Coherent forests
Monroe Eskew
Abstract
A forest is a generalization of a tree, and here we consider the Aronszajn and Suslin properties for forests. We focus on those forests satisfying coherence, a local smallness property. We show that coherent Aronszajn forests can be constructed within ZFC. We give several ways of obtaining coherent Suslin forests by forcing, one of which generalizes the well-known argument of Todorčević that a Cohen real adds a Suslin tree. Another uses a strong combinatorial principle that plays a similar role to diamond. We show that, starting from a large cardinal, this principle can be obtained by a forcing that is small relative to the forest it constructs.
We consider a type of structure called a forest, a generalization of a tree. Forests contain many trees, but can be much wider than a single tree. Thomas Jech had previously studied the same type of object under the name “mess” [5]. The nicer choice of terminology is due to Christoph Weiß [10]. In contrast to the work of Weiß, we will focus on forests that do not contain long branches.
The notions of being Aronszajn and Suslin carry over from trees to forests. In this paper, we explore several ways of obtaining large Aronszajn and Suslin forests that also satisfy a certain local smallness property called coherence. We show that large coherent Aronszajn forests can be constructed within ZFC and by forcing. Next, we explore a constraint imposed by the P-ideal dichotomy that shows the optimality of some of these results. Finally, we give three ways of forcing large coherent Suslin forests. The first is a modification of Jech’s method of forcing by local approximations. The second generalizes the argument of Todorčević that a Cohen real adds a Suslin tree. Here, we compose a Cohen-generic function with certain kind of coherent Aronszajn forest, and show that while the structure remains non-trivial, all large antichains destroyed. The third method uses a guessing principle that plays a similar role to diamond in the construction of Suslin trees. We show that this principle can be obtained from a Mahlo cardinal κ using a forcing of size κ, yet results in a coherent Suslin forest of size 2κ. In other work [2], this last result is applied to the study of saturated ideals.
A (κ,X,μ)-forest is a collection of functions F satisfying:
- (1)
{dom(f):f∈F}=Pκ(X).
2. (2)
(∀f∈F)ran(f)⊆μ.
3. (3)
For z∈Pκ(X), let Fz={f∈F:dom(f)=z}. A forest must satisfy that for z0⊆z1 in Pκ(X), Fz0={f↾z0:f∈Fz1}.
Forests are full of trees. If F is a (κ,X,μ)-forest, and S={xα:α<κ} is an enumeration of distinct elements of X, then TS={f∈F:(∃β<κ)dom(f)={xα:α<β}} forms a tree of height κ under the subset ordering.
A (κ,X,μ)-forest F is called thin if for all z∈Pκ(X), ∣Fz∣<κ. A collection of functions F is called κ-coherent if for all f,g∈F, ∣{x∈dom(f)∩dom(g):f(x)=g(x)}∣<κ. If F is a (κ+,X,μ)-forest we say it is coherent if it is κ-coherent. Clearly, if μ≤κ=κ<κ, then any coherent (κ+,X,μ)-forest is thin.
A chain in a forest is a subset which is linearly ordered under ⊆. Two elements f,g in a forest F are said to be compatible when they have a common extension h∈F. An antichain in a forest is a subset of pairwise incompatible elements. We say that a (κ,X,μ)-forest F is Aronszajn if it contains no well-ordered chain of length κ. We say it is Suslin if it contains no antichain of cardinality κ. If F is a (κ,X,μ)-forest with μ≥2, closed under finite modifications, then F is Suslin only if it is Aronszajn. This is because we can “split off” from any chain of length κ to get an antichain of size κ.
Proposition 0.1
If F is a (κ,X,μ)-forest, then for any z∈Pκ(X), Fz is a maximal antichain.
Let f∈F, z∈Pκ(X). By clause (3) of the definition of forests, there is g∈F such that f⊆g and dom(g)=dom(f)∪z. Then g↾z∈Fz, so g is a common extension of f and something in Fz. □
The following lemma will be useful in several constructions:
Lemma 0.2
Suppose F is a coherent (κ+,X,μ)-forest, and F is closed under <κ modifications. Then two functions in F have a common extension in F if and only if they agree on their common domain.
Let f,g∈F agree on dom(f)∩dom(g). Let h∈F be such that dom(h)=dom(f)∪dom(g). By coherence, we can change the values of h on a set of size <κ to get h′:dom(h)→μ with h′↾dom(f)=f, and h′↾dom(g)=g. By the closure of F, h′∈F. □
1 Aronszajn forests
The first theorem of this section generalizes of an argument of Koszmider [6].
Lemma 1.1
Let κ be a regular cardinal, and suppose F={fα:α<κ} is a κ-coherent set of partial functions from κ to μ.
- (a)
There is a function f:κ→μ such that {f}∪F is κ-coherent.
2. (b)
If μ=κ and each fα is <κ to 1, then there is a <κ to 1 function f:κ→κ such that {f}∪F is κ-coherent.
For each α, let Dα=dom(fα)∖⋃β<αdom(fβ). Let E=κ∖⋃αDα. For the first claim, choose any function g:E→μ, and let
[TABLE]
For any α, {β:f(β)=fα(β)}=⋃γ<α{β∈Dγ∩dom(fα):fγ(β)=fα(β)}. This is a union of <κ sets of size <κ, so has size <κ.
