Measure, category and projective wellorders

We show that each admissible assignment of א1 and א2 to the cardinal invariants in the Cichoń Diagram is consistent with the existence of a projective wellorder of the reals. 2010 Mathematics Subject Classification 03E17 (primary); 03E35 (secondary)


Introduction
There are various ways of forcing ∆ 1  3 wellorders of the reals.In [13], relying on the method of almost disjoint coding, L. Harrington produces a generic extension in which there is a boldface ∆ 1  3 wellorder of the reals and MA holds.Similar techniques can be found in J. Bagaria and H. Woodin [2].Later work by R. David [4] and the second author [10,Theorem 8.52] made use of the method of Jensen coding to obtain such wellorders when ω 1 is inaccessible to reals.More recently, the present authors, A. Törnquist and L. Zdomskyy have developed and used further techniques to produce generic extensions in which there are lightface ∆ 1  3 wellorders of the reals in the presence of a large continuum, as well as other combinatorial properties hold.For example, in V. Fischer and S. D. Friedman [5] the method of coding with perfect trees is used to obtain the consistency of the existence of a lightface ∆ 1  3 wellorder on the reals with each of the following inequalities between some of the well-known combinatorial cardinal characteristics of the continuum: d < c, b < a = s, b < g.In V. Fischer, S. D. Friedman and L. Zdomskyy [7] the method of almost disjoint coding is used to show that the existence of a lightface ∆ 1  3 wellorder of the reals is consistent with b = c = ℵ 3 and the existence of a Π 1 2 definable ω -mad subfamily of [ω] ω .The same method has been used in V. Fischer, S. D. Friedman and A. Törnquist [6] to show the existence of a generic extension in which there is a lightface ∆ 1 Furthermore it is well-known that the iterations of posets which do not add a certain type of real, for example dominating reals, might very well add such reals (see U. Abraham [1]).Thus we need a poset with strong combinatorial properties which guarantee not only that the poset but also that its iterations do not add undesirable reals.
To achieve our goal, we use the method of coding with perfect trees.The method was introduced in V. Fischer and S. D. Friedman [5], which to the best knowledge of the authors is the first work discussing cardinal characteristics in the context of projective wellorders of the reals.As shown in [5], the poset of coding with perfect trees C(Y) is ω ω -bounding and proper (see also Lemma 3.3) and so its countable support iterations preserve the ground model reals as a dominating family.As we will see in this paper, C(Y) has other strong combinatorial properties which guarantee for example that its iterations do not add Cohen and random reals (see Lemmas 3.4 and 3.6).The fact that the combinatorial properties of the coding with perfect trees poset are strong enough to obtain every admissible constellation is one of the main results of this paper.
Of course there are cases in which other methods can be used as well.For example it is well-known that finite support iterations of σ -centered posets do not add random reals.Relying on this fact, in two instances we provide alternative proofs for obtaining the corresponding admissible assignments in the presence of a ∆ 1  3 wellorder using the method of almost disjoint coding (see also [7]).However, we have to point out that whenever we choose to use a different method to force the projective wellorder of the reals, we have to guarantee that the corresponding iteration does not add undesirable reals, and so guarantee that the iterands themselves satisfy a number of strong combinatorial properties.The task of verifying what kind of reals are added by a certain partial order, and what kind of reals are not added is in general highly nontrivial and lies at the heart of many open problems in the field.
The poset which forces the definable wellorder of the reals and is introduced in [5] can be presented in the form P α , Qα : α < ω 2 where α is a two-step iteration: an arbitrary S-proper poset Q 0 α of size at most ℵ 1 , for some stationary S ⊆ ω 1 chosen in advance, followed by a three step iteration The poset K 0 α shoots closed unbounded sets through certain components of a countable sequence of stationary sets (see [5,Definition 3]), K 1 α is a poset known as localization (see [5,Definition 1]), and K 2 α is the forcing notion for coding with perfect trees (see [5,Definition 3]).The poset Q(T) for shooting a club through a stationary, co-stationary set T is ω 1 \T -proper and ω -distributive.The localization poset L(φ) is proper and does not add new reals.The only poset of these three forcing notions which does add a real is the coding with perfect trees partial order.The freedom at each stage α of using an arbitrary S-proper poset Q 0 α allows us to provide in addition each admissible Vera Fischer, Sy David Friedman and Yurii Khomskii ℵ 1 -ℵ 2 assignment to the characteristics in the Cichoń diagram.
