SOUSLIN TREES WHICH ARE HARD TO SPECIALISE

JAMES CUMMINGS

Abstract. We construct some κ^{+}-Souslin trees which cannot be specialised by any forcing which preserves cardinals and cofinalities. For κ a regular cardinal we use the principle, for κ singular we use squares and diamonds.

1. Introduction

We start by recalling a few basic definitions concerning trees. For more information, see [4].

Definition 1.1. Let κ be a regular cardinal.

1. (T, <_{T}) is a tree iff <_{T} is a partial ordering of T such that { y : y <_{T} x } is well ordered by <_{T} for all x ∈ T.

2. Let T be a tree. Then (a) If x ∈ T, ht_{T} (x) is the order type of ({ y : y <_{T} x }, <_{T}).

(b) T_{α} = { x ∈ T : ht_{T} (x) = α }.

(c) ht(T) is the least α such that T_{α} = ∅.

(d) T ↾ α = ⋃_{β<α} T_{β}.

(e) A cofinal branch of T is a set B ⊆ T such that B is linearly ordered by

<_{T}, and ∀α < ht(T) ∃b ∈ B ht_{T} (b) ≥ α.

3. T is a κ-tree iff ht(T) = κ and |T_{α}| < κ for all α < κ.

4. T is a κ-Aronszajn tree iff T is a κ-tree with no cofinal branch.

5. T is a special κ^{+}-tree iff T is a κ^{+}-tree and there exists F : T ⟶ κ such that x <_{T} y ⟹ F (x) ≠ F (y).

6. X ⊆ T is an antichain in T iff for all x, y ∈ X with x ≠ y we have x ≰_{T} y and y ≰_{T} x.

7. T is a κ-Souslin tree iff T is a κ-Aronszajn tree and T has no antichain of size κ.

It is easy to see that a special κ^{+}-tree is an κ^{+}-Aronszajn tree and that a κ^{+}- Souslin tree is a non-special κ^{+}-Aronszajn tree. Sometimes it is possible to make trees special by mild forcing, for example if T is an ℵ_{1}-Aronszajn tree then it is proved in [1] that there exists a c.c.c forcing ℙ_{T} which adds a specialising function for T.

We will construct some λ^{+}-Souslin trees which are hard to specialise. The key idea will be that of an ascent path, which is due to Laver.

Definition 1.2. Let κ and λ be cardinals with κ = cf(κ) < λ. Let T be a λ^{+}-tree. Then a κ-ascent path in T is a sequence ⟨x_{α} : α ∈ A⟩ such that 1. A is unbounded in λ^{+}.

1991 Mathematics Subject Classification. Primary 03E05; Secondary 03E35.

Key words and phrases. Souslin trees, ascent paths, squares and diamonds.

Supported by grant number 517/94 of the Israel Science Foundation.

2. For each α ∈ A, x_{α} is a function from κ to T_{α}.

3. If α, β ∈ A with α < β then ∃γ < κ ∀δ ≥ γ x_{α}(δ) <_{T} x_{β}(δ).

The connection between ascent paths and questions about specialisation is given by the following theorem from [6], a paper in which some Aronszajn trees with ascent paths are constructed. The theorem is due to Shelah, building on work of Laver and Todorčević.

Theorem 1. Let T be a λ^{+}-tree with a κ-ascent path, where κ ≠ cf(λ). Then T is not special.

Since the hypotheses of this theorem will remain true in any extension of the universe with the same cardinals and cofinalities, it follows that trees with ascent paths are rather hard to specialise.

Baumgartner proved that if ♢_{ω_{2}}(E) and □_{ω_{1}}(E) hold for some stationary set E ⊆ ℵ_{2}, then it is possible to construct an ℵ_{2}-Souslin tree with an ω-ascent path. The construction appears in [2], and is a variation on Jensen’s construction of an ℵ_{2}-Souslin tree from the same hypotheses.

There is another well-known construction of an ℵ_{2}-Souslin tree, in which the tree is built using the hypotheses that CH and ♢(S_{1}^{2}) hold (where S_{1}^{2} is the set { α < ℵ_{2} : cf(α) = ℵ_{1} }). In this construction the resulting tree is countably closed, because the stages at which antichains are guessed and dealt with have cofinality ℵ_{1}, while all cofinal branches are completed at stages of cofinality ω. In Theorem 2 we will show that it is possible to construct a countably closed ℵ_{2}-Souslin tree with an ω-ascent path; unfortunately we will need rather stronger assumptions, namely we will need to assume CH plus the principle .

Theorem 2 has an easy generalisation to successors of regular cardinals, which is given in Theorem 3. We will prove in Theorems 4 and 5 that under the right hypotheses (square and some consequences of GCH) we can build Souslin trees that are hard to specialise at successors of singulars. Here we will use the idea of Baumgartner’s construction, plus some results of Shelah concerning diamond principles that follow from the GCH.

