ, and so Suppose thatτ is a name for an element of V , that p 0 ∈ Q, that M 0 ∈ w p0 and n 0 ≥ 1 are such that nor(p 0 (h)) ≥ n 0 + 2 for all h ∈ w p0 \ M 0 . Then there is a condition q such that: • q ≤ p 0 .
• Below M 0 , q and p 0 are identical, 30 i.e.: Proof. We may assume that p 0 is pruned. Our proof will consist of several steps: 1. Using halving; the mini-steps.
Suppose that we are given p ∈ Q, M ∈ w p , and n ≥ 1 such that nor(p(h)) > n for all h ∈ w p \ M . We show how to construct an extension of p, denoted r(p, M, n).
First enumerate poss(p, <M ) as (η 1 , . . . , η m ). Note that m ≤ maxposs(<M ). Setting p 0 = p, we inductively construct conditions p 1 , . . . , p m and the auxiliary conditionsp 1 , . . . ,p m so that for each k < m the following holds: 29 The last ≤ holds since r(m) contains the same subatoms as d (on the common support; however the support of r(m) may be larger, therefore the last inequality is not necessarily an equality). 30 supp(q) can be larger than supp(p), so below M 0 there will be new parts of the trunk t q .
(1)p k+1 is p k where we replace everything below M (and in supp(p)) with η k+1 . Remarks: • By (3) below, we will get min(wp k+1 ) = M . • If k = 0, thenp 1 is just p ∧ η 1 . But for k > 0, η k+1 will not be in poss(p k , <M ), so we cannot use the notationp k+1 = p k ∧ η k+1 . • Note that generally supp(p k ) will be larger than supp(p), so we do not replace the whole trunk below M by η k+1 , but just the part in supp(p). (2) p k+1 ≤p k+1 . Note that we do not have p k+1 ≤ p k , for trivial reasons: their trunks are incompatible. Remarks: • So by strengtheningp k+1 to p k+1 , we do not increase the overall trunklength min(w). • Note that we do not assume that w p k+1 = w p k \ M , i.e., generally the w-sets will become thinner due to gluing. (4) supp(p k+1 , M ) = supp(p, M ).
• Remark: This only holds at level M : Generally, supp(p k+1 ) will be larger than supp(p k ).
• (halve) p k+1 = half(p k+1 , ≥M ). More explicitly: If the deciding case is possible, then we use it. Only if it is not possible, we halve.
We then define r = r(p, M, n) as follows: Below M , r is identical to p; and above (including) M , r is identical to p m (the last one of the p k constructed above). In more detail: • w r = (w p ∩ M ) ∪ (w p m \ M ); i.e., below M the levels of r are the ones of p; and above (including) M the levels of r are the ones of p m .
• This determines the domain of t r ; and we set t r to be t p m restricted to this domain. r = r(p, M, n) has the following properties: • r ∈ Q, r ≤ p.
• If η ∈ poss(r, <M ) and if there is a s ≤ r ∧ η such that s essentially decidesτ , min(w s ) = M and nor(s(ℓ)) > 0 for all ℓ ≥ M in w s , then r ∧ η essentially decidesτ .
Proof of (5.3.2). η extends some η k+1 ∈ poss(p, <M ); so s ≤ r ∧ η ≤ p k+1 ≤p k+1 . All we have to show is that p k+1 was constructed using the "decide" case. Assume towards a contradiction that the "halve" case was used. Then s is stronger than half(p k+1 , ≥M ), so we can unhalve it (using Lemma 5.2.3) to get some s ′ ≤p k+1 with large norm such that s ′ ≤ * s, showing that we could have used the "decide" case after all. This ends the proof of (5.3.2).
2. Iterations of the mini-steps; the condition q. Given p 0 , M 0 , n 0 as in the statement of the Lemma, we inductively construct conditions p k and natural numbers M k for k ≥ 1. Given p k and M k , our construction of p k+1 and M k+1 is as follows: Choose M k+1 ∈ w p k bigger than M k such that nor(p k (h)) > k + n 0 + 3 for all h ∈ w p k \ M k+1 . Then set p ′ k+1 = r(p k , M k+1 , k + n 0 + 3), and construct p k+1 by gluing together everything between (including) M k and (excluding) M k+1 .
The sequence of conditions (p k ) k∈ω converges to a condition of Q, which we will denote by q. Note that r ≤ q implies that w r is a subset of (w p0 ∩ M 0 ) ∪ {M 0 , M 1 , M 2 , . . . } (as we have glued everything between each M i and M i+1 ).
It is clear that q ≤ p 0 , and that nor(q, h) > n 0 + 1 for all h ∈ w q \ M 0 . We will later show that q essentially decidesτ (thus proving the lemma).
The following property will be central: Assume that η ∈ poss(q, <M ℓ ) for some ℓ ∈ ω, and r ≤ q ∧ η essentially decidesτ and min(w r ) = M ℓ and each r(m) has norm > 1 for each m ∈ w r . Then q ∧ η essentially decidesτ . Proof of (5.3.3): η (or rather: a restriction of η to supp(p)) was considered as a possible trunk η k+1 in the "mini-step" when constructing r(p ℓ−1 , M ℓ , ℓ + n 0 + 2). So we can use (5.3.2). This ends the proof of (5.3.3). 3. Using bigness to thin out q to prove essentially deciding.
We now repeat the construction of the proof of Lemma 5.1.5, but this time we do not homogenize on the potential values of somex, but rather on whether q η essentially decidesτ or not.
