Towers, mad families, and unboundedness

We show that Hechler’s forcings for adding a tower and for adding a mad family can be represented as finite support iterations of Mathias forcings with respect to filters and that these filters are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}$$\end{document}B-Canjar for any countably directed unbounded family \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}$$\end{document}B of the ground model. In particular, they preserve the unboundedness of any unbounded scale of the ground model. Moreover, we show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {b}}=\omega _1$$\end{document}b=ω1 in every extension by the above forcing notions.


Introduction
In this paper, we analyze Hechler's forcings from [7] for adding a tower (see Sect. 3) and for adding a mad family (see Sect. 4), after giving some preliminaries on B-Canjar filters in Sect. 2.
The forcings consist of finite conditions approximating a generic tower or a generic mad family, respectively. We first show that the poset for adding a tower can be represented as a finite support iteration, where each iterand adds a single real to the tower (which diagonalizes the initial part of the tower). In fact, each such iterand is equivalent to Mathias forcing with respect to the filter generated by the respective initial part of the tower (see Lemma 3.5). For the forcing adding a mad family, the situation is analogous (see Lemma 4.5); in this case, the filter is generated by the complements of the elements of the initial part of the mad family. It follows from these representations that the above forcing notions are σ -centered (see Corollary 3.6 and Corollary 4.6) in many cases of interest.
The main results of this paper show that the above posets preserve the unboundedness of any countably directed unbounded family of the ground model (see Theorem 3.7 and Theorem 4.7); in particular, any unbounded ground model scale is preserved. We actually prove that, for a given countably directed unbounded family B of the ground model, all the filters which are involved in the representation of Hechler's posets are B-Canjar, i.e., the corresponding Mathias forcings preserve the unboundedness of B. To verify B-Canjarness, we use a combinatorial characterization from [5] (see Theorem 2.3), together with a genericity argument. In Sect. 5, we conclude that b = ω 1 holds true in every extension by one of Hechler's forcings, using that they can be decomposed into a forcing which adds an unbounded family of size ω 1 and a forcing which preserves the unboundedness of this family (see Corollaries 5.1 and 5.2). Finally, in Sect. 6, we list some open questions.
In [3], the authors of this paper define a forcing which adds a refining matrix of regular height λ, i.e., a refining system of mad families of height λ without common refinement. There is always a refining matrix of height h (which is the minimal possible height), where h is the well-known distributivity number. In order to get a model with a refining matrix of regular height λ > h, it is shown that the forcing to add the refining matrix keeps the bounding number b (and hence h) small: to this end, the forcing is represented as an iteration of Mathias forcings with respect to filters, which are shown to be B-Canjar, where B is the family of ground model reals; this ensures that B is unbounded in the final model, witnessing that b is small.
Since a refining matrix consists of mad families as well as (along the branches of its corresponding tree) towers, the forcing used in [3] is an elaborate combination of Hechler's poset for adding a mad family and a tower, respectively. The proof that the forcing from [3] preserves the unboundedness of the ground model reals is a more complicated version of the proofs given in this paper.

B-Canjar filters
In this section, we will give the necessary preliminaries about B-Canjar filters and the preservation of unboundedness. Definition 2.1 Let F ⊆ P(ω) be a filter containing the Fréchet filter. Mathias forcing with respect to F (denoted by M(F)) is the set of pairs (s, A) with s ∈ 2 <ω and A ∈ F, where the order is defined as follows: (1) t s, i.e., t extends s, for each n ≥ |s|, if t(n) = 1, then n ∈ A.
Note that M(F) is σ -centered: for s ∈ 2 <ω , the set {(s, A) | A ∈ F} is clearly centered (i.e., finitely many conditions have a common lower bound). Also note that Mathias forcing with respect to a countably generated filter has a countable dense subset, and therefore is forcing equivalent to Cohen forcing C. For f , g ∈ ω ω , we write f ≤ * g if f (n) ≤ g(n) for all but finitely many n ∈ ω. We say that B ⊆ ω ω is an unbounded family, if there exists no g ∈ ω ω with f ≤ * g for all f ∈ B. The bounding number b is the smallest size of an unbounded family in ω ω . A family B ⊆ ω ω is called countably directed if the following closure property holds: A filter F is Canjar if M(F) does not add a dominating real over the ground model (i.e., the ground model reals remain unbounded). We are interested in the following generalization of Canjarness: preserves the unboundedness of B (i.e., B is still unbounded in the extension by M(F)).