The paper is organized as follows: in section 2 we establish the relevant preservation theorems for S-proper rather than proper iterations, in section 3 we study the combinatorial properties of the coding with perfect trees poset C(Y) and in section 4 we show that each admissible assignment is consistent with the existence of a ∆ 1 3 -w.o. on R.

Preservation theorems
Throughout this section S denotes a stationary subset of ω 1 .
For T ⊆ ω 1 a stationary, co-stationary set let Q(T) denote the poset of all countable closed subsets of ω 1 \T with extension relation given by end-extension.Note that if G is a Q(T)-generic set, then G is a closed unbounded subset of ω 1 which is disjoint from T .Thus Q(T) destroys the stationarity of T .One of the main properties of Q(T) which will be used throughout the paper is the fact that Q(T) is ω -distributive and so does not add new reals (see T. Jech [15]).
Since Q(T) destroys the stationarity of T , it is not proper.However Q(T) is ω 1 \Tproper.
Definition 2.1 Let T ⊆ ω 1 be a stationary set.A poset Q is T -proper, if for every countable elementary submodel M of H(Θ), where Θ is a sufficiently large cardinal, such that M ∩ ω 1 ∈ T , every condition p ∈ Q ∩ M has an (M, Q)-generic extension q.
The proofs of the following two statements can be found in M. Goldstern [11].Preserving V ∩ 2 ω as a dominating or as an unbounded family: A forcing notion P is said to be ω ω -bounding if the ground model reals V ∩ ω ω form a dominating family in V P .This property is preserved under countable support iteration of proper forcing notions.A forcing notion P is said to be weakly bounding if the ground model reals V ∩ ω ω form an unbounded family in V P .In contrast to the ω ω -bounding property, this property of weak unboundedness is not preserved under countable support iterations of proper posets.There are well-known examples of two-step iterations of weakly bounding posets, which add a dominating real over V (see [1]).An intermediate property, which preserves the ground model reals as an unbounded family in countable support iterations of proper posets, is the almost ω ω -boundedness.A forcing notion P is said to be almost ω ω -bounding if for every P-name for a real ḟ , ie a P-name for a function in ω ω , and for every condition p ∈ P, there is a real g ∈ ω ω ∩ V such that for every A ∈ [ω] ω ∩ V there is an extension q ≤ p such that q ∃ ∞ i ∈ Ǎ( ḟ (i) ≤ ǧ(i)).These are our main tools in providing that the ground model reals remain a dominating or an unbounded family in the various models which we are to consider in section 4.
The proofs of the two preservation theorems below follow very closely the proofs of the classical preservation theorems concerning preservation of the ω ω -bounding and the almost ω ω -bounding properties respectively under countable support iterations of proper forcing notions (see [1] or [11]).
Following standard notation we denote by M and N the ideals of meager and null subsets of the real line, respectively.Thus add(M), cov(M), non(M), cof(M) and add(N ), cov(N ), non(N ), cof(N ) denote the above defined cardinal invariants for the ideals M and N .
To preserve small witnesses to non(M), non(N ) and cof(N ) we will use preservation theorems which follow the general framework developed by M. Goldstern in [12].
Definition 2.8 ([3, Definition 6.1.6])Let be the union of an increasing sequence n n∈ω of two place relations on ω ω such that • the sets C = dom( ) and {f ∈ ω ω : f n g}, where n ∈ ω , g ∈ ω ω , are closed and have absolute definitions, that is, as Borel sets they have the same Borel codes in all transitive models.
Let N be a countable elementary submodel of H(Θ) for some sufficiently large Θ containing .We say that g ∈ ω ω covers N if ∀f ∈ N ∩ C(f g).