For the second claim, choose any <κ to 1 function g:E→κ, and let
[TABLE]
For any α, {β:f(β)=fα(β)}⊆⋃γ≤α{β∈Dγ:fγ(β)<γ or fγ(β)=fα(β)}. By the hypotheses, this set has size <κ. For each α, f−1(α)⊆g−1(α)∪⋃{fγ−1(β):γ,β≤α}, so f is <κ to 1. □
Theorem 1.2
Let κ be a regular cardinal. For every ζ<κ, there is a coherent (κ+,κ+ζ,κ)-forest consisting of <κ to 1 functions.
We will prove by induction the following stronger statement: For every ζ<κ and every sequence ⟨(Xα,Fα):α<κ⟩ such that:
- (1)
each Xα⊆κ+ζ,
2. (2)
each Fα is a (κ+,Xα,κ)-forest of <κ to 1 functions,
3. (3)
⋃αFα is κ-coherent,
there is a coherent (κ+,κ+ζ,κ)-forest F⊇⋃αFα consisting of <κ to 1 functions.
For ζ=0, pick a collection {fα:α<κ} such that for each α, fα∈Fα, and dom(fα)=Xα. By Lemma 1.1(b), there is a <κ to 1 function f:κ→κ that coheres with each fα, and we can take F={g:dom(g)⊆κ and ∣{x:f(x)=g(x)}∣<κ}.
Assume ζ=η+1 and the statement holds for η. For each β<κ+ζ, let Fαβ=⋃α{f↾β:f∈Fα}. We will construct F⊇⋃Fα as the union of a ⊆-increasing sequence ⟨Gβ:β<κ+ζ⟩ such that for each β, Gβ is a coherent (κ+,β,κ)-forest of <κ to 1 functions containing ⋃αFαβ. Let G0={∅}. Given Gβ, let Gβ+1={f:dom(f)⊆(β+1), ran(f)⊆κ, and f↾β∈Gβ}.
Suppose β is a limit ordinal of cofinality ≤κ, and let ⟨γi:i<δ≤κ⟩ be cofinal in β. The collection ⋃i<δGγi∪⋃α<κFαβ is κ-coherent, because (∀α<κ)(∀f∈Fαβ)(∀i<δ)(f↾γi∈Fαγi⊆Gγi). Since β has cardinality ≤κ+η, the inductive assumption implies that we can extend to a forest Gβ with the desired properties.
Suppose β is a limit ordinal of cofinality >κ. Let Gβ=⋃γ<βGγ. Then Gβ is a forest with the desired properties because ⋃α<κFαβ=⋃γ<β(⋃α<κFαγ). Finally, we let F=⋃β<κ+ζGβ.
Now assume ζ is a limit ordinal of cofinality <κ, and the statement holds for all η<ζ. Let ⟨γi:i<δ=cf(ζ)⟩ be an increasing cofinal sequence in ζ. Like above, recursively build an increasing sequence ⟨Gi:i<δ⟩ such that each Gi is a (κ+,κ+γi,κ)-forest of <κ to 1 functions extending ⋃αFαγi. This is done by applying the inductive hypothesis for κ+γi to ⋃αFαγi∪⋃j<iGj. We may also assume each Gi is closed under <κ modifications. Simply let F be the collection of functions f such that dom(f)⊆κ+ζ, and (∀i<δ)f↾γi∈Gγi. Clearly F⊇⋃αFα.
First note that if f∈F were not <κ to 1, then there would be some i<δ such that f↾κ+γi is not <κ to 1, which is false. If f,g∈F were to disagree at κ many points, then there would be some i<δ such that f↾κ+γi and g↾κ+γi disagree at κ many points, which is false. Second, we check that for any z∈Pκ+(κ+ζ), there is an f∈F such that dom(f)=z. We can recursively build a sequence ⟨gi:i<δ⟩ such that for all i<j<δ, gi∈Gi, dom(gi)=z∩κ+γi, and gi⊆gj. If we have built such a sequence up to j<δ, then ⋃i<jgi∈Gj, because for any h∈Gj with domain z∩κ+γj, the set of disagreements with ⋃i<jgi has size <κ. Let f=⋃i<δgi. □
Koszmider showed that in the case κ=ω, if λ is a singular cardinal of cofinality ω, and □λ and λω=λ+ hold, then the induction can push through λ as well. The argument generalizes almost verbatim to show for any regular κ, the induction can go forward at λ of cofinality κ, under the assumptions □λ and λκ=λ+. (The reader may want to verify this.) As a consequence, we get that in L, for every regular κ and every λ≥κ, there is a coherent, (κ+,λ,κ)-forest of <κ to 1 functions.
Recall that a partial order P is called κ-Knaster if for any A⊆P of size κ, there is B⊆A of size κ that consists of pairwise compatible elements.
Corollary 1.3
For every regular cardinal κ and every ζ<κ, there is a coherent (κ+,κ+ζ,κ)-forest, which is Aronszajn, does not have the 2<κ or the κ+ chain condition, but is (2κ)+-Knaster. If ζ is finite or 2<κ<κ+ω, then the forest is (2<κ⋅κ+)+-Knaster.
Let F be given by Theorem 1.2. We may assume F is closed under <κ modifications. To see the failure of the 2<κ chain condition, note that for any z⊆κ+ζ of size κ, Fz is an antichain of size 2<κ.