A combinatorial characterization of B-Canjarness
Later, we will prove that certain filters are B-Canjar. A combinatorial characterization of Canjarness has been given by Hrušák-Minami [8], which has been generalized to B-Canjarness for well-ordered unbounded families B by Guzmán-Hrušák-Martínez [4]. This has been extended to countably directed unbounded families by Guzmán-Kalajdzievski [5].
Let F be a filter on ω; recall that a set X ⊆ [ω] <ω is in (F <ω ) + if and only if for each A ∈ F there is an s ∈ X with s ⊆ A. Note that if G ⊆ F are filters and X ∈ (F <ω ) + , then X ∈ (G <ω ) + . GivenX = X n | n ∈ ω (with X n ⊆ [ω] <ω for each n ∈ ω), and f ∈ ω ω , let It is well-known that Cohen forcing C preserves 1 the unboundedness of every unbounded family. As mentioned above, Mathias forcing with respect to a countably generated filter is forcing equivalent to C, and hence any countably generated filter is B-Canjar for every unbounded family B. To illustrate the characterization of B-Canjarness from Theorem 2.3, we want to provide the following easy combinatorial proof:

Lemma 2.4 Let B be a countably directed unbounded family. Then every countably generated filter is B-Canjar.
Proof Let F be a filter generated by {a n | n < ω}, i.e., A ∈ F if and only if a n ⊆ A for some n ∈ ω. LetX = X n | n ∈ ω ⊆ (F <ω ) + . For every n ∈ ω, let s n ∈ X n with s n ⊆ k<n a k (such s n exists since X n ∈ (F <ω ) + ). Let g ∈ ω ω be such that g(n) = max(s n ) for every n ∈ ω. Since B is unbounded, we can pick f ∈ B such that f (n) > g(n) for infinitely many n. It is easy to check that s n ∈X f for infinitely many n, and this implies thatX f ∈ (F <ω ) + , as desired.
Later, we will actually use the following lemma (which is again based on the characterization from the above Theorem 2.3) to show that a filter is B-Canjar. Lemma 2.5 Let V ⊆ W be models of ZFC, and assume that B ⊆ ω ω ∩ V is unbounded and countably directed in W , and that F ∈ W is a filter on ω. Moreover, assume the following: for each sequence X n | n ∈ ω ⊆ (F <ω ) + there exists a sequence s n | n ∈ ω , as well as a model V with V ⊆ V ⊆ W such that (1) s n | n ∈ ω ∈ V , (2) s n ∈ X n for each n ∈ ω, (3) for each D ∈ [ω] ω ∩ V and for each A ∈ F, there exists n ∈ D such that s n ⊆ A.
Proof We want to show that F is B-Canjar by proving its characterization given by Theorem 2.3. So suppose a sequence X n | n ∈ ω ⊆ (F <ω ) + is given. By the hypothesis of the lemma, we can fix s n | n ∈ ω and V satisfying (1)-(3). Due to (1), there is g ∈ V such that s n ⊆ g(n) for each n ∈ ω. Since B is unbounded in W , there is an f ∈ B such that g * f (i.e., g(n) < f (n) for infinitely many n ∈ ω); to finish the proof, we want to show that is in (F <ω ) + . So fix A ∈ F. We will find s ∈X f with s ⊆ A. Note that both f (which is actually in V ) and g are in V , so there is an infinite set D ∈ V such that g(n) ≤ f (n) for each n ∈ D. Now use (3) to obtain an n ∈ D with s n ⊆ A; observe that s n ∈ X n by (2), and s n ⊆ g(n) ≤ f (n), hence s n ∈X f , as desired.