• We set d u u to be the collection of Sacks columns q(u). We set B u u to be the set of η ∈ poss(q, <u) such that q η essentially decidesτ .
• By downwards induction on u ′ ∈ sblvls(q), (M 0 , −1) ≤ u ′ < u, we construct d u u ′ and B u u ′ such that the following is satisfied: d u u ′ is a strengthening of the subatom (or: collection of Sacks columns) q(u ′ ), the norm decreases by at most 1. -(Homogeneity) B u u ′ is a subset of poss(q, <u ′ ), such that for each η ∈ B u u ′ and each ν ∈ poss(d u u ′ ) η ⌢ ν ∈ B u u ′ +1 ; and analogously for each η ∈ poss(q, <u ′ ) \ B u u ′ and each ν ∈ poss(d u u ′ ), η ⌢ ν / ∈ B u u ′ +1 . (Just as in the case of rapid reading, we can find these objects using bigness: Assume that u ′′ is the sblvls(q)-successor of u ′ ; by induction there is a function F which maps each η ∈ poss(q, <u ′′ to {∈ B, / ∈ B}; we thin out q(u ′ ) to d u u ′ such that for each ν ∈ poss(q, <u ′ ) each extension of ν compatible with d u u ′ has the same F -value F * (ν); this in turn defines B u u ′ .) • Assume that v < u as above, that η ∈ poss(q, <v), that q η essentially decidesτ and that η ′ ∈ poss(q, <u) extends η. Then trivially q η ′ also essentially decidesτ . So we get: If q η essentially decidesτ for η ∈ poss(q, <v), then η ∈ B u v for any u > v.
• We now show the converse: (5.3.5) Whenever η ∈ B u u ′ for some sublevel u ′ of the form (M ℓ ′ , −1) ≤ u for some ℓ ′ , then q ∧ η essentially decidesτ . (Equivalently: q η essentially decidesτ , as q η = * q ∧ η.) Proof: We can modify q to a stronger condition r using η as trunk and using d u u ′′ for all u ′ ≤ u ′′ ≤ u. Any η ′ ∈ poss(r, <u) is in B u u , so q η ′ = * r η ′ essentially decidesτ . So r essentially decidesτ . Also, each compound creature in r has norm > 1, so we can use (5.3.3). This ends the proof of (5.3.5).
• So to show that q essentially decidesτ , it is enough to show that for all η ∈ poss(q, <(M 0 , −1)) there is a u such that η ∈ B u (M0,−1) . • As in the rapid reading case, we choose an "infinite branch" This defines a condition q 1 ≤ q.
So fix such an η. Find any r ≤ q 1 ∧ η decidingτ . Without loss of generality, min(w r ) = M ℓ for some ℓ, and each compound creature in r has norm at least 1. Let η ′ > η be the trunk of r (restricted to supp(q) and M ℓ ). According to (5.3 . So according to (5.3.5), q ∧ η essentially decidesτ .

5.4.
Properness, ω ω -bounding, rapid reading, no randoms. A standard argument now gives the following: Theorem 5.4.1. Q satisfies (the finite/ω ω -bounding version of ) Baumgartner's Axiom A, in particular it is proper and ω ω -bounding and (assuming CH in the ground model) preserves all cofinalities. Also, Q rapidly reads everyr ∈ 2 ω . Proof. We already know that we can rapidly read each real if we can continuously read it.
We define q ≤ n p as: q ≤ p and there is an h ∈ w q , h ≥ n, such that q and p are identical below h and nor(q(ℓ)) > n for all ℓ ≥ h.
It is clear that any sequence p 0 ≥ 0 p 1 ≥ 1 p 2 ≥ 2 . . . has a limit; and Lemma 5.3.1 shows that for any nameτ of an ordinal, n ∈ ω and p ∈ Q, there is a q ≤ n p such that modulo q there are only finitely many possibilities forτ .
Rapid reading gives us: Lemma 5.4.2. Every new real is contained in a ground model null set, i.e., no random reals are added. So assuming CH in the ground model, we will have cov(N ) = ℵ 1 in the extension.
6. The specific forcing and the main theorem 6.1. The forcing. Recall that Ξ is partitioned into Ξ sk , Ξ li and Ξ ls . We now further partition Ξ ls into Ξ nn and Ξ cn . So every ξ ∈ Ξ has one of the following four types: • type sk (Sacks) for ξ ∈ Ξ sk , • type cn (cofinality null) for ξ ∈ Ξ cn , • type nn (non null) for ξ ∈ Ξ nn , and • type nm (non meager) for ξ ∈ Ξ li . So nm is the only lim-inf type.
Let κ t be the size of Ξ t .
In the inductive construction of Q in Section 4, several assumptions are made in the subatom stages u. We will satisfy those assumptions in the following way: For each type t ∈ {cn, nn, nm} we assume that we have a family of subatomic families K ′ t,b indexed by a parameter b, such that for each b ∈ ω, K ′ t,b is a subatomic family living on some POSS ′ t,b satisfying b-bigness. Actually, we will require a stronger variant of b-bigness such that we can find an homogeneous successor subatom while decreasing the norm not by 1 but by at most 1/b. I.e., we require: Additionally we require that (6.1.2) there is at least one subatom in Then we set for each subatomic sublevel u = (ℓ, j) where v is the largest 31 subatomic sublevel smaller than u. So the sequence b(u) is strictly (actually: very quickly) increasing. According to the definition 4.0.2 of B(u), we also get: Lemma 6.1.4. b(u) ≥ 2·maxposs(<u), and even b(u) ≥ 2 (number of sublevels below u)·maxposs(<u) .