Let {αβ:β<κ+} be any enumeration of distinct ordinals in κ+ζ, and for each γ<κ+, let fγ∈F have domain {αβ:β<γ}. Since each f∈F maps into κ, there is a ξ<κ and a stationary subset S0⊆{γ<κ+:cf(γ)=κ} such that for all γ∈S0, fγ+1(αγ)=ξ. Since each f∈F is <κ to 1, each set {β<γ:fγ+1(αβ)=ξ} is bounded below γ when cf(γ)=κ. Thus there is an η<κ+ and a stationary S1⊆S0 such that for all γ∈S1, {β<γ:fγ+1(αβ)=ξ}⊆η. Therefore, for any γ0<γ1 in S1∖η, fγ0+1(αγ0)=fγ1+1(αγ0). This shows that F does not have the κ+ chain condition.
It also shows that F is Aronszajn. For otherwise, let ⟨fα:α<κ+⟩ be a strictly increasing ⊆-chain in F. Let {ξβ:β<κ+}=⋃αdom(fα), and for each γ let gγ=(⋃αfα)↾{ξβ:β<γ}. Then ⟨gγ:γ<κ+⟩ is a strictly increasing chain, but by the above paragraph, it contains an antichain of size κ+, contradiction.
To show the (2κ)+-Knaster property, let {fα:α<(2κ)+}⊆F. Let T0⊆(2κ)+ have size (2κ)+ and be such that {dom(fα):α∈T0} forms a delta-system with root r. Let T1⊆T0 have size (2κ)+ and be such that for a fixed g, fα↾r=g for all α∈T1. The union of any two of these is in F.
For the case where ζ<ω or 2<κ<κ+ω, let θ=(2<κ⋅κ+)+. First note that it is easy to see by induction that for every n<ω, Pκ+(κ+n) has a cofinal subset of size κ+n. Let A={fα:α<θ}⊆F, and let S=⋃αdom(fα).
Suppose first that ∣S∣<θ. There is an R⊆Pκ+(S) that covers {dom(fα):α<θ} and has cardinality ∣S∣. Therefore, by the coherence of F, there is a G⊆F of cardinality ≤∣S∣⋅2<κ<θ such that for all α<θ, there is g∈G with fα⊆g. Therefore there is a g0∈G which is a common lower bound to θ many fα.
Now suppose that ∣S∣=θ. Since θ is regular and θ>κ+, we can use the delta-system argument to get an S0⊆S of cardinality less than θ and a T0⊆θ of cardinality θ such that for all α0,α1∈T0, dom(fα0)∩dom(fα1)⊆S0. By the above paragraph, there is a T1⊆T0 of cardinality θ such that for any α0,α1∈T1, fα0 and fα1 agree on their common domain contained in S0. □
One may ask whether the condition “<κ to 1” can be strengthened to “1 to 1” in Theorem 1.2. But this cannot always be achieved:
Proposition 1.4
If there is a coherent (κ+,λ,κ)-forest consisting of injective functions, then there are λ many almost disjoint subsets of κ.
Let F be such a forest, and for each z∈Pκ+(λ), choose fz∈F with domain z. Let S be a collection of λ many pairwise disjoint subsets of λ, each of cardinality κ. For x=y in S, ran(fx) is almost disjoint from ran(fy). This is because the sets A=ran(fx∪y↾x) and B=ran(fx∪y↾y) are disjoint, and ∣A△ran(fx)∣<κ, and ∣B△ran(fy)∣<κ. □
A positive answer in the following special case is well-known (see [7], Chapter II, Theorem 5.9 and exercise 37):
Theorem 1.5
Let κ be a regular cardinal. There is a κ-coherent collection of functions {fα:α<κ+}, such that each fα is an injection from α to κ.
A more general positive answer can be forced:
Theorem 1.6
Assume κ is a regular cardinal with 2<κ=κ, and λ≥κ. There is a κ-closed, κ+-c.c. partial order that adds a coherent (κ+,λ,κ)-forest of injective functions.
Such a forest will be Aronszajn because a chain of length κ+ would give an injection from κ+ to κ. Unlike the forests of Theorem 1.2, it will never have the λ chain condition.
Let P be the collection partial functions p that assign to <κ many z⊆λ of size ≤κ, a partial injective function from z to κ defined at <κ many points. Let p≤q when:
- (a)
dom(p)⊇dom(q).
2. (b)
For all z∈dom(q), p(z)⊇q(z).
3. (c)
If z0,z1∈dom(q), α∈z0∩z1∖(dom(q(z0))∪dom(q(z1)), and α∈dom(p(z0)), then α∈dom(p(z1)) and p(z0)(α)=p(z1)(α).
It is easy to check that ≤ is transitive and that ⟨P,≤⟩ is κ-closed. To check the chain condition, let A⊆P have size κ+. Since κ<κ=κ, we can find a B⊆A of size κ+ such that {dom(p):p∈B} forms a delta-system with root R. Again since κ<κ=κ, there is a C⊆B of size κ+ and a collection of functions {fz:z∈R} such that ∀p∈C, ∀z∈R, p(z)=fz. If p,q∈C, then p∪q is a common extension.
If G⊆P is generic, then for all z∈Pκ+(λ)V, G gives an injective function fz:z→κ as ⋃{p(z):z∈p∈G}. For z0,z1∈Pκ+(λ)V, there is some p∈G such that z0,z1∈dom(p). p forces that fz0 and fz1 agree outside dom(p(z0))∪dom(p(z1)). Finally, by the κ+-c.c., Pκ+(λ)V is cofinal in Pκ+(λ)V[G]. So we can define a (κ+,λ,κ)-forest F as {f:f is an injection into κ, (∃z)dom(f)⊆z∈Pκ+(λ)V, and f disagrees with fz at <κ many points}. □
2 Influence of the P-ideal dichotomy
In the previous section, we saw that coherent, Aronszajn (ω1,ωn,ω)-forests can be constructed in ZFC for every natural number n. Here we show that the third coordinate is optimal, in the sense that for n<ω and λ≥ω1, ZFC cannot prove the existence of a coherent, Aronszajn (ω1,λ,n)-forest. Let us recall the relevant notions:
An ideal I⊆P(X) is a P-ideal when Pω(X)⊆I⊆Pω1(X), and for any {zn:n<ω}⊆I, there is z∈I such that zn∖z is finite for all n.