Following [3, Definition 6.1.7],we say that a poset P S-almost-preservesiff the following holds: if N is a countable elementary submodel of H(Θ) for some sufficiently large Θ, containing P, C, and ω 1 ∩ N ∈ S, g covers N , and p ∈ P ∩ N , then there is an (N, P)-generic condition q extending p such that q "g covers N[ Ġ]".Similarly, we say that the forcing notion P S-preservesif P satisfies [3, Definition 6.1.10]with respect only to countable elementary submodels whose intersection with ω 1 is an element of the stationary set S. More precisely, P S-preservesif whenever N is a countable elementary submodel of H(Θ) for some sufficiently large Θ which contains P and as elements and such that ω 1 ∩ N ∈ S, whenever g covers N and p n n∈ω is a sequence of conditions interpreting the P-names ḟi i≤k ∈ N for functions in C as the functions f * i i≤k , then there is an N -generic condition q ≤ p 0 such that q P "g covers N[ Ġ]" and Furthermore we obtain the following analogue of Goldstern's preservation theorem (see [12] or [3,Theorem 6.1.3]).Theorem 2.9 Let S be a stationary set and let P α , Qα : α < δ be a countable support iteration such that for all α < δ , α " Qα S-preserves-".Then P δ S-preserves-.
Of particular interest for us are the relations random , Cohen and ∆ defined in Definitions 6.3.7,6.3.15, and on page 303, respectively, of [3].For convenience of the reader we define these relations below: random : Denote by Ω the set of all clopen subsets of 2 ω .Then let . Note that f random x if and only if x / ∈ A f and that x covers N with respect to random if and only if x is random over N .
Cohen : Let and that for every dense open set H ⊆ 2 ω there is an . Then f Cohen x if and only if x ∈ A f .Therefore x covers N with respect to Cohen if and only if x is a Cohen real over N .
Each of those relations satisfies the properties of Definition 2.8.Thus Theorem 2.9 implies the following two theorems (analogous to Theorems 6.1.13and 6.3.20,respectively, from [3]).
Vera Fischer, Sy David Friedman and Yurii Khomskii Theorem 2.10 If P α , Qα : α < δ is a countable support iteration and for each α < δ , α " Qα S-preserves-random ", then P δ preserves outer measure.That is for Theorem 2.11 If P α , Qα : α < δ is a countable support iteration and for each α < δ , α " Qα S-preserves-Cohen ", then P δ preserves non meager sets.That is for every set A ⊆ 2 ω which is not meager, V P δ A is not meager.In particular Recall Theorem 2.12 If P α , Qα : α < δ is a countable support iteration and for each α < δ , α " Qα S-preserves-∆ ", then P δ has the Sacks property and so preserves the base of the ideal of measure zero sets.
No random and no amoeba reals: Some of the preservation theorems which we use to show that certain iterations do not add amoeba or random reals, are based on a general framework due to H. Judah and M. Repický [14].
Definition 2.13 ([3, Definition 6.1.17])Let be the union of an increasing chain n n∈ω of two place relations on ω ω such that • for all n ∈ ω and all h ∈ ω ω the set {x : h n x} is relatively closed in the range of , , and is absolute for all transitive models containing f and g.
A real x is said to be -dominating over V if for all y ∈ V ∩ dom( ), y x.
We have the following S-proper analogue of Judah and Repický's preservation theorem (see [3,Theorem 6.1.18]).
Theorem 2.14 If P α , Qα : α < δ , δ limit, is a countable support iteration of Sproper posets, such that for all α < δ , P α does not add a -dominating real, then P δ does not add a -dominating real.
Note that x ∈ 2 ω random -dominates V if and only if x is random over V .Furthermore the relation random satisfies the conditions of definition 2.13 and so by the above theorem we obtain the following S-proper analogue of Theorem 6.3.14 from [3].
Theorem 2.15 If P α , Qα : α < δ , δ limit, is a countable support iteration of Sproper forcing notions and for each α < δ , P α does not add random reals, then P δ does not add a random real.
Note that ∆ also satisfies the conditions of Definition 2.13.Then by Theorem 2.14 above, as well as [3, Theorem 2.3.12]we obtain the following analogue of [3, Theorem 6.3.41].
Other preservation theorems: We say that a forcing notion P is S-(f , h)-bounding, if it satisfies [3, Definition 7.2.13] but instead of proper we require that P is Sproper.That is, we say that Theorem 2.17 If P α , Qα : α < δ , δ limit, is a countable support iteration such that for all α, α " Qα is S-(f , h)-bounding", then P δ is S-(f , h)-bounding.
We will also use preservation theorems for the so called (F, g)-preserving posets.