Then we set (for all ξ ∈ Ξ t ) K ξ,u := K ′ t,b(u) . This way we automatically satisfy requirements (b) and (c) of item ( * 5) on page 23. And since there are only four, i.e., finitely many, types, there is automatically a bound M on | POSS ξ,u | as required in (d).
Strong bigness gives us the following property: Lemma 6.1.5. Let I be a finite set of subatomic sublevels (and thus I is naturally ordered). Let v be the minimum of I. For each u ∈ I let ξ u ∈ non-sk and x u a subatom in K ξu,u . Let F : u∈I poss(x u ) → b(v). Then there are y u < x u with nor(y u ) ≥ nor(x u ) − 1 /b(u) and such that F ↾ u∈I poss(y u ) is constant. Proof. We construct y u by downwards induction on u ∈ I: Let u ′ be the maximum of I, then F can be written as function from poss(x u ′ ) to b(v) P , where P = u∈I\{u ′ } poss(x u ). As |P | is less than the number of sublevels below u ′ times maxposs(<u ′ ), we get |P | < b(u ′ ), and thus can use strong bigness to get y u ′ < x u ′ . Now continue by induction.
The families K ′ t,r that we will actually use are described in Section 10 for t = cn, Section 8 for t = nn, and Section 7 for t = nm.
In addition, we will define there for each K ′ t,b a number H ′ (t, =b), and in the inductive construction, we define H as follows: Definition. H(<(0, −1)) := 3. If u = (ℓ, j) is a sublevel with immediate predecessor u ′ , we define H(<u) = H(≤u ′ ) in cases by: • So in particular, if p rapidly readsr, then for all t ∈ {nm, nn, cn} and all subatomic sublevels u Note that once we fix the parametrized subatomic families K ′ t,b and H ′ (t, =b) (and the cardinalities κ t ), we have specified everything required to construct Q, and Q will satisfy Baumgartner's Axiom A, will be ω ω -bounding, and, assuming CH, will have the ℵ 2 -cc. We also get rapid reading.
6.2. The main theorem. We will show: Theorem 6.2.1. Assume (in V ) CH, κ nm ≤ κ nn ≤ κ cn ≤ κ sk and κ ℵ0 t = κ t for t ∈ {nm, nn, cn, sk}. Then there is a forcing Q which forces (1)  As mentioned above, we fix disjoint index sets Ξ t (t ∈ {sk, cn, nn, nm}) of respective sizes κ t , and we construct Q as described above. Then the following points are obvious or have already been shown: (1) d = ℵ 1 , since Q is ω ω -bounding. And it was already shown in Lemma 5.4.2 that no random reals are added, so cov(N ) = ℵ 1 . (5) If α = β ∈ Ξ sk , then the generic reals at α and β are forced to be different, so we have at least κ sk many reals. Every real in the extension is read continuously, so by Lemma 5.1.4 there are at most κ ℵ0 sk = κ sk many reals. (•) The "moreover" part is clear because Q satisfies Baumgartner's Axiom A and has the ℵ 2 -cc. In the rest of the paper, we will describe the families K ′ t,b and H ′ (t, =b) and prove the remaining parts of the main theorem: 6.3. The Sacks part: cof(N ) ≤ κ cn . We will show that every null set added by Q is contained in a null set which is already added by the non-Sacks part.
We will first show that the quotient Q/Q Ξnon-sk (in other words: the extension from the universe obtained not using the sacks coordinates to the full generic extension) has the Sacks property.
Recall that the Sacks property states (or, depending on the definition, is equivalent to): Every function in ω ω in the extension is caught by an (n + 2)-slalom from the ground model. (I.e., there is a function S : ω → [ω] <ω in the ground model with |S(n)| ≤ n + 2, and f (n) ∈ S(n) for all n.) The Laver property is similar, but only applies to functions f in the extension which are bounded by a ground model function.
We get Lemma 6.3.1.
(1) Laver property is equivalent to: Wheneverr ∈ 2 ω is in the extension and G : ω → ω in the ground model, then there is in the ground model a tree T (without terminal nodes) such thatr ∈ [T ] and |T ↾ 2 G(n) | < n + 2 for all n.  (1), we only show how to get the Laver property (which is enough for this paper, and the other direction is similarly easy). Suppose that g : ω → ω is given. Enumerate {(n, m) : m ≤ g(n)} in lexicographic order as (n i , m i ). Define a function G : ω → ω by G(n) = min{i : n i > n} = n + 1 + k≤n g(k).
(For convenience we will think of G(−1) = 0.) Note that according to the enumeration given above, every function r : ω → 2 determines a subset of n<ω (g(n) + 1) by {(n i , m i ) : r(i) = 1}. Accordingly, certain functions r induce a function bounded by g: those functions r such that given any n there is a unique m ≤ g(n) such that (n, m) is in the subset determined by r as described above. (Equivalently, for each n there is a unique G(n − 1) ≤ i < G(n) such that r(i) = 1.) Given such an r, by val(r, n) we denote m i where G(n − 1) ≤ i < G(n) is such that r(i) = 1.
Note that given any function f bounded by g there is a unique function r f : ω → 2 (which determines a function bounded by g as described above) such that val(r f , n) = f (n) for all n.