The P-ideal dichotomy (PID) is the statement that for any P-ideal I on a set X, either
- (1)
there is an uncountable Y⊆X such that Pω1(Y)⊆I, or
2. (2)
there is a partition of X into {Xn:n<ω} such that for all n and all z∈I, z∩Xn is finite.
PID is a consequence of the Proper Forcing Axiom, and is also known to be consistent with ZFC+GCH relative to a supercompact cardinal [9]. The restriction of PID to ideals on sets of size ω1 is known to be consistent without the use of large cardinals, both with and without GCH [1].
Using a coherent, Aronszajn (ω1,ω1,ω)-forest F, we can obtain a coherent, Aronszajn ω1-tree T of binary functions by taking the collection of characteristic functions of members of F whose domain is an ordinal, considering the functions as subsets of α×ω for α<ω1. A cofinal branch would be a function g:ω1×ω→2 with g↾(α×ω)∈T for all α<ω1, and this would code an uncountable well-ordered chain in F. Further, using a regressive function argument, we can see that the closure of T under finite modifications remains Aronszajn. On the other hand, forests are more flexible. If we take such a tree T, close it under subsets to get a forest F, then it may be that there is an uncountable well-ordered chain C⊆F, but with dom(⋃C) a proper subset of ω1×ω. This is what happens under PID.
Theorem 2.1
Assume PID, and let F be a coherent (ω1,λ,n)-forest closed under finite modifications, for some λ≥ω1, n<ω. Then F is not Aronszajn.
First we prove this for n=2. Let F be a coherent (ω1,λ,2)-forest closed under finite modifications. Let I be the collection of z⊆λ such that for some f∈F, z⊆{α:f(α)=1}.
We claim I is a P-ideal. Let {zn:n<ω}⊆I, and for each n, choose fn∈F witnessing zn∈I. Let f∈F have domain ⋃ndom(fn), and let z={α:f(α)=1}. For any n, f disagrees with fn on a finite set, so there can only be finitely many α∈zn∖z.
Assume that alternative (1) of PID holds, and let Y⊆λ be uncountable such that Pω1(Y)⊆I. Enumerate Y as ⟨yα:α<ω1⟩. For each α<ω1, let fα be the function that has fα(yβ)=1 for β<α, and is undefined elsewhere. Since F is closed under subsets, each fα∈F, and these form an uncountable well-ordered chain.
Assume alternative (2) of PID holds. Let Xn⊆λ be uncountable such that for all z∈I, Xn∩z is finite. Let g have constant value 0 on Xn. If f∈F and dom(f)⊆Xn, then {α:f(α)=1} is finite. Thus for any countable z⊆Xn, g↾z∈F, so again we have an uncountable well-ordered chain.
Now assume the result holds for n, and let F be a coherent (ω1,λ,n+1)-forest. Let r(k)=0 for k<n, and r(n)=1. Consider the forest G={r∘f:f∈F}, and let g0,g1 be the functions on λ with constant value 0 and 1 respectively. By the above argument, there is some uncountable Y⊆λ such that either g0↾z∈G for all countable z⊆Y, or likewise for g1. The latter case shows that F is not Aronszajn. In the former case, we have that for all countable z⊆Y, there is a function fz∈F with domain z that only takes values below n. If H={g:(∃z∈Pω1(Y))g:z→n and {α:g(α)=fz(α)} is finite}, then H is a coherent (ω1,Y,n)-forest contained in F. By induction, H contains an uncountable well-ordered chain. □
3 Suslin forests
Lemma 3.1
Let κ be a regular cardinal. All Suslin (κ,λ,μ)-forests are κ-distributive.
Let F be a Suslin (κ,λ,μ)-forest, and let ⟨Aα:α<δ<κ⟩ be a sequence of maximal antichains contained in F. By the Suslin property, each Aα has size <κ, so if z=⋃α{dom(f):f∈Aα}, ∣z∣<κ. By maximality, for every α<δ and every g∈Fz, there is an f∈Aα such that g is compatible with f. But since dom(f)⊆dom(g), this means f⊆g. Thus Fz refines each Aα. □
The boolean completion of a Suslin (κ,λ,μ)-forest is a κ-Suslin algebra, which is a complete boolean algebra with that is both κ-c.c. and κ-distributive. The cardinality of this algebra is at least λ. Therefore the existence of varieties Suslin forests is constrained by the following (see [4], Theorem 30.20):
Theorem 3.2** (Solovay)**
If B is a κ-Suslin algebra, then ∣B∣≤2κ.
Large Suslin forests can be obtained by forcing. In [5], Jech defined a class of partial orders Pλ such that under CH, Pλ is countably closed, ω2-c.c., and adds a Suslin (ω1,λ,2)-forest. However, this forest fails to be coherent. Modifying his forcing slightly, we obtain:
Theorem 3.3
Assume κ is a regular cardinal, 2<κ=κ, and 2κ=κ+. Then for all λ>κ, there is a κ+-closed, κ++-c.c. forcing of size λ<κ that adds a coherent, Suslin (κ+,λ,2)-forest.