Preservation of unboundedness at limits
We will also use the following theorem by 2 Judah-Shelah [9] about preservation of unboundedness in finite support iterations: Theorem 2.6 Suppose {P α ,Q α | α < δ} is a finite support iteration of c.c.c. partial orders of limit length δ, and B ⊆ ω ω is unbounded and countably directed. Moreover, suppose that ∀α < δ P α "B is an unbounded family". Then P δ "B is an unbounded family".

Hechler's tower forcing
In this section, we analyze Hechler's forcing from [7] to add a tower. First, we give some basic definitions: For a, b ∈ [ω] ω , we say that b ⊆ * a if b \ a is finite, i.e., ⊆ * denotes almost-inclusion. For a sequence a ξ | ξ < δ ⊆ [ω] ω , we say that b ∈ [ω] ω is a pseudo-intersection of a ξ | ξ < δ if b ⊆ * a ξ for each ξ < δ. We say that a ξ | ξ < δ is a tower of length δ if a η ⊆ * a ξ for any η > ξ, and it does not have an infinite pseudo-intersection. The tower number t is the smallest length of a tower.
The definition of the forcing we are giving here is not exactly as in [7], but it is easy to see that it is equivalent. Let λ be a regular uncountable cardinal. Definition 3.1 TOW λ is defined as follows: p ∈ TOW λ if p is a function with finite domain, dom( p) ⊆ λ, and for each α ∈ dom( p), we have for each β ∈ dom( p) with β < α, |s β | ≥ |s α |, whenever β ∈ dom( f α ), and n ∈ ω with n ∈ dom(s β ) ∩ dom(s α ) and n ≥ f α (β), we have The order on TOW λ is defined as follows: q ≤ p ("q is stronger than p") if and for each α ∈ dom( p), we have Given a generic filter G for TOW λ , we define, for each α < λ, It is not difficult to verify that the generic object a α | α < λ added by TOW λ is a tower of length λ.

Complete subforcings
We will now show that Hechler's forcing for adding a tower of length λ has many complete subforcings. Let us start with a useful definition: The set of full conditions is dense: For every condition p ∈ TOW λ there exists a full condition q with q ≤ p and dom(q) = dom( p). In particular the set of full conditions is dense in TOW λ .
was not defined before. It is easy to see that this extension yields a condition which fulfills (3) It is easy to see that this is a condition and it is full.
Then TOW C is a complete subforcing of TOW α . Moreover, if p ∈ TOW α is a full condition, then p C is a reduction of p to TOW C .
In particular, TOW β is a complete subforcing of TOW α for each β < α ≤ λ, and, if p ∈ TOW α is a full condition (in this case, it is easy to see that it is actually not necessary to assume that p is full), then p β is a reduction of p to TOW β . In fact, this is all we are going to need in this paper. Nevertheless, we decided to prove the more general version (for sets C which are not an ordinal) because it might be useful for future applications.
Proof of Lemma 3. 4 We first show that TOW C ⊆ ic TOW α , i.e., incompatible conditions from TOW C are incompatible in TOW α . Let p 0 , p 1 ∈ TOW C and q ∈ TOW α with q ≤ p 0 , p 1 . We have to show that there exists a condition q ∈ TOW C with q ≤ p 0 , p 1 . Let q := q C. It is very easy to check that q is as we wanted.
Let p ∈ TOW α . We want to define a reduction of p to TOW C . Let p ≤ p be a full condition (see Lemma 3.3), and let N p ∈ ω be such that |s Let q ≤ red( p) with q ∈ TOW C ; by appending 0's if necessary we can assume that there is N q ∈ ω such that N q ≥ N p and |s q β | = N q for all β ∈ dom(q) (we do not need to assume that q is full). We have to show that q is compatible with p. To show this, we define a witness r as follows. Let dom(r ) The only non-trivial part in showing that r is a condition in TOW α is verifying Definition 3.1 (5). So assume that β < γ , β ∈ dom( f r γ ), n ≥ f r γ (β) and s r γ (n) = 1. We have to show that s r β (n) = 1. In case both γ and β belong to dom(q), this just follows from the fact that q is a condition; otherwise, both γ and β belong to dom( p ), and at least one of the two does not belong to C. If n < N p , we get that s r β (n) = 1 by definition of r and the fact that p is a condition. So we can assume that n ∈ [N p , N q ), and it remains to check the following three cases.
, it follows that s q β (n) = 1, and hence s r β (n) = 1 by definition of r . It is straightforward to check that r ≤ q and r ≤ p ≤ p.
Moreover, because conditions in TOW λ have finite domain, for each limit ordinal α ≤ λ; in other words, TOW α is the direct limit of the forcings TOW δ for δ < α. So TOW λ is forcing equivalent to the finite support iteration of the quotients TOW α+1 /TOW α for α < λ.
Recall that M(F) denotes Mathias forcing with respect to the filter F (see Definition 2.1). We are now going to show that TOW α+1 /TOW α is forcing equivalent to M(F α ) for a filter F α . Work in an extension by TOW α , and note that, for each β < α, a set a β has been added by TOW α . Let i.e., F α is the filter generated by (the Fréchet filter and) the ⊆ * -decreasing sequence {a β | β < α} added by TOW α . Note that each element of F α is a superset of a β \ N for some β < α and N ∈ ω.
The quotient TOW α+1 /TOW α adds the set a α . The following lemma will provide a dense embedding from TOW α+1 /TOW α to M(F α ) which preserves (the finite approximations of) the generic real a α . Therefore, a α is also the generic real for M(F α ). Recall that the generic real for M(F) is a pseudo-intersection of F, and the definition of F α ensures that a pseudo-intersection of it is almost contained in a β for each β < α, as it is the case for the real a α .