Suppose thatf is a name for a function bounded by the ground model function g. Letr f be a name for the function ω → 2 as described above, and let T be the tree guaranteed to exist by the assumption (using the function G defined from g above). We may assume that all branches x of T determine a function bounded by g as described above. Now define a slalom S by S(n) = {val(x, n) : x ∈ [T ]}. It is clear that S catchesf .
We now prove our version of the Laver property for the quotient. As the whole forcing is ω ω -bounding, this implies the Sacks property. (1) Assume that p is a condition,r ∈ 2 ω a name and G : ω → ω is in V . Then there is a q ≤ p and a nameT ⊆ 2 <ω (of a tree without terminal nodes) such that: q continuously readsT not using any Sacks indices; q forces r ∈ [T ]; and |T ↾ 2 G(n) | < n + 2 for all n.
(2) Therefore the quotient Q/Q Ξnon-sk has the Laver property (and thus the Sacks property).
Proof. If G 1 (n) ≤ G 2 (n) for all n, andT witnesses the conclusion of the lemma for G 2 , thenT also witnesses the lemma for G 1 . So we may without loss of generality increase the function G whenever this is convenient. We can assume that p rapidly readsr, i.e., poss(p, <n) determinesr ↾ H(<n) for all n ∈ w p .
We can then assume that there is a strictly increasing function G ′ such that G ′ (n) ∈ w p and G(n) = H(<G ′ (n)) for all n (as we can increase G). Also, to simplify notation, we can assume that So each η ∈ poss(p, <G ′ (n)) determines a value forr ↾ G(n), which we call R n (η). We view η as a pair (η sk , η non-sk ) for η t := η ↾ Ξ t for t ∈ {non-sk, sk}. Accordingly we write R n (η sk , η non-sk ). If we fix η sk , then R n (−, η sk ) can be viewed as a name (for an element of 2 G(n) ) which does not depend on the Sacks part, in the following way: If there is a η non-sk compatible with the generic filter such that (η non-sk , η sk ) = η ∈ poss(p, <G ′ (n)), then the value is R n (η) (and otherwise ∅, say).
Below we will construct q ≤ p by gluing and by strengthening Sacks columns (and we will leave the support, the subatoms and the halving parameters unchanged).
We set h 0 = 0; so G ′ (h 0 ) = min(w p ) and q below G ′ (h 0 ) has to be identical to p. And (⋆ 0 ) holds as S 0 is a singleton.
Assume we have already constructed h n and q below G ′ (h n ), satisfying (⋆ ℓ ) for ℓ ≤ h n .
(1) For any I and s ⊆ 2 I , we write nor * Sacks (s) for nor B(G ′ (hn)),G ′ (hn) Sacks (s), see 2.3.5. (I.e., the Sacks norm that would be assigned to a Sacks column starting at G ′ (h n ) which has the same nor split as s.) Let Σ := supp(p, G ′ (h n )) ∩ Ξ sk , the set of Sacks indices active at the current level. Let s be minimal such that nor * Sacks (2 s ) ≥ n, and define h ′ by (6.3.3) h ′ := (h n + 1) · 2 s·|Σ| .
By the definition of s, we have nor Sacks (q(ξ, h n )) ≥ n, and therefore nor(q(h n )) ≥ min(n, nor(p(h n ), . . . , nor(p(h n+1 − 1)))). So in particular the q we get after the induction will be an element of Q.
By Lemma 6.3.1(3), we conclude: (1) IfÑ is the name of a null set and p a condition, then there is a q ≤ p and some name of a null setÑ ′ not depending on any Sacks indices such that q forcesÑ ⊆Ñ ′ .
6.4. Lim-inf and lim-sup: non(M) ≤ κ nm . The following does not require any knowledge about the particular subatoms used in the forcing construction, the only relevant fact is that the nm indices are the only ones that use a lim-inf construction. Proof. We claim that the set of all reals that can be read continuously from nmindices is not meager. This set has size ≤ κ nm by Lemma 5.1.4.
LetM be a name for a meager set. We can find namesT n ⊆ 2 <ω for nowhere dense trees such thatM = n∈ω [T n ] is forced. We want to show that we can continuously read a realr / ∈M using only the nm-indices. As Q is ω ω -bounding andT n is nowhere dense, there is in V a function f n : ω → ω such that for each ν ∈ 2 k there is a ν ′ ∈ 2 fn(k) extending ν and not inT n .
We fix some p ∈ Q forcing the above, and assume that p is pruned and continuously readsT n for each n. We will construct (in V ) a q ≤ p and anr continuously read by q only using nm indices, such that q forcesr / ∈M . Assume we have already constructed q below some k n ∈ w q , and that we already have some h n ∈ ω and a namel n for an element of 2 hn that is decided by poss(q, <k n ) ↾ Ξ nm . (The realr will be the union of thel n .) We also assume that is already guaranteed thatl n is not inT 0 ∪ · · · ∪T n−1 ).
Enumerate poss(q, <k n ) as η 0 , . . . , η K−1 . Set k 0 := k n , h 0 := h n ,l 0 :=l n , and we define q ′ below k 0 to be q. By induction on r ∈ K we now deal with η r : Assume we are given a namel r for an element of 2 h r that is decided by poss(q ′ , <k r ) ↾ Ξ nm , and that we have constructed q ′ below k r ∈ w p , in a way that between k 0 and k r on the non-nm indices, all subatoms and Sacks columns in q ′ are singletons.