Let P be the set of all partial functions f from λ to 2 of size ≤κ, and say f≤g when dom(f)⊇dom(g) and ∣{α:f(α)=g(α)}∣<κ. κ+-closure follows from Lemma 1.1(a), and the κ++-c.c. follows from a delta-system argument. If G is P-generic over V, in V[G] let F={f:(∃g∈G)dom(g)=dom(f) and ∣{α:f(α)=g(α)}∣<κ}. Clearly F is coherent. The argument that F is Suslin in V[G] proceeds as in [5]. □
By adapting an argument of Todorčević that appears in [8], we can obtain large Suslin forests in a different way:
Theorem 3.4
Assume κ is a regular cardinal, 2<κ=κ, and there is a coherent (κ+,λ,κ)-forest of injective functions. Then adding a Cohen subset of κ adds a coherent, Suslin (κ+,λ,2)-forest.
Let F be a coherent (κ+,λ,κ)-forest of injections closed under <κ modifications to other injections. Let g:κ→2 be an Add(κ) generic function over V. Consider the family G0={g∘f:f∈F}. Since Add(κ) is κ+-c.c., Pκ+(λ)V is cofinal in Pκ+(λ)V[g], so G0 generates a forest G when we close under subsets. G inherits coherence from F. We claim G is Suslin.
First we note that G is closed under <κ modifications. If f∈F, then by the argument for Proposition 1.4, κ∖ran(f) has size κ. By a density argument, {α∈κ∖ran(f):g(α)=i} has size κ for both i=0,1. So if g∘f∈G, and x⊆domf has size <κ, we can switch values of g∘f on x by choosing distinct ordinals {αi:i∈x}⊆κ∖ran(f) such that g(αi)=g(f(i))+1mod2. If f′=f except that f′(i)=αi for i∈x, then f′∈V by κ-closure, so g∘f′∈G. So by Lemma 0.2, members of G have a common extension when they agree on their common domain.
Towards a contradiction, suppose A={g∘fα:α<κ+} is an antichain in G0, and let p0∈Add(κ) force this. Since ∣Add(κ)∣=κ, there is some p1≤p0 such that p1⊩g˙∘fˇ∈A˙ for κ+ many f∈F. Let A0={f:p1⊩g˙∘fˇ∈A˙}, and let Z=⋃{dom(f):f∈A0}.
Case 1: ∣Z∣≤κ. Let h∈F be such that dom(h)=Z. There are at most κ many <κ modifications of h, so there are f0,f1∈A0 such that both agree with the same modification of h. But p1 forces that g∘f0 and g∘f1 are compatible, contradiction.
Case 2: ∣Z∣=κ+. Let ⟨αi:i<κ+⟩ be an enumeration of Z. Let β0=sup(dom(p1))+1, and for each f∈A0, let Xf={α:f(α)<β0}. Since each f is injective, each ∣Xf∣<κ. For each Xf, let ⟨Xf(i):i<βf⟩ be an enumeration of Xf that agrees in order with the above enumeration of Z.
Case 2a: There is no i<κ such that ∣{Xf(i):f∈A0}∣=κ+. Then there is a γ<κ+ such that for all f∈A0, {i:αi∈Xf}⊆γ. Since κ<κ=κ, we may choose some A1⊆A0 such that for all f∈A1, Xf is the same set S, and further that f↾S is the same for all f∈A1.
Let f0,f1∈A1, and let D={α∈dom(f0)∩dom(f1):f0(α)=f1(α)}. ∣D∣<κ, D∩S=∅, and if α∈D, then f0(α),f1(α)≥β0. Thus we can define a q≤p1 such that for all α∈D, q∘f0(α)=q∘f1(α)=0. q forces that g∘f0 and g∘f1 are compatible, contradiction.
Case 2b: There is some i<κ such that ∣{Xf(i):f∈A0}∣=κ+. Let i0 be the least such ordinal. We choose a sequence ⟨fα:α<κ+⟩. Let f0∈A0 be arbitrary. Let f1 be such that Xf1(i0) has index in the enumeration of Z above {i:αi∈dom(f0)}. Keep going in this fashion such that for β<γ<κ+, Xfγ(i0) has index greater than sup{i:αi∈dom(fβ)}. By the minimality of i0, there is C⊆κ+ of size κ+ and a set S⊆Z such that for all α∈C, {Xfα(i):i<i0}=S, and fα↾S is the same.
Now let β<γ be in C, and let D={α∈dom(fβ)∩dom(fγ):fβ(α)=fγ(α)}. As before, ∣D∣<κ and D∩S=∅. If α∈D, then fγ(α)≥β0, because Xfγ∩dom(fβ)=S. We construct q≤p1 such that for all α∈D, q∘fγ(α)=q∘fβ(α). Let D0={α∈D:fβ(α)∈dom(p1)}, and let q0=p1∪{⟨fγ(α),p1∘fβ(α)⟩:α∈D0}. We are free to do this because fγ is injective and fγ(α)∈/dom(p1) for α∈D.
Note that for all α∈D, q0 is defined at fγ(α), only if it is defined at fβ(α). But it may be that for some α∈D0 and some α′∈D∖D0, fγ(α)=fβ(α′). Assume we have a sequence q0≥...≥qn such that:
- (1)
for all k≤n, D∩fγ−1[dom(qk)]⊆D∩fβ−1[dom(qk)],
2. (2)
for all k≤n, qk∘fγ↾(D∩fγ−1[dom(qk)])=qk∘fβ↾(D∩fγ−1[dom(qk)]),
3. (3)
if k+1≤n, then D∩fγ−1[dom(qk+1)]=D∩fβ−1[dom(qk)].