Lemma 3.5 TOW α+1 /TOW α is densely embeddable into M(F α ).
Proof We work in a fixed extension by TOW α with generic filter G. The embedding ι is defined as follows: To see that it is a dense embedding, we have to check the following conditions:

It follows that
Therefore, and by the above, s α s, A ⊆ A, and s α (n) = 0 for all n ≥ |s|. So ι( p) = (s α , A ) ≤ (s, A).
We prove (2) The only non-trivial part in showing that q is a condition in TOW α+1 /G is verifying Definition 3.1(5) for α. We can assume without loss of generality that s p α Let n ≥ |s As a side result, let us mention that Hechler's forcing for adding a tower is σcentered: Proof Since Mathias forcing with respect to a filter is always σ -centered (see the remark after Definition 2.1) and TOW α+1 /TOW α is densely embeddable into such a forcing by the above lemma, also TOW α+1 /TOW α is σ -centered.
So TOW λ is a finite support iteration of σ -centered forcings of length at most c. As a matter of fact, the finite support iteration of σ -centered forcings of length strictly less than c + is σ -centered (the result was mentioned without proof in [10, proof of Lemma 2]; for a proof, see [1] or [6, Lemma 5.3.8]).

The filters are B-Canjar
Finally, we show that Hechler's forcing TOW λ preserves the unboundedness of countably directed unbounded families B. More precisely, let V be the ground model over which we force with TOW λ , and let B ∈ V be a countably directed unbounded family of reals; we want to show that B is still unbounded in the extension by TOW λ . Since there always exists an unbounded family B of size b which is countably directed, TOW λ does not increase the bounding number b (for more details, see Sect. 5; in fact, we argue there that we even get b = ω 1 whenever we force with TOW λ ).
In Sect. 3.2, we have defined filters F α for α < λ and have shown that TOW λ is equivalent to the finite support iteration of the Mathias forcings M(F α ). So we can finish the proof by showing that the filters F α are B-Canjar (and M(F α ) therefore preserves the unboundedness of B), and using Theorem 2.6 at limits. In fact, we show the following: Proof First note that B is countably directed in the extension by TOW α for each α ≤ λ, since TOW α has the c.c.c. (and thus all countable sets of ground model objects are covered by a countable set of the ground model).
Proof of (1): In case α = α + 1 is a successor ordinal, use the fact that (1) holds for α by induction, so B is unbounded in the extension by TOW α ; recall that, by Lemma 3.5, TOW α = TOW α * M(F α ); since (2) holds for α by induction, M(F α ) preserves the unboundedness of B, hence the same is true for TOW α , as desired.
In case α is a limit ordinal, we use the fact that TOW α is the finite support iteration of c.c.c. forcings, as well as that (1) holds for each α < α; so we can apply Theorem 2.6 to conclude (1) for α.
In case cf(α) > ω, we proceed as follows (this is going to be the main technical part of the proof): in order to show that F α is B-Canjar, it is sufficient to establish the hypothesis of Lemma 2.5.