For convenience of the reader we state the definition of (F, g)-preserving (see [3,Definition 7.2.23]).Let g be a given real and for n ∈ ω let P n = {a ⊆ g(n + 1) : Let F be a family of strictly increasing functions.For every f ∈ F choose a function f + ∈ F and assume that for all f ∈ F , n ∈ ω we have that f (n) < g(n)/2 n .A forcing notion P is said to be (F, g)-preserving if for every f ∈ F and every P-name Vera Fischer, Sy David Friedman and Yurii Khomskii Ṡ which has the property that for all n, P Ṡ(n) ⊆ P n and P norm( Ṡ(n)) < f (n), there exists a function T ∈ V such that for all n, T(n) ⊆ P n , norm(T(n)) < f + (n) and Note that the countable support iteration of (F, g)-preserving posets is (F, g)-preserving (see [3,Theorem 7.2.29]) and that (F, g)-preserving posets do not add Cohen reals (see [3,Theorem 7.2.24]).

Coding with perfect trees
Let Y ⊆ ω 1 be such that in L[Y] cofinalities have not been changed, and let μ = {µ i } i∈ω 1 be a sequence of L-countable ordinals such that µ i is the least ordinal µ with µ > {µ j : Fix L[Y] as the ground model.The poset C(Y), to which we refer as coding with perfect trees, consists of all perfect trees T ⊆ 2 <ω such that every branch r through T codes Y below |T|.For T 0 , T 1 conditions in C(T) define T 0 ≤ T 1 if and only if T 0 is a subtree of T 1 . 1elow we summarize some of the main properties of the poset C(Y).Note that T 0 ≤ T 1 if and only if [T 0 ] ⊆ [T 1 ], where [T] denotes the set of infinite branches through T .For n ∈ ω , let T 0 ≤ n T 1 if and only if T 0 ≤ T 1 and T 0 , T 1 have the same first n splitting levels.(For the notion of n-splitting level of a tree see for example [15].)For T a perfect tree and m ∈ ω let S m (T) be the set of nodes on the m-splitting level of T (and so |S m (T)| = 2 m ), and for t ∈ T let T(t) = {η ∈ T : t ⊆ η or η ⊆ t}.Note that by Π 1  1 absoluteness, r codes Y below |T| even for branches through T in the generic extension.
We will refer to x F and f F as representatives of the meager set F .
Recursively we will define a sequence of conditions τ = {T n } n∈ω , such that for every n, the condition T n is an element of N , T n+1 ≤ n+1 T n , |T n | ≥ i n and (1) T 2n C(Y) "c / ∈ F(ẋ n , ḟn )", where F(ẋ n , ḟn ) denotes a name for the meager set corresponding to the names ẋn , ḟn , , where Ġ is the canonical C(Y)-name for the generic filter.
Furthermore the entire sequence τ will be an element of Thus its fusion T * will also be an element of L µi [Y ∩ i], and so a condition in C(Y) which extends T and has the desired properties.
We will need the following two claims: Proof Let N 0 be a sufficiently elementary submodel of N such that N "N 0 is countable" and all relevant parameters are elements of N 0 , that is R, C(Y), μ, ḟ , ẋ, n and α are elements of N 0 .Let N 0 denote the transitive collapse of N 0 and let j = ω 1 ∩ N 0 .Note that N 0 is of the form L µ [Y ∩ j] for some µ, and since L µ [Y ∩ j] "j is uncountable" and L µ j [Y ∩ j] "j is countable" we have that On the other hand, since L µ j [Y ∩ j] is definable from Y, j, and µ j , and all of those are in N , we obtain that L µ j [Y ∩ j] ∈ N .Let j = {j m } m∈ω be an increasing cofinal in j sequence, which is an element of The condition R will be obtained as the fusion of a sequence R m m∈ω such that the entire sequence is definable in L µ j [Y ∩ j] and for all m, R m ∈ N 0 (and so R m ∈ N 0 ).Let R 0 = R.For every s ∈ Split n (R 0 ) and every t ∈ Succ s (R 0 ) find R 0 t ≤ R 0 (t) which decides ẋ |t| and ḟ |t|.By elementarity we can assume that R 0 t ∈ N 0 and so R 0 t ∈ N0 .Since the set of conditions in C(Y) of height strictly greater than α and j 0 is dense, again by elementarity we can assume that Claim Let R , ẋ, ḟ , n, α, N be as above and let c be a Cohen real over N .Then there is a condition R ∈ N such that R ≤ n R , |R | ≥ α, |R | and R forces that c does not belong to the meager set determined by ẋ, ḟ .