If D∩fγ−1[dom(qn)]=D∩fβ−1[dom(qn)], let qn+1=qn. Otherwise, let Dn+1=D∩fβ−1[dom(qn)], and let qn+1=qn∪{⟨fγ(α),qn∘fβ(α)⟩:α∈Dn+1}. Clearly the induction hypotheses are preserved for n+1.
Put qω=⋃qn. (Note in the case κ=ω, D is finite, so qω=qn for some n.) By (1) and (3), D∩fβ−1[dom(qω)]=D∩fγ−1[dom(qω)], so call this set Dω. Let q=qω∪{⟨fβ(α),0⟩:α∈D∖Dω}∪{⟨fγ(α),0⟩:α∈D∖Dω}. This q forces that g∘fβ and g∘fγ are compatible, again in contradiction to the assumption about p1. □
Corollary 3.5
Assume κ is a regular cardinal, 2<κ=κ, and λ>κ. Then there is a κ-closed, κ+-c.c. forcing that adds a coherent, Suslin (κ+,λ,2)-forest.
Apply Theorems 1.6 and 3.4. □
Large Suslin forests can also be obtained from combinatorial principles rather than forcing. As reported by Jech [3] [4] [5], Laver proved in unpublished work that the existence of Suslin (ω1,ω2,2)-forests follows from Silver’s principle W and ♢, both of which hold in L. Unfortunately, Laver’s proof seems to be lost to history. In trying to reconstruct it, we encountered technical issues that led to the development of a new combinatorial principle, which we prove consistent from a Mahlo cardinal, that can be used to construct large Suslin forests. The main appeal for us is that, unlike the above forcing constructions, it allows a Suslin (κ,κ+,2)-forest to be generically added to any model with sufficiently large cardinals using a forcing of size κ rather than κ+.
Let us establish some notation concerning trees. Suppose T is a κ-tree and α<κ. Tα is the set of nodes at level α. If b is a cofinal branch in T, πα(b) is the node at level α in b. If β<α, and x∈Tα, πα,β(x) is the node in Tβ below x.
Wκ(λ) is the statement that there is a κ-tree T, a set of cofinal branches B, and a sequence ⟨Wα:α<κ⟩ with the following properties:
- (1)
∣B∣=λ.
2. (2)
For each α, ∣Wα∣<κ, and Wα⊆P(Tα).
3. (3)
For every z∈Pκ(B), there is an α<κ such that for all β≥α, πβ[z]∈Wβ.
Let T, B, ⟨Wα:α<κ⟩ be as above. If z∈Pκ(B), say “z is captured at α” when for all β≥α, πβ[z]∈Wβ and πβ↾z is injective. If z∈Wα and γ<α, say “z is captured at γ” when for all β such that γ≤β<α, πα,β[z]∈Wβ and πα,β↾z is injective.
Wκ∗(λ) asserts Wκ(λ), and that there exists a stationary S⊆κ and a sequence ⟨Aα:α<κ⟩ with each Aα⊆Wα2, such that the following additional clauses hold:
- (4)
κ=μ+ for a regular cardinal μ, and each Wα is a μ-complete subalgebra of P(Tα) containing all singletons.
2. (5)
For all α∈S, {z∈Wα:z is captured below α} is closed under arbitrary <μ sized unions and taking subsets which are in Wα.
3. (6)
If f:κ→Pκ(B)2 is such that ∣⋃α<κf0(α)∪f1(α)∣=κ, let ⟨bα:α<κ⟩ enumerate the elements of ⋃α<κf0(α)∪f1(α). The set of α∈S with the following properties is stationary:
- (a)
{bβ:β<α} is captured at α.
2. (b)
If z⊆{πα(bβ):β<α} is captured below α, then sup{β:πα(bβ)∈z}<α.
3. (c)
{⟨πα[f0(β)],πα[f1(β)]⟩:β<α}=Aα.
It is easy to see that Wκ(λ) implies 2<κ=κ, and in fact Wκ(κ) is equivalent to 2<κ=κ. If κ=μ+ and S forms part of the witness to Wκ∗(λ), then clause (4) implies μ<μ=μ, and clause (6) can be used to show ♢κ(S). On the other hand, it follows from the next theorem that Wκ∗(λ) prescribes no value for 2κ, besides that λ≤2κ.
Theorem 3.6
Suppose κ is a Mahlo cardinal and μ<κ is regular. If G∗H⊆Col(μ,<κ)∗Add(κ) is generic, then V[G∗H] satisfies Wκ∗(2κ).
In V, let T be the complete binary tree on κ, and let B be the set of all branches. For α<κ, let Gα=G∩Col(μ,<α), and let Wα=P(Tα)V[Gα]. Let S={α<κ:α is inaccessible in V}. In V[G], fix enumerations ⟨sβα:β<μ⟩ of the Wα2, and in V[G∗H], let Aα={sβα:H(α+β)=1}. Let us check each condition.
- (1)
(2κ)V=(2κ)V[G∗H], so V[G∗H]⊨∣B∣=2κ.
2. (2)
Since κ is inaccessible, each Wα is collapsed to μ.
3. (3)
Suppose z∈Pκ(B). There is some α<κ such that z∈V[Gα]. For β≥α, πβ[z]∈Wβ.