Set h r+1 := f n (h r ). Choose k r+1 ∈ w p bigger than k r and large enough to determineX :=T n ↾ h r+1 . I.e., there is a function F from poss(p, <k r+1 ) to potential values ofX. We now define q ′ between k r and k r+1 : The nm-subatoms are unchanged (i.e., the ones of p), for the other subatoms and Sacks columns, we choose arbitrary singletons. A ν ∈ poss(p, <k r+1 ) consists of: the part below k r called A, then non-nm-part above k r called B, and the nm-part above k r called C. So we can writeX = F (A, B, C). If we assume that the generic chooses η r (i.e., A = η r ) and then follows the singleton values of q on the non-nm-part (which determines B to be some B q ), thenX can be written as nm-name. More formally: We can defineX ′ as F (η r , B q , −), which is a nm-name and forced by q to beX.
Also, we know that p forces that there is an element ℓ ′ ∈ 2 h r+1 which extendsl r (which by induction is already determined by the nm-part of η r ) and which is not inX. So (in V ) we can pick for all choices of C an ℓ ′ (C) ∈ 2 h r+1 \ F (η r , B q , C) extendingl r . Thenl r+1 = ℓ(−) is a nm-name determined below k r+1 , and q forces thatl n+1 extendsl n , and q ∧ η r forces thatl n+1 / ∈T n . We repeat the construction for all r ∈ K, and set ℓ n+1 := ℓ K , h n+1 := h K and set k n+1 to be the w p -successor of k K , where we use the Sacks columns and subatoms of p between k K and k n+1 . We now glue the condition between k n and k n+1 . This results in a condition that still has "large" norm, as described in Lemma 3.5.6. 7. The nm part 7.1. The subatomic creatures for type nm. We now describe the subatomic family K ′ nm,b used at nm-indices (depending on the parameter b). Definition 7.1.1.
(1) Fix a finite index set I ⊆ ω which is large enough so that item (4) below is satisfied. For notational simplicity, we assume that I is disjoint to all intervals already used. 32 Clearly, the norm satisfies strong b-bigness (i.e., satisfies the requirement (6.1.1)).
Note 7.1.2. We just used the simplest possible norm here. It turns out that the details of the definition of this norm are not relevant, as long as the norm has bigness. Later in section 11 we will use a different norm to get a different constellation of cardinal characteristics.
7.2. The generic object. Recall that (according to Section 6.1) when constructing the forcing at subatomic sublevels u, we use for all ξ ∈ Ξ nm the subatomic family K ξ,u = K ′ nm,b(u) living on some interval I, which we will call I nm,u . Fix α of type nm. Recall that the generic objectỹ α assigns to each subatomic sublevel u the element of POSS α,u chosen by the generic filter.
We define the nameM α of a meager set as follows: (7.2.1) A real r ∈ 2 ω is inM α iff for all but finitely many levels ℓ there is a subatomic sublevel u = (ℓ, j) such that r ↾ I nm,u =ỹ α (u).
Lemma 7.3.1. Letr be a name of a real, p a condition that rapidly readsr not using 33 α ∈ Ξ nm . Then p forces thatr ∈M α .
Proof. It is enough to prove that some q ≤ p forces thatr ∈M α : Assume that p does not forcer ∈M α , then some p ′ ≤ p forces the negation; p ′ still rapidly readsr not using α, so if we know that there is a q ≤ p ′ as claimed, we get a contradiction. We can assume that p is pruned and that α ∈ supp(p). We will construct a q purely stronger than p (in particular with the same w, halving parameters, and trunk). Actually, we will only strengthen one subatom at index α for each level h ≥ min(w p ).
For all h ≥ min(w p ) (not necessarily in w p ), there are several j ∈ J h such that nor(x) > 1 for the subatom x = p(α, (h, j)). For each such h we pick exactly one subatomic sublevel u(h) = (h, j), with x(h) the according subatom.
is larger than max(I), it would also follow from: "the minimum of I is bigger than H(<u ′ ), where u ′ is the predecessor of the current sublevel". 33 cf. 5.1.1 According to (7.2.2),r ↾ I nm,u is decided ≤ u and therefore even below u (since α is the active index at sublevel u; according to modesty no other index can be active; andr does not depend on α). Therefore there are at most maxposs(<u) many possibilities forr ↾ I nm,u . According to (7.2.3) there has to be at least one element s of poss(x(h)) which differs from all of these possibilities. So we can in q replace the subatom x(h) with the singleton {s}. Then the norms in q will still be large. (If A ⊆ J h witnesses the large nor liminf of p, then A \ {j} for u(h) = (h, j) witnesses that the nor liminf of q decreases only slightly.) So q is constructed by strengthening each x(h) in this way. Clearly q ≤ p is still a valid condition, and forcesr ∈M α , asr ↾ I nm,(h,u(h)) disagrees withỹ α for all h ≥ min(w p ). Proof. Assume that κ nm > ℵ 1 (otherwise there is nothing to show). Fix a condition p and κ < κ nm and names (r i ) i∈κ of reals. It is enough to show that there is an α ∈ Ξ nm such that p forces that {r i : i ∈ κ} is a subset of the meager setM α .
For each i fix a maximal antichain A i below p such that each a ∈ A i rapidly reads r i . Due to ℵ 2 -cc, and since κ nm > ℵ 1 and κ nm > κ, we can find an index α ∈ Ξ nm not appearing in the support of any condition in any A i . According to the previous lemma, every element a ∈ A i (and hence also p itself) forces thatr i ∈M α . 8. The nn part 8.1. The subatomic creatures for type nn. We describe the subatomic families K ′ nn,b , depending on a parameter b. Definition 8.1.1.