Let W be the extension of V by TOW α ; note that F α , which is generated by the Fréchet filter and {a β | β < α}, lies in W . Now observe that we have already proven (1) for α (without having used (2) for α), i.e., we know that B is unbounded in W . Now suppose that X n | n ∈ ω ⊆ (F α <ω ) + is given. We will find s n | n ∈ ω and V with V ⊆ V ⊆ W such that Lemma 2.5(1)-(3) hold. Since the X n 's are essentially reals, the forcing TOW α has the c.c.c., and cf(α) > ω, we can fix γ < α such that X n | n ∈ ω belongs to the extension of V by TOW γ ; let V be the extension by TOW γ +1 ; clearly, V ⊆ V ⊆ W . For each n ∈ ω, we have a γ \ n ∈ F α and X n ∈ (F α <ω ) + ; therefore, for each n, there exists an s ∈ X n such that s ⊆ a γ \n. The same holds in V since X n ∈ V for each n and a γ ∈ V . Since X n | n ∈ ω ∈ V , we can pick a sequence s n | n ∈ ω ∈ V such that s n ∈ X n and s n ⊆ a γ \ n for every n.
It remains to show that Lemma 2.5(3) holds true. So fix D ∈ [ω] ω ∩ V ; we have to prove that each element of F α contains (as a subset) an s n for some n ∈ D, i.e., that the following holds for each β < α: ∀k ∈ ω ∃n ∈ D s n ⊆ a β \ k. (1) In case β ≤ γ , this is easy: fix k ∈ ω; recall that a γ ⊆ * a β , so we can pick n ≥ k with n ∈ D such that a γ \ n ⊆ a β ; but then s n ⊆ a γ \ n ⊆ a β \ k, as desired.
In case β > γ , we show (1) by induction on β: assume we have shown it for every β < β; we will show it for β.
Fix k ∈ ω, and work in the extension by TOW β (note that D belongs to the extension by TOW γ +1 , hence also to the extension by TOW β due to β ≥ γ + 1); observe that a β is added in the step from β to β + 1, i.e., by the quotient forcing TOW β+1 /TOW β (which is equivalent to M(F β )). We finish the proof by showing that the set , and note that β < β. Moreover, let be large enough such that a β \ ⊆ a β for each β ∈ dom( f ), and let L := max( , k, |s|). Use (1) for β and L to pick n ∈ D such that s n ⊆ a β \ L; because L ≥ , it follows that s n ⊆ a β \ L for each β ∈ dom( f ). Now strengthen p as follows. Extend s to s * in such a way that s * (m) = 1 if m ∈ s n and s * (m) = 0 if m / ∈ s n and m ≥ |s| (this is legitimate, because s n is a subset of each a β with β ∈ dom( f )); then it is easy to find a condition q ∈ TOW β+1 /TOW β such that q ≤ p and q(β) = (s * , f ). Note that q s n ⊆ a β \ k, as desired.

Hechler's mad family forcing
In this section, we analyze Hechler's forcing from [7] to add a mad family. Again, we start with some basic definitions: For a, b ∈ [ω] ω , we say that a and b are almost disjoint if a ∩ b is finite. Moreover, we say that A ⊆ [ω] ω is an almost disjoint family if a and a are almost disjoint whenever a, a ∈ A with a = a . An almost disjoint family A is maximal (called mad family) if for each b ∈ [ω] ω there exists a ∈ A such that |b ∩ a| = ℵ 0 . The almost disjointness number a is the smallest size of an infinite mad family.
The definition of the forcing we are giving here is not exactly as in [7], but it is easy to see that it is equivalent. Let λ be a regular uncountable cardinal.
whenever β ∈ dom(h α ), and n ∈ ω with n ∈ dom(s β ) ∩ dom(s α ) and n ≥ h α (β), we have The order on MAD λ is defined as follows: q ≤ p ("q is stronger than p") if and for each α ∈ dom( p), we have Given a generic filter G for MAD λ , we define, for each α < λ,