4. (4)
The regularity of μ is preserved, and clearly each Wα contains all singletons. Let ⟨aξ:ξ<δ⟩⊆Wα with δ<μ. Each aξ∈A is τξGα for some Col(μ,<α)-name τξ. By the μ-closure of Col(μ,<κ), ⟨τξ:ξ<δ⟩∈V, so ⟨aξ:ξ<δ⟩∈V[Gα].
5. (5)
By the Mahlo property, S is stationary, and by the κ-c.c. of Col(μ,<κ) and κ-closure of Add(κ), it remains stationary in V[G∗H]. Suppose α∈S.
- (a)
Unions: Let A∈Pμ(Wα) have the property that all a in A are captured below α. As above, A∈V[Gα]. Now in V[Gα], α=μ+ and ∣Tβ∣=μ for β<α. So if πα,β↾a is injective, then V[Gα]⊨∣a∣<α, and thus V[Gα]⊨∣⋃A∣<α. For distinct x,y∈⋃A, let γx,y<α be the least γ such that πα,γ(x)=πα,γ(y). We have γ=sup{γx,y:x,y∈⋃A}<α. Hence if γ≤β<α and all a∈A are captured at β, then ⋃A is captured at β.
2. (b)
Subsets: Suppose z0∈Wα is captured below α, and z1∈Wα is a subset of z0. Then V[Gα]⊨∣z1∣<α, so by the α-c.c. of Col(μ,<α), there is some β<α such that z1∈V[Gβ]. Thus z1 is captured below α.
6. (6)
First work in V[G]. Let f˙ be an Add(κ)-name for a function from κ to Pκ(B)2, and let ⟨b˙α:α<κ⟩ be as in clause (6). Let C˙ be a name for a club, and let p0∈Add(κ) be arbitrary. Build a continuous decreasing chain of conditions below p0, ⟨pα:α<κ⟩⊆Add(κ), and a continuous increasing chain of ordinals, ⟨ξα:α<κ⟩⊆κ, with the following properties: For all α,
pα+1⊩ξα∈C˙,
pα+1 decides f˙↾dom(pα) and {b˙β:β<α},
dom(pα+1) is an ordinal >max{dom(pα),ξα,α}, and
ξα+1>dom(pα+1).
Let g:κ→Pκ(B)2 and {bα:α<κ} be the objects defined by what the chain ⟨pα:α<κ⟩ decides. For each α<κ, there is a predense set Eα⊆Col(μ,<κ) of size <κ such that g(α) and bα are decided by elements of Eα. There is a club D∈V such that ∀α∈D, ∀β<α, Eβ⊆Col(μ,<α). For α∈D, g↾α and {bβ:β<α} are in V[Gα].
Back in V[G], for α<κ, let γα be the least γ≥α such that πγα↾{bβ:β<α} is injective. If α is closed under β↦γβ, then γα=α. As S is stationary, there is α∈S∩D such that γα=α, ξα=α, and pα⊩α∈C˙. We have that {bβ:β<α} is captured at α, and that {⟨πα[g0(β)],πα[g1(β)]⟩:β<α}⊆Wα2. Since α is inaccessible in V, if z⊆{πα(bβ):β<α} is captured below α, then V[Gα]⊨∣z∣<α, so {β:πα(bβ)∈z} is bounded below α.
Let q≤pα be such that for β<μ, q(α+β)=1 if sαβ=⟨πα[g0(β)],πα[g1(β)]⟩, and q(α+β)=0 otherwise. Then q⊩α∈C˙∩S, and that items (a), (b), and (c) in clause (6) hold at α. As p0 was arbitrary, clause (6) is forced. □
Can Wκ∗(λ) be forced without the use of large cardinals? Can it be forced in a cardinal-preserving way? Does L⊨ “For all regular κ, Wκ+∗(κ++)”?
Theorem 3.7
Wκ∗(λ)* implies there is a coherent, Suslin (κ,λ,2)-forest.*
Let κ=μ+, and let T, B, ⟨Wα:α<κ⟩, ⟨Aα:α<κ⟩, and S⊆κ witness Wκ∗(λ). We will construct a sequence of functions ⟨fα:α<κ⟩ on the nodes of T that will generate a coherent family of functions on B with the desired properties. Each fα will have domain Tα and range contained in {0,1}.
Let f0 be a function from T0 to 2. Assume we have have constructed a sequence of functions ⟨fβ:β<α⟩, with each fβ:Tβ→2, satisfying the following property:
- (∗)
If r∈Wβ is captured at γ<β, then fβ↾r disagrees with fγ∘πβ,γ↾r on a set of size <μ.
Let Rα={r∈Wα:r is caputured below α}. Consider the set Fα of partial functions on Tα of the form fγ∘πα,γ↾r for r∈Rα and γ witnessing its membership in Rα. Assume γ0<γ1 and fγ0∘πα,γ0↾r0 and fγ1∘πα,γ1↾r1 are in Fα. By hypothesis (∗), fγ1 disagrees with fγ0∘πγ1,γ0 at less than μ many points in πα,γ1[r0]. Therefore, there are less than μ many points in r0∩r1 at which fγ0∘πα,γ0 and fγ1∘πα,γ1 disagree. So Fα is a μ-coherent family.
Assume first that α∈/S. Using Lemma 1.1(a), let fα:Tα→2 be such that {fα}∪Fα is μ-coherent. Then (∗) holds for ⟨fβ:β≤α⟩.