(1) Fix an interval I large enough such that (4) is satisfied (and in particular |I| > b). As in the nm subatoms, we assume that this interval I is disjoint to all intervals previously chosen.  Note that nor 0 of the subatom with full possibility set is approximately 2 |I| /2 b . In particular, for large I the norm gets large, i.e., we can satisfy (4). (1) The subatomic family has strong b-bigness (i.e., satisfies the requirement (6.1.1)). Assume that all C i have nor 0 at most r, witnesses by X i ⊆ 2 I . Then . So there is at least one i with nor(C i ) ≥ nor(C) − 1 /b, as required.
The generic object. The following paragraph is just as in the nm case 7.2: According to Section 6.1, when constructing the forcing at subatomic sublevels u, we use for all ξ ∈ Ξ nn the subatomic family K ξ,u = K ′ nn,b(u) living on some interval I, which we temporarily call I nn,u . Also, if p rapidly readsr, thenr ↾ I nn,u is decided below ≤ u.
Fix α of type nn. Recall that the generic objectỹ α assigns to each subatomic sublevel u the elementR α,u of POSS α,u chosen by the generic filter. SoR α,u is a subset of 2 Inn,u of relative size (1 − 1/2 b(u) ).
Proof. As in 7.3.1, it is enough to find a q ≤ p forcing r ∈ N α ; and we assume that p is pruned and that α ∈ supp(p). We construct q purely stronger than p by induction, only modifying subatoms at index α (and decreasing their subatom norms by at most 1): Pick a subatomic sublevel u (higher than any sublevel previously considered) where α is active with the subatom C "living" on I := I nn,u .
r ↾ I is decided ≤ u and therefore even below u (asr is read from p not using α; and due to modesty α is the only index active at sublevel u). So the set E of possibilities forr ↾ I has size at most maxposs(<u), and we can remove them all from the subatom at C while decreasing the norm by at most 1, according to Lemma 8.1.2(2) and (6.1.4).
Repeat this for infinitely many sublevels u.

Some simple facts about counting
We now list some simple combinatorial properties that will be used for the definitions and proofs in the cn-part. Lemma 9.1.1. For δ ∈ (0, 1) and ℓ ∈ ω there are M (δ, ℓ) ∈ ω and ε ∩ (δ, ℓ) > 0 such that: Whenever we have a probability space Ω and a family (A i : i < M ) of sets of measure ≥ δ, we can find a subfamily of ℓ many sets whose intersection has measure at least ε ∩ (δ, ℓ).
Proof. By straightforward counting. 34 We write χ B for the characteristic function of B. Assume we have M many sets A i , and set X ⊂ Ω to contain all points that lie in at least ℓ many of the A i . Then then there are at least δ /2 "many" points in X. We can assign to each point x ∈ X a subset M x of M (of size at least ℓ) by This partitions X into at most 2 M many sets; and at least one of the pieces has to have size at least . We can use this notion to define a norm on natural numbers: So we get the following: Fix a measure space Ω and a sequence (T i ) i∈A of sets of mea- . Note that without loss of generality the function ε ∩ satisfies: ε ∩ (δ, ℓ 1 ) ≥ ε ∩ (δ, ℓ 2 ) whenever ℓ 2 > ℓ 1 > 0. We write down the following trivial consequence of (9.1.3) for later reference: Assume that A is a subset of some finite set POSS. Fix a measure space Ω and a sequence (T i ) i∈A of sets of measure 34 Originally we used a stronger statement for which we only had a more complicated proof.
We are grateful to William B. Johnson for pointing out in http://mathoverflow.net/q/108380 that the statement in the current form has the obvious straightforward proof. 9.2. Most large subsets do not cover a half-sized set. Let Ω be the set of subsets of some finite set A ∈ ω of relative size 1 − ǫ (for 0 < ǫ < 1 /4). (Since A ∈ ω, we can write A for the cardinality |A|.) I.e.: x ∈ Ω implies x ⊆ A and |x| = A · (1 − ǫ). We can assume A ≫ 1 /ǫ and that A · ǫ is an integer.
Let T ⊆ A be of relative size ≥ 1/2, i.e., |T | ≥ A /2. Let Ω T be the elements of Ω that cover T , i.e., x ∈ Ω T iff x ∈ Ω and T ⊆ x.
We will use the following easy fact from combinatorics:  9.3. Providing bigness. In this section, we write log to denote log 2 . Apart from unimportant rounding effects, log of nor ÷ satisfies 2-bigness (and the same for nor ∩ ). Instead of thinking about such effects, we just define for any norm a 2-big version. Actually, we define a 2-big version of the combinations of two norms (of course, any finite number of norms can be combined in this way): Definition 9.3.1. Assume that nor 1 , nor 2 : ω → ω are weakly increasing and converge to infinity.
• lognor(x) is a well-defined natural number for all x, i.e., there is a maximal m such that lognor(x) ≥ m holds. • lognor is weakly increasing and diverges to infinity.
Monotonicity follows from the monotonicity of nor 1 and nor 2 . We now prove that by induction on m that there are only finitely many x with lognor(x) < m. For m = 0 this is obvious, as all x satisfy lognor(x) ≥ 0. For m > 0: lognor(x) < m iff either nor 1 (x) < m or nor 2 (x) < m or lognor(⌊ x 2 ⌋) < m − 1 or there is some y and some i ∈ {1, 2} with nor i (y) ≥ nor i (x)−1 and lognor(y) < m−1; for each case there are only finitely many possibilities.