Lemma 4.4 Let C ⊆ α ≤ λ.
Then MAD C is a complete subforcing of MAD α . Moreover, if p ∈ MAD α is a full condition, then p C is a reduction of p to MAD C .
Before proving the lemma, let us recall that in the context of Hechler's forcing to add a tower, we only used a special instance of Lemma 3.4, namely that TOW β is a complete subforcing of TOW α , whereas here, we are going to use the more general version for sets C ⊆ α which are not ordinals. For Sect. 4.2, we need again only the special case of β < α; the more general version is needed in Sect. 4.3. In Sect. 3.3, when dealing with TOW λ , we do not need such a more general version, for the following reason: the filter F γ +1 is always countably generated (just because {a γ \ n | n ∈ ω} is a basis, due to the fact that a γ ⊆ * a β for each β < γ ), and so the analogue of the set C ⊆ α needed in Theorem 4.7 can be replaced by any upper bound which is a successor ordinal. This is not possible when dealing with MAD λ since then F β is never countably generated unless β < ω 1 .
Proof of Lemma 4. 4 We first show that MAD C ⊆ ic MAD α , i.e., incompatible conditions from MAD C are incompatible in MAD α . Let p 0 , p 1 ∈ MAD C and q ∈ MAD α with q ≤ p 0 , p 1 . We have to show that there exists a condition q ∈ MAD C with q ≤ p 0 , p 1 . Let q := q C. It is very easy to check that q is as we wanted.
Let p ∈ MAD α . We want to define a reduction of p to MAD C . Let p ≤ p be a full condition (see Lemma 4.3), and let N p ∈ ω be such that |s We have to show that q is compatible with p. To show this, we define a witness r as follows.  (4). So assume that β < γ , β ∈ dom(h r γ ), n ≥ h r γ (β) and s r γ (n) = 1. We have to show that s r β (n) = 0 if it is defined. In case both γ and β belong to dom(q), this just follows from the fact that q is a condition; otherwise, both γ and β belong to dom( p ), and at least one of the two does not belong to C. If n < N p , we get that s r β (n) = 0 by definition of r and the fact that p is a condition. But if n ≥ N p , then either s r β (n) or s r γ (n) is not defined, and there is nothing to show. It is straightforward to check that r ≤ q and r ≤ p ≤ p.

Iteration via filtered Mathias forcings
For α < λ, MAD α is a complete subforcing of MAD α+1 by Lemma 4.4, so we can form the quotient MAD α+1 /MAD α . For a generic filter G for MAD α , the quotient is defined by MAD α+1 /MAD α = {p ∈ MAD α+1 | ∀q ∈ G p ⊥ q}. Note that using Lemma 4.4 a full condition p ∈ MAD α+1 belongs to MAD α+1 /MAD α if and only if p α ∈ G.
Moreover, because conditions in MAD λ have finite domain, for each limit ordinal α ≤ λ; in other words, MAD α is the direct limit of the forcings MAD δ for δ < α. So MAD λ is forcing equivalent to the finite support iteration of the quotients MAD α+1 /MAD α for α < λ.
Recall that M(F) denotes Mathias forcing with respect to the filter F (see Definition 2.1). We are now going to show that MAD α+1 /MAD α is forcing equivalent to M(F α ) for a filter F α . Work in an extension by MAD α , and note that, for each β < α, a set a β has been added by MAD α . Let i.e., F α is the filter generated by (the Fréchet filter and) the complements of the members of the almost disjoint family {a β | β < α} added by MAD α .
The quotient MAD α+1 /MAD α adds the set a α . The following lemma will provide a dense embedding from MAD α+1 /MAD α to M(F α ) which preserves (the finite approximations of) the generic real a α . Therefore, a α is also the generic real for M(F α ). Recall that the generic real for M(F) is a pseudo-intersection of F, and the definition of F α ensures that a pseudo-intersection of it is almost disjoint from a β for each β < α, as it is the case for the real a α .