Proof Just as in the previous claim let N 0 be a sufficiently elementary submodel of N such that N "N 0 is countable" and all relevant parameters are elements of N 0 .Let N 0 denote the transitive collapse of N 0 .Let j = ω 1 ∩ N 0 and let j = {j m } m∈ω be an increasing and cofinal in j sequence which is an element of L µ j [Y ∩ j].The condition R will be obtained as the limit of a fusion sequence R m m∈ω which is definable in L µ j [Y ∩ j] and whose elements are in N 0 .Let R 0 = R .For every s ∈ Split n (R 0 ) and every t ∈ Succ t (R 0 ) find a branch b t ∈ N 0 ∩ [R 0 ] such that t ⊆ b t .Then b t gives an interpretation of the names ẋ, ḟ as reals x t and f t in N 0 .Since c is Cohen over N , it is Cohen over N 0 and so there is j t > |t| such that Take any k t > j t .Let R 1 = s∈Split n (R 0 ) t∈Succs(R 0 ) R 0 (b t k t ).Thinning out once again we can assume that |R 0 (b Suppose R m is defined.Again, for every s Then b t gives an interpretation x t , f t of ẋ, ḟ as reals x t , f t in N 0 .Using the fact that c is Cohen over N 0 we can find {l t a } 1≤a≤m such that |t| < l t 1 , l t a < l t a+1 for a < m such that for every j ∈ {l t a } 1≤a≤m , Take any k t > l t m .Let R m+1 = s∈Split n+m (Rm) t∈Succs(Rm) R m (b t k t ).Passing to an extension if necessary we can assume that |R m (b t k t )| > j m , α and so that With this we can proceed with the construction of the fusion sequence T n n∈ω .Let T 0 = T .Reproducing the proof of [5,Lemma 7] find Using the previous two claims find a condition In order to show that the coding with perfect trees forcing notion preserves random , we will use the fact that C(Y) is weakly bounding and that C(Y) preserves positive outer measure (see below).
Lemma 3.5 Suppose that A is a set of positive outer measure.Then C(Y) µ * (A) > 0.
Proof Suppose not.Then there is a condition Let N be a countable elementary submodel of L Θ [Y] for some sufficiently large Θ such that T, C(Y), A are elements of N .Then there is a sequence İn n∈ω ∈ N of names for rational intervals such that T lim m→∞ n>m µ( İn ) = 0 and T A ⊆ n∈ω m≥n İm .Then in particular, there is a C(Y)-name for a function ġ in ω ω such that for all n, T m≥ġ(n) µ( İm ) < 2 −(n 2 +n) .Since C(Y) is ω ω -bounidng (see Lemma 3.3), there is R ≤ T and a ground model real g, ie function in ω ω such that for all n ∈ ω , R ġ(n) < ǧ(n).Then in particular, for all n ∈ ω , R g(n)≤i<g(n+1) µ( İi ) < 2 −(n 2 +n) .Let i = ω 1 ∩ N and let ī = {i n } n∈ω be an increasing and cofinal in i sequence, which belongs to Recursively define a fusion sequence R n n∈ω as follows.Let R 0 = R. Suppose R n has been defined.For every n-splitting node t of R n find R t ≤ R n (t) such that for some finite sequence I n t,j g(n)≤j<g(n+1) of rational intervals, for all j : g(n) ≤ j < g(n + 1) we have R t İj = Ǐn t,j .By elementarity we can assume that R t is a condition which is an element of N which is also of height ≥ i n , and that I n t,j g(n)≤j<g(n+1) ∈ N .Let R n+1 = t∈Split n (Rn) R t and let J n = t∈Split n (Rn) g(n)≤j<g(n+1) I n t,j .Note that Since J := n m≥n J m is a measure zero set, there is x ∈ A\J .However İm and so R * x ∈ J , which is a contradiction.

Lemma 3.6
The coding with perfect trees forcing notion C(Y) preserves random .