2-bigness and the last item follow directly from the definition. b-bigness is Lemma 2.1.7.
10. The cn part 10.1. The subatomic creatures for type cn. We now describe the subatomic families K ′ cn,b used for the cn-indices. Definition 10.1.1.
(1) Fix an interval I which is large enough to satisfy (4). In particular, |I| > b. Again, we assume that this interval is disjoint to all intervals previously chosen.
(2) The basic set of all possibilities and the set of subatoms is the same as in the nn-case 8.1.1 (but the norm will be different). So POSS consists of all subsets X of 2 I with relative size 1 − 1/2 b : (4) We require nor(POSS) > b (thus satisfying (6.1.2)).
Note that (in contrast to the nn case) this norm is a counting norm, i.e., nor(C) only depends on |C|, not on the "structure" of C.
10.2. The generic object. Just as in the nn-case, we set I nn,u to be the I used for K ′ nn,b(u) ; and we defineÑ α analogously to the nn-case. 35 As before,Ñ α is a name for a null set, and a real r is inÑ α iff there are infinitely many sublevels u such that r ↾ I cn,u is not in the possibility X of K ′ cn,u = K α,u that is chosen by the generic filter.
This time, the purpose ofÑ α is not to cover all reals not depending on α, but rather to avoid being covered by any null set not depending on α.
Lemma 10.2.1. Fix a subatomic sublevel u, an index α ∈ Ξ cn and a subatom C ∈ K ′ cn,u = K α,u . (1) Given T ⊆ 2 Icn,u of relative size ≥ 1 /2 we can strengthen C to D, decreasing the norm by at most 1 /2 min(I) ·b(u) such that T ⊆ X for all X ∈ POSS(D). (2) Fix a probability space Ω and a function F that maps every X ∈ poss(C) to F (X) ⊆ Ω of measure ≥ 1 /b(u). Then we can strengthen C to D, decreasing the norm by at most 1 /2 min I ·b(u) such that X∈poss(D) F (X) has measure at least 1 /b(u+1). Here, u + 1 denotes the smallest subatomic sublevel above u.
2) The cardinality of K cn,b(u) is less than b(u + 1). is a null set (closed under rational translations). Conversely, for every null set N there is such a T with N ⊆ N T . The relative measure of s in T (for s ∈ 2 n , n ∈ ω) is defined as µ([T ]∩[s])·2 n . For completeness, we say that the relative measure of s is 0 if s / ∈ T . (Analogously, we can define the relative measure of a node s in a finite tree T ⊆ 2 ≤m with no terminal 35 Of course, generally Icn,u = Inn,u, soÑα for α ∈ Ξnn lives on a different domain thanÑ β for β ∈ Ξcn. Assume we already have r ↾ n ∈T for some n. SinceT has no nodes of relative norm 0, there is a h ′ > n and an t ′ ∈ T ∩ 2 h ′ extending r ↾ n with relative measure ≥ 1 /2 (see 10.3.2). Pick a sublevel u such that: min(I) =: h > h ′ for I := I cn,u , and u was considered in our construction of q. There is still some t ∈ 2 h ′ extendingr ↾ n of relative measure 1 /2. Set T * :=T ↾ max(I) + 1. Note that in our construction of q, when considering u, we dealt with the pair (T * , t), and thus made sure for all X ∈ poss(q(α, u)) (so in particular for the one actually chosen by the generic filter) there is some t ′ ∈ 2 I such that t ⌢ t ′ ∈ T * and t ′ / ∈ X. So we can just set r ↾ max I := t ⌢ t ′ .
Proof. This is very similar to the proof of 7.3.2: Assume that there is a ℵ 1 ≤ κ < κ cn and a p forcing that (Ñ * i ) i∈κ is a basis of null sets. As described above, we can assume that eachÑ * i = NT i for some pruned-1 /2 treeT i of measure 1 /2. For each i, fix a maximal antichain A i below p of conditions rapidly readingT i . X := i∈κ,q∈Ai supp(q) has size κ, so there is an α ∈ Ξ cn \ X. Each a ∈ A i rapidly readsT i not using α. So by the preceding lemma,Ñ α ⊆ NT i is forced by a (and therefore by p, as A i is predense below p).
10.5. non(N ) ≤ κ nn . We want to show that the set X of reals reals that are added by (or more precisely: rapidly read from) the nm and nn parts (i.e., not depending on the cn and Sacks parts) is not null.
Let Q Ξnon-sk be the set of conditions p with supp(p) ∩ Ξ sk = ∅. Recall that according to Lemma 3.6.1, Q Ξnon-sk is a complete subforcing of Q (and satisfies ω ω -bounding, rapid reading, etc). We have seen in 6.3 that the quotient of Q and Q Ξnon-sk satisfies the Sacks property, and in particular that every null set N in the Q-extension is contained in a null-set N ′ ⊇ N in the intermediate Q Ξnon-sk -extension.
So it is enough to show that X is still non-null in the Q Ξnon-sk -extension; in other words, we can in the rest of the paper ignore the Sacks indices altogether (i.e., work in Q Ξnon-sk , or in other words assume that Ξ sk = ∅).