Lemma 4.5 MAD α+1 /MAD α is densely embeddable into M(F α ).
Proof We work in a fixed extension by MAD α . The embedding ι is defined as follows: To see that it is a dense embedding, we have to check the following conditions: (1) (Density) For every condition (s, A) ∈ M(F α ) there exists a condition p such that ι( p) ≤ (s, A). (2) (Incompatibility preserving) If p and p are incompatible, then so are ι( p) and ι( p ).
To show (1): Let (s, A) ∈ M(F α ). Since A ∈ F α , there exist finitely many {β i | i < m} ⊆ α and N ∈ ω such that i<m (ω \a β i )\ N ⊆ A. Extend s with 0's to s α such that |s α | = max(|s|, N ) and define h α by dom(h α ) = {β i | i < m} and h α (β i ) = |s α | for each i < m. Let p := {(α, (s α , h α ))} ∪ {(β i , ( , ∅)) | i < m}. Clearly, p α belongs to any generic filter for MAD α , and therefore p ∈ MAD α+1 /MAD α . Now, preserves the unboundedness of B), and using Theorem 2.6 at limits. In fact, we show the following: Theorem 4.7 Let B be a countably directed unbounded family. Then MAD λ preserves the unboundedness of B. More precisely, Proof First note that B is countably directed in the extension by MAD α for each α ≤ λ, since MAD α has the c.c.c. (and thus all countable sets of ground model objects are covered by a countable set of the ground model).
Proof of (1): In case α = α + 1 is a successor ordinal, use the fact that (1) holds for α by induction, so B is unbounded in the extension by MAD α ; recall that, by Lemma 4.5, MAD α = MAD α * M(F α ); since (2) holds for α by induction, M(F α ) preserves the unboundedness of B, hence the same is true for MAD α , as desired.
In case α is a limit ordinal, we use the fact that MAD α is the finite support iteration of c.c.c. forcings, as well as that (1) holds for each α < α; so we can apply Theorem 2.6 to conclude (1) for α.
In case α ≥ ω 1 , we proceed as follows (this is going to be the main technical part of the proof): in order to show that F α is B-Canjar, it is sufficient to establish the hypothesis of Lemma 2.5.
Let W be the extension of V by MAD α ; note that F α , which is generated by the Fréchet filter and {ω \ a β | β < α}, lies in W . Now observe that we have already proven (1) for α (without having used (2) for α), i.e., we know that B is unbounded in W . Now suppose that X n | n ∈ ω ⊆ (F α <ω ) + is given. We will find s n | n ∈ ω and V with V ⊆ V ⊆ W such that Lemma 2.5(1)-(3) hold. Since the X n 's are essentially reals and the forcing MAD α has the c.c.c., we can pick a countable "support" C ⊆ α, i.e., a set C such that X n | n ∈ ω belongs to the extension by MAD C (which is a complete subforcing of MAD α by Lemma 4.4); let V be the extension by MAD C ; clearly, V ⊆ V ⊆ W . Enumerate C by {γ | < ω} and let c := ω \ a γ for each ∈ ω. For each n ∈ ω, we have ≤n c \ n ∈ F α and X n ∈ (F α <ω ) + ; therefore, for each n, there exists an s ∈ X n such that s ⊆ ≤n c \ n. The same holds in V since X n ∈ V for each n and c ∈ V for each . Since X n | n ∈ ω ∈ V and c | ∈ ω ∈ V , we can pick a sequence s n | n ∈ ω ∈ V such that s n ∈ X n and s n ⊆ ≤n c \ n for every n. It remains to show that Lemma 2.5(3) holds true. So fix D ∈ [ω] ω ∩ V ; we have to prove that each element of F α contains (as a subset) an s n for some n ∈ D, i.e., that the following holds for each finite sequence β i | i < N ⊆ α: We first observe that (2) holds in case that {β i | i < N } ⊆ C: fix k ∈ ω, and note that there is m ∈ ω such that for each n ≥ m, we have hence there is such an n in the infinite set D, as desired.
We now show (2) for arbitrary {β i | i < N } ⊆ α, using a genericity argument. Let N C := {i ∈ N | β i ∈ C}, and N α\C := {i ∈ N | β i / ∈ C}, so N = N C∪ N α\C . Fix k ∈ ω, and work in V , the extension by MAD C (note that D ∈ V ); observe that the a β i 's for i ∈ N α\C are added by the quotient forcing MAD α /MAD C . We finish the proof by showing that the set Since (2) holds for β i 's in C (as shown above), we can pick n ∈ D such that Now extend p to q by extending all the s p β i with i ∈ N α\C with 0's up to the maximum of s n (recall that we can always 3 extend with 0's, because this does not harm the requirement related to almost disjointness). So we get that q forces m ∈ ω \ a β i for all i ∈ N α\C and all m ∈ s n , and hence as desired.