Now assume α∈S. Let Hα be the closure of Fα under <μ modifications. Consider Hα as a partial order with f≤g iff f⊇g. The set Aα⊆Wα2 codes a set of relations from subsets of Tα to 2. If ⟨a0,a1⟩∈Aα, construct a relation h by putting ⟨x,i⟩∈h iff x∈ai, and call the set of all such things Aα′. It may be the case that every member of Aα′ is a function and a member of Hα, and that Aα′ is a maximal antichain in Hα. If not, ignore all these considerations, and let fα be as in the case α∈/S, so that (∗) is preserved.
Suppose Aα′ is a maximal antichain in Hα. Enumerate Rα as ⟨rβ:β<μ⟩. By clauses (4) and (5) of the definition of W∗, Rα is closed under unions of size <μ. Hα is also a μ-closed partial order. If ⟨hi:i<β<μ⟩ is a decreasing sequence, then ⋃i<βdom(hi)=r∈Rα, so let γ witness this. By (∗), each hi disagrees with fγ∘πα,γ on a set of size <μ, and so ⋃i<βhi does as well by the regularity of μ.
Setting sβ=⋃ξ<βrξ, we have ⟨sβ:β<μ⟩ is an increasing cofinal sequence in Rα. For β<μ, let γβ be the least γ<α that witnesses sβ∈Rα. Let ⟨tβ:β<μ⟩ enumerate all <μ sized subsets of Tα, such that each subset is repeated μ many times. For a partial function f:Tα→2 and β<μ, let f/tβ be f with its output values switched at the points in dom(f)∩tβ.
We will define fα inductively as ⋃β<μhβ. Let h0=∅. Assume ⟨hi:i<β⟩ has been chosen so that:
- (1)
for i<j<β, hi⊆hj;
2. (2)
for i<β, dom(hi)=sξi where ξi≥i, and ξi>ξj for j<i;
3. (3)
for i<β, there is a∈Aα′ such that hi+1/ti is a common extension of hi/ti and a.
Given hi, there is some a∈Aα′ that is compatible with hi/ti. Let ξi+1>ξi be such that sξi+1⊇dom(a)∪sξi, and let g∈Hα be a common extension of a and hi/ti with domain sξi+1. Let hi+1=g/ti. Clearly (1)–(3) are preserved at successor steps. At limit steps β, we set hβ=⋃i<βhi. This is in Hα as well by μ-closure, and the preservation of (1)–(3) is trivial.
The point is this: For every t∈Pμ(Tα), fα/t extends some a∈Aα′. For let i<μ be large enough that sξi⊇t and ti=t. Then by (3), hi+1/t extends some a∈Aα′, and hi+1/t=(fα/t)↾sξi+1. We also check that (∗) is preserved at α: Every r∈Rα is covered by some sξi, and fα↾sξi=hi, which coheres with fγ∘πα,γ↾sξi when sξi is captured at γ.
Now we define the forest. For z∈Pκ(B), let γz be the least γ<κ such that z is captured at γ. Let fz:z→2 be fγz∘πγz↾z. Let F be the closure of {fz:z∈Pκ(B)} under <μ modifications. Note that by (∗), if β≥γz, then fβ∘πβ↾z disagrees with fz at <μ many points. Hence F is a coherent (κ,B,2)-forest.
Finally, we verify the κ-c.c. First note that F satisfies the κ+-c.c. by a delta-system argument. So assume towards a contradiction that A={aα:α<κ} is a maximal antichain. Let zα=dom(aα), and code each aα as ⟨zα0,zα1⟩, where zαi={b:aα(b)=i}. Let ⟨bα:α<κ⟩ enumerate the elements of ⋃α<κzα. Define:
C0={α<κ:⋃β<αzβ={bβ:β<α}}.
C1={α<κ:{aβ:β<α} is a maximal antichain contained in {f∈F:(∃η<α)dom(f)⊆{bβ:β<η}}}.
C2={α<κ:(∀β<α)γzβ0,γzβ1,γzβ<α}.
It is easy to see that C0, C1, and C2 are club. By clause (6) of the definition of W∗, let α∈S∩C0∩C1∩C2 be such that {bβ:β<α}=⋃β<αzβ is captured at α, all z⊆{πα(bβ):β<α} captured below α have sup{β:πα(bβ)∈z}<α, and Aα={⟨πα[zβ0],πα[zβ1]⟩:β<α}.
We claim Aα′ is a maximal antichain in Hα. For β<α, zβ is captured below α since α∈C2, so the function coded by ⟨πα[zβ0],πα[zβ1]⟩ is in Hα. If h∈Hα is incompatible with every member of Aα′, then consider z={bβ:β<α and πα(bβ)∈dom(h)}, and let f=h∘πα↾z. Clauses (4) and (5) imply πα[z] is captured below α, so sup{β:bβ∈z}<α. Since α∈C1, f is compatible with some aβ with β<α. But aβ is coded and projected down as ⟨πα[zβ0],πα[zβ1]⟩∈Aα, so h is compatible with some member of Aα′ after all.
Since {bβ:β<α} is captured at α, the construction has sealed this antichain. Consider any other f∈F such that dom(f)⊇{bβ:β<α}. Then f↾{bβ:β<α} is a <μ modification of fα∘πα↾{bβ:β<α}. By the above argument, all <μ modifications of fα extend a member of Aα′, and so f is compatible with some aβ, β<α. This contradicts the assumption that A={aγ:γ<κ} is an antichain. □