We have seen that the sets of the form N T for pruned-1 /2 trees T form a basis of null sets; so we just have to show the following: Lemma 10.5.1. LetT * be a pruned-1 /2 tree rapidly read by p. Then there is a q ≤ p continuously reading somer ∈ 2 ω not using the cn part, such that q forces r ∈ [T * ]. (As described above, the Sacks part is not used at all.) Asr ∈ [T * ] impliesr / ∈ NT * , andr only depends on the nm and nn parts, we get: Corollary 10.5.2. Q forces non(N ) ≤ κ nn .
• Z are the possibilities of d * between k * and k * * (which we can restrict to the lim-inf part, as there are only singletons in the lim-sup-part). (7) Fix a ν ∈ Z. We will now perform an induction on the (subatomic) sublevels u between k and k * , starting with the lowest one, (k, 0). We assume that we have arrived in this construction at sublevel u with the active subatom C, and that we already have constructed the following: • The (final) subatoms for all sublevels v below u (and above k), with subatom-norm at most 2 smaller than the norms of the original subatoms (i.e., those in d * ). • (Preliminary) subatoms for all sublevels u ′ above (including) u (and below k * ), where the norm of the subatom at u ′ has been reduced from the original one by at most K /b(u ′ ), where K is the number of steps already performed in the current induction (i.e., K is the number of subatomic sublevels between k and u). So our current C is one of these "preliminary subatoms". • A function F u that maps each possibility η ∈ X × Y to a subsets F u (η) of 2 m ; such that for all η -F u (η) is forced to be a subset ofL ′ by the condition q * modulo the fixed ν ∈ Z, modulo η and modulo the already constructed subatoms (the final ones as well as the preliminary ones). 38 -F u (η) ⊆ 2 m is of relative size ≥ 1 /b(u).
-F u (η) does not depend on any cn-indices below u. The first sublevel, (k, 0), is clear: there are no sublevels below where we have to define final subatoms, the preliminary subatoms above are just the original ones, and F (k,0) is just given by the nameL ′ . Now we perform the inductive step. If our subatom C is not of cn-type, we do nothing 39 and go to the next step. So let us assume that the current (preliminary) C is of cn-type.
38 See (5.1.9) for a definition of "modulo". If η is not a compatible with the currently constructed (final and preliminary) subatoms, then F u (η) is irrelevant. 39 slightly more formally: we make the current preliminary subatom final, and set F u+1 := F u 40 We are concerned only about the η still are compatible with the currently constructed preliminary/final subatoms.
Note that there are less than b(u + 1) many possibilities for D(η + ), cf (10.2.2). Finally we can use bigness of the Y + -part, as stated in Lemma 6.1.5, to find successor subatoms at all sublevels above u, resulting in a new set of possibilitiesỸ + ⊆ Y + such that for each η + ∈Ỹ + we get the same D := D(η + ). This D will be the (final) subatom at our current level u.
As above, this is a set of measure ≥ 1 /b(u+1), does not depend on the cn-part ≤ u, and it is forced (modulo D) to be a subset ofL ′ .
We have now chosen the new final subatom D, the new preliminary subatoms and F u+1 in a way that we can perform the next step of the iteration. (8) We perform the whole inductive construction of (7) for every ν ∈ Z independently (i.e., we start at the original d * for each ν ∈ Z). So for every ν we get a different sequenceD(ν) of subatoms between k and k * . Using bigness (again as in Lemma 6.1.5), we can thin out the subatoms between k * and k * * , resulting in Z ′ ⊆ Z, such that for each ν ∈ Z ′ we get the same sequenceD(ν) =:D which finally defines the compound creature d * * stronger than d * .
We set q n+1 to be q * with d * strengthened to d * * , and we set i n+1 := m and k n+1 := k * * . (9) Now work modulo q n+1 . So the final function F of the induction in (7) gives us a name for a subsetL ′′ ⊆L ⊆ 2 m of positive relative size (in 2 m ), and the nameL ′′ does not depend on any cn indices: Not on any below k, since we started with the nameL ′ which did not depend on such subatoms; not on any between k and k * , as we removed this dependence sublevel by sublevel during the induction; and not on any cn subatoms between k * and k * * , as cn indices are of lim-sup type, and we have only singleton subatoms for the lim-sup part between k * and k * * . So we can pick a non-cn-name s n+1 for an arbitrary (the leftmost, say) element ofL. (10) q n+1 forces that s n+1 is inL, i.e., a "fat" node, more specifically:T ′ := T [sn+1] n has a measure greater than 1−ǫ 2 m . The treeT ′ is read continuously by q n and therefore also by q n+1 . In particular, for each ℓ > m the finite treeT ′ ∩ 2 ℓ is decided below some ℓ ′ . For η ∈ poss(q n+1 , <ℓ ′ ) let T ℓ,η be the according value ofT ′ ∩ 2 ℓ (a subset of 2 ℓ with at least 2 ℓ · 1−ǫ 2 m elements). We call η and η ′ equivalent if they differ only on the cn part below k * * . Each equivalence class has size ≤ maxposs(<k * ), as there are only singleton values in the lim-sup part between k * and k * * . We assign to each equivalence class [η] the tree T ℓ,[η] := η ′ ∈[η] T ℓ,η ′ . Then T ℓ, [η] has size at least 2 ℓ · 1−maxposs(<k * )·ǫ 2 m (and of course does not depend on the cn-part below k * * ). So the family T ℓ,[η] defines a continuous name for a treeT n+1 not depending on the cn-part below k * * with root s n+1 and measure > 1 /2 m+1 , as required.