Conclusion
In this section, we present some facts about cardinal characteristics which easily follow from our analysis of TOW λ and MAD λ .
First note that any unbounded scale, i.e., any unbounded set B = { f i | i < κ} such that f i ≤ * f j for i ≤ j, is countably directed, because its length κ has uncountable cofinality. Therefore, by Theorems 3.7 and 4.7, any unbounded scale of the ground model remains unbounded in the extension by TOW λ and MAD λ , respectively. It is easy to see that there exists always an unbounded scale of length b. Assume V | "b = κ". Then V TOW λ | "there exists an unbounded scale of length κ and there exists a tower of length λ". In particular, this implies that V TOW λ | "b ≤ κ". The same argument works for MAD λ , therefore V MAD λ | "b ≤ κ and there exists an unbounded scale of length κ and a mad family of size λ".
Note that the above shows that b = ω 1 holds in the extension by TOW λ (or MAD λ ) provided that b = ω 1 holds in the ground model. But in fact the following argument shows that no assumption about b in the ground model is necessary for this conclusion. The forcing TOW λ can be decomposed into TOW ω 1 * (TOW λ /TOW ω 1 ). By Sect. 3.2, TOW ω 1 is equivalent to an iteration of length ω 1 of Mathias forcings with respect to countably generated filters, therefore it is equivalent to the Cohen forcing which adds ω 1 many Cohen reals. Since these ω 1 many Cohen reals form an unbounded family, it follows that V TOW ω 1 | "b = ω 1 ". In V TOW ω 1 , let B be an unbounded family of size ω 1 which is countably directed. The quotient TOW λ /TOW ω 1 is equivalent to a finite support iteration of Mathias forcings with respect to filters which are B-Canjar (which follows as in the proof of Theorem 3.7), therefore B is unbounded in V TOW λ , thus, using that t ≤ b, we get the following: The analogous argument works for MAD λ , so we get the following: Corollary 5.2 Let λ be a regular uncountable cardinal. Then the following holds in V MAD λ : (1) t = b = ω 1 .
(2) There exists 5 a mad family of size λ.

Questions
Finally, let us list a few questions, which the anonymous referee suggested to add to the paper. Note that TOW λ and MAD λ are forcing equivalent in case λ ≤ ω 1 , 4 The generic object added by TOW ω 1 is a tower of length ω 1 in V TOW ω 1 , but it is clearly not a tower in V TOW λ any more. 5 Of course, there also exists a tower of length ω 1 , as in V TOW λ . because in this case both can be written (see Sects. 3.2, 4.2) as finite support iterations of Mathias forcings with respect to countably generated filters (which are just Cohen forcing). We strongly conjecture, however, that this is not the case for larger λ: Question 6.1 Are TOW λ and MAD λ forcing equivalent for λ > ω 1 ?
The above question could be settled by showing that MAD λ adds an object which is not added by TOW λ , or vice versa: Question 6.2 Let λ > ω 1 . Does TOW λ add a mad family of size λ? Does MAD λ add a tower of length λ?