Proof The proof proceeds similarly to the proof that Laver forcing preserves random (see [3,Theorem 7.3.39]).Let N be a countable elementary submodel of L Θ [Y] for some sufficiently large Θ, let ḟ0 be an element of Ċrandom ∩N , and let τ = T n n∈ω ∈ N be an approximating sequence for ḟ0 below T for some T ∈ C(Y) ∩ N .Let f * 0 be the approximation of ḟ0 determined by τ .Note that f * 0 ∈ N ∩ ω Ω.Let x be a random real over N .We have to show that there is an extension T * of T which is an (N, C(Y))-generic condition, such that T * "x is random over N[ Ġ]" and such that for all n ∈ ω , T * (f Let D denote the collection of all dense subsets of C(Y) which are in N .Since x is random over N and f * 0 ∈ N there is n 0 such that for all k ≥ n 0 , x / ∈ f * 0 (k).For every n ≥ n 0 let Y n n be the set of all reals z ∈ 2 ω such that there is Z ≤ T n such that φ n (z, Z) holds, where φ n (z, Z) is the conjunction of the following three formulas: . Since the quantifiers of φ 1 , φ 2 , φ 3 are relativized to subsets of N , all three of these formulas are Borel.
For a partial order P and p ∈ P let P(p) = {q ∈ P : q ≤ p}.Recall that a forcing notion P is weakly homogenous if for every p, q ∈ P there are p ≤ p and q ≤ q such that P(p ) ∼ = P(q ).To see that C(Y) is weakly homogeneous consider arbitrary T 0 and T 1 in P. Proof Fix z ∈ 2 ω and let G be an N[z]-generic filter for Coll(22 ℵ 0 , ℵ 0 ) (the algebra for collapsing 2 2 ℵ 0 onto ℵ 0 ).Now we have z The second equivalence follows from absoluteness of Σ 1  1 formulas and the third from homogeneity of Coll( 2 ] B where ṙ is the canonical name for a random real.For a random real z over N we have, Note that in particular µ(B n ) ≥ 1 − 2 −n . 2 Using the fact that x is random over N we obtain that there is L ω 2 with parameter ω 1 , such that F −1 (a) is unbounded in ω 2 for every a ∈ L ω 2 and whenever M, N are suitable models such that ω M 1 = ω N 1 then F M , SM agree with In addition, if M is suitable and ω M 1 = ω 1 , then F M , SM equal the restrictions of F , S to the ω 2 of M .Let S be a stationary subset of ω 1 which is ∆ 1 -definable over L ω 1 and almost disjoint from every element of S.
Recursively define a countable support iteration P α : α ≤ ω 2 , Qα : α < ω 2 such that P = P ω 2 will be a poset adding a ∆ 1 3 -definable wellorder of the reals.We can assume that all names for reals are nice in the sense of [5] and that for α < β < ω 2 all P α -names for reals precede in the canonical wellorder < L of L all P β -names for reals which are not P α -names.For each α < ω 2 define < α as in [5]: that is, if x, y are reals in L[G α ] and σ α x , σ α y are the < L -least P γ -names for x, y respectively, where γ ≤ α, define x < α y if and only if σ α x < L σ α y .Note that < α is an initial segment of < β .If G is a P-generic filter, then < G = {< G α : α < ω 2 } will be the desired wellorder of the reals.
In the recursive definition of P ω 2 , P 0 is defined to be the trivial poset and Qα is of the form Q0 α * Q1 α , where Q0 α is an arbitrary P α -name for a proper forcing notion of cardinality at most ℵ 1 and Q1 α is defined as in [5] and so carries out the task of forcing the ∆ 1 3 -w.o. of the reals.Note that Q 1 α is the iteration of countably many posets shooting clubs through certain stationary, co-stationary sets from S (and so each of those is S-proper and ω -distributive), followed by a "localization" forcing which is proper and does not add new reals, followed by coding with perfect trees.In the following we will use the fact that Q0 α is arbitrary, to force the various ℵ 1 -ℵ 2admissible assignments to the cardinal characteristics of the Cichón diagram in the presence of a ∆ 1 3 wellorder of the reals.Proof For even α let Q0 α be the random real forcing B, and for α odd let Q0 α be the Blass-Shelah forcing notion Q defined in [3, 7.4.D].Since all iterands are almost ω ω -bounding, by Lemma 2.7 the ground model reals remain an unbounded family and so a witness to b = ℵ 1 .On the other hand Q adds an unbounded real and Q "2 ω ∩ V ∈ N ", which implies that V Pω 2 d = non(N ) = ℵ 2 .Since cofinally often we add random reals, we have that cov(N ) = ℵ 2 in the final extension.To show that no Cohen reals are added by the iteration, use the fact that all iterands are (F, g)-preserving, as well as [3, Theorems 7.2.29 and 7. Proof For α even let Q0 α = PT f ,g , and for α odd let Q0 α = PT, where PT f ,g and PT are defined in [3, Definition 7.3.43 and Definition 7.3.3]respectively.Since PT f ,g 2 ω ∩V ∈ M and PT adds an unbounded real, V Pω 2 non(M) = d = ℵ 2 .All iterands are almost ω ω -bounding and so b remains small.All iterands S preserve random , and so by Theorem 2.10 P ω 2 preserves outer measure and so V Pω 2 non(N ) = ℵ 1 .To see that the iteration does not add random reals, note that PT and C(Y) have the Laver property and so are (f , g)-bounding for all f , g.On the other hand PT f ,g is (f , h)-bounding for some appropriate h, which implies that all iterands are S-(f , h)bounding.Then by Theorem 2.17, P ω 2 is S-(f , h)-bounding, which implies that is does not add random reals.Proof For α even let Q0 α be the rational perfect tree forcing PT, and for α odd let Q0 α be the random real forcing B. Then V Pω 2 cov(N ) = d = 2 ℵ 0 .By [3, Theorem 6.3.12]B preserves random , by [3,Theorem 7.3.47]PT preserves random and by Lemma 3.6 Sacks coding preserves random .Then Theorem 2.10, V Pω 2 2 ω ∩ V / ∈ N .All iterands are almost ω ω -bounding, and so by Theorem 2.7 the ground model reals remain an unbounded family in V Pω Proof For α even let Q0 α be Cohen forcing, and for α odd let Q0 α be PT f ,g (see [3,Definition 7.3.3]).Since PT f ,g 2 ω ∩ V ∈ M, V Pω 2 non(M) = ℵ 2 .Since cofinally often we add Cohen reals, clearly cov(M) = ℵ 2 in the final generic extension.All involved partial orders are almost ω ω -bounding and so V Pω 2 b = ω 1 .To see that the iteration does not add random reals, proceed by induction using Theorem 2.15 at limit steps.
Alternative Proof: The result can be obtained using finite support iteration of ccc posets.We will slightly modify the coding stage of the construction of [7].Let P α , Qβ : α ≤ ω 2 , β < ω 2 be a finite support iteration such that P 0 is the poset defined in [7, Lemma 1].Suppose P α has been defined.If α is a limit, α = ω 1 • α + ξ where ξ < ω 1 and α > 0, define Q α as in Case 1 of the original construction.If α is not of the above form, ie α is a successor or α < ω 1 , let Qα be a name for the following poset adding an eventually different real: where t 0 , t 1 ≤ s 0 , s 1 if and only if s 0 is an initial segment of t 0 , s 1 ⊆ t 1 , and for all ξ ∈ s 1 and all j ∈ [|s 0 |, |t 0 |) we have t 0 (j) = x ξ (j), where x ξ is the ξ -th real in L[G α ]∩ω ω according to the wellorder <Gα α .The sets Ȧα are defined as in [7].With this the definition of P ω 2 is complete.Following the proof of the original construction one can show that P ω 2 does add a ∆ 1 3 -definable wellorder of the reals (note that in our case V Pω 2 c = ℵ 2 .)Since the eventually different forcing adds a Cohen real and makes the ground model reals meager, we obtain that V Pω 2 cov(M) = non(M) = ℵ 2 .Since all iterands of our construction are σ -centered, by [3, Theorems 6.5.30 and 6.5.29]P ω 2 does not add random reals and so V Pω 2 cov(N ) = ℵ 1 .The ground model reals remain an unbounded family and so a witness to b = ℵ 1 in V Pω 2 .We should point out that the coding techniques of [7] allow one to obtain the consistency

Lemma 3 . 4
The coding with perfect trees forcing notion C(Y) preserves Cohen .Proof Let N be a countable elementary submodel of L Θ [Y] for some sufficiently large Θ, such that C(Y), μ are elements of N .Let c be a Cohen real over N .Let T be a condition in C(Y) ∩ N .It is enough to show that there is a condition T * which is a (N, C(Y))-generic extension of T and which forces that "c is Cohen over N[ Ġ]".Let {ẋ n , ḟn } n∈ω and {D n } n∈ω enumerate names for representatives of all meager sets in N C(Y) and all dense subsets of C(Y) in N , respectively.Let N denote the transitive collapse of N , let i = ω 1 ∩ N .Note that N = L µ [Y ∩ i] for some µ and since and let {ẋ, ḟ } be C(Y)-names in N (for reals), representing a meager set in N C(Y) , let n ∈ ω and let α ∈ N ∩ ω 1 such that α > |R|.Then there is a condition R in N such that R ≤ n R, |R | ≥ α and every branch through R decides ẋ, ḟ .
and T 2n forces that c does not belong to the meager set corresponding to {ẋ n , ḟn }.Obtain T 2n+1 as in the base case.With this the fusion sequence T n n∈ω is defined.Let T * = n∈ω T n .Note that |T * | = i and so in particular T ∈ C(Y).Clearly, T * is (N, C(Y))-generic and T * C(Y) "c is Cohen over N[ Ġ]".
Without loss of generality |T 0 | ≤ |T 1 |.The properties of C(Y) imply that T 0 has an extension T 0 such that |T 0 | = |T 1 |.Then the order preserving bijection between T 0 and T 1 extends to a partial order isomorphism between C(Y)(T 0 ) and C(Y)(T 1 ), and so C(Y) is weakly homogenous.Now using this fact and the fact that C(Y) preserves positive outer measure (see Lemma 3.5), one can easily modify the proof of [3, Lemma 7.3.41] to obtain that for every n ≥ n 0 , the inner measure µ * (Y n n ) ≥ 1 − 2 −n .This implies that Y * := n≥n 0 Y n n is a set of measure 1. Claim (see [3, Lemma 7.3.42])There is a sequence B k : k ≥ n 0 ∈ N of Borel sets such that for all n, B n ∈ N and B n Y n n ⊆ (N ∩ N).

Theorem 4 . 1
The constellation determined by cov(M) = cov(N ) = ℵ 2 and b = ℵ 1 is consistent with the existence of a ∆1  3 wellorder of the reals.Proof Perform the countable support iteration described above, which forces a ∆ 1 3w.o. of the reals and in addition specify Q0 α as follows.If α is even let α Q0 α = B be the random real forcing, and if α is odd let α Qα = C be the Cohen forcing.Then in V Pω 2 cov(M) = cov(N ) = ℵ 2 .At the same time, since the countable support iteration of S-proper, almost ω ω -bounding posets is weakly bounding, the ground model reals remain an unbounded family and so a witness to b = ℵ 1 .Theorem 4.2 The constellation determined by d = ℵ 2 , non(M) = non(N ) = ℵ 1 is consistent with the existence of a ∆ 1 3 wellorder of the reals.Vera Fischer, Sy David Friedman and Yurii Khomskii Proof In the forcing construction described above, which forces a ∆ 1 3 -w.o. of the reals, define Q0 α to be the rational perfect tree forcing PT defined in [3, Definition 7.3.43].To claim that d = ℵ 2 in the final generic extension, note that PT adds an unbounded real.It remains to show that non(M) = non(N ) = ℵ 1 .By [3, Theorem 7.3.46] the rational perfect tree forcing preserves Cohen , and by Lemma 3.4 the coding with perfect tress C(Y) also preserves Cohen .Therefore by Theorem 2.11 in V Pω 2 the set 2 ω ∩ V is non meager and so V Pω 2 non(M) = ℵ 1 .By [3, Theorem 7.3.47], the rational perfect tree forcing preserves random and by Lemma 3.6 the prefect tree coding C(Y) preserves random .Therefore by Theorem 2.10 in the final extension 2 ω ∩ V is a non null set and so V Pω 2 non(N ) = ℵ 1 .Theorem 4.3 The constellation determined by cov(N ) = d = non(N ) = ℵ 2 , b = cov(M) = ℵ 1 is consistent with the existence of a ∆ 1 3 wellorder of the reals.

Theorem 4 . 5
The constellation determined by cov(N ) = d = ℵ 2 and b = non(N ) = ℵ 1 is consistent with the existence of a ∆ 1 3 wellorder of the reals.
Assume CH .Let P α : α ≤ δ be a countable support iteration of length δ < ω 2 of S-proper posets of size ω 1 .Then CH holds in V P δ .
preserves the stationarity of every stationary subset S of ω 1 which is contained in S. Lemma 2.3 If P α : α ≤ δ , Qα : α < δ is a countable support iteration of S-proper posets then P δ is S-proper.Lemma 2.4 Assume CH .Let P α : α ≤ δ be a countable support iteration of length δ ≤ ω 2 of S-proper posets of size ω 1 .Then P δ is ℵ 2 -c.c.