2. Successors of regulars

In the constructions of this section we will use the combinatorial principle (introduced in [3]).

Definition 2.1. A -sequence is a sequence ⟨S_{α}, C_{α} : α < ℵ_{2}⟩ where 1. S_{α} ⊆ α and C_{α} is club in α.

2. ot(C_{α}) ≤ ℵ_{1}.

3. β ∈ lim(C_{α}) implies C_{β} = C_{α} ∩ β and S_{β} = S_{α} ∩ β.

4. If C is club in ℵ_{2} and X ⊆ ℵ_{2} then there exists β with cf(β) = ℵ_{1} such that C_{β} ⊆ C and S_{β} = X ∩ β.

It is shown in [3] that holds in L. can also be forced rather easily; the conditions are just sequences of the form ⟨S_{α}, C_{α} : α ≤ β⟩ obeying clauses 1-3 of the definition, and it is easy to see that the forcing is (ℵ_{1} + 1)-strategically closed and that the generic sequence is a -sequence.

Theorem 2. Assume that CH and hold. Then there is a countably closed ℵ_{2}- Souslin tree with an ω-ascent path.

Proof. We will build a countably closed ℵ_{2}-Souslin tree T together with a function α ⟼ x_{α} which assigns to each α < ℵ_{2} an ω-sequence x_{α} of points from T_{α}. Suppose that ⟨S_{α}, C_{α} : α < ℵ_{2}⟩ is a -sequence. We will maintain the following hypotheses during the inductive construction of T and x⃗.

1. The elements of T are ordinals less than ℵ_{2}, and |T_{α}| ≤ ℵ_{1} for all α. The ordinals appearing in T ↾ α form an initial segment of ℵ_{2}.

2. α < β < ℵ_{2} ⟹ ∃n ∀m ≥ n x_{α}(m) <_{T} x_{β}(m).

3. T ↾ α is countably closed; that is to say, if x_{0} <_{T} x_{1} <_{T} x_{2} … and β = sup_{n} ht(x_{n}) < α then there exists y ∈ T_{β} such that ∀n x_{n} <_{T} y.

4. T ↾ α is normal, that is

(a) Every element x with ht(x) + 1 < α has two immediate successors in T_{ht(x)+1}.

(b) If ρ < σ < α and x ∈ T_{ρ} then there is y ∈ T_{σ} with x < y.

(c) If λ is a limit ordinal less than α, and x, y ∈ T_{λ} with x ≠ y, then

{ z : z <_{T} x } ≠ { z : z <_{T} y }.

5. α ∈ lim(C_{β}) ⟹ ∀m x_{α}(m) <_{T} x_{β}(m).

Before we begin the construction, a few words of motivation: Hypotheses 1–3 are completely natural in the light of our final goal, and Hypothesis 4 is a standard one in inductive constructions of trees. Hypothesis 5 is intended to help us maintain Hypothesis 2, which might otherwise be impossible at limit stages of cofinality ℵ_{1}. Hypothesis 3 will be maintained by completing all cofinal branches through T ↾ α at every limit stage α of cofinality ω; this will not cause T_{α} to become too large because we are assuming CH.

The main difficulty in the construction is that maintaining Hypothesis 5 will force us to complete certain branches at stages of cofinality ℵ_{1}. This will potentially cause conflict with our strategy for making T Souslin, which (as usual) will involve sealing off antichains by not completing certain branches at stages of cofinality ℵ_{1}. We will use the powerful guessing properties of the -sequence to resolve the conflict.

We describe the inductive construction of T and x⃗.

1. α = 0. T_{0} has a single point and x_{0} is constant with value that point.

2. α = β +1. Put two points of T_{α} over each point of T_{β} and for each n designate one of the successors of x_{β}(n) to be x_{α}(n).

3. cf(α) = ω and lim(C_{α}) is unbounded in α. Put a point of T_{α} over each cofinal branch of T ↾ α. If γ, δ ∈ lim(C_{α}) with γ < δ then γ ∈ lim(C_{δ}), so that x_{γ}(n) <_{T} x_{δ}(n). So for each n the sequence ⟨x_{γ}(n) : γ ∈ lim(C_{α})⟩ is a cofinal branch, and we designate the point of T_{α} on top of that branch as x_{α}(n).

Hypothesis 5 is satisfied by construction. Hypothesis 2 is also satisfied because if β < α then for some γ ∈ lim(C_{α}) we have β < γ < α; by construction x_{β}(n) <_{T} x_{γ}(n) for all large n and x_{γ}(n) <_{T} x_{α}(n) for all n.

4. cf(α) = ω and lim(C_{α}) is bounded in α, say the maximum point is γ. Notice that for all β ∈ lim(C_{α}) and all n we have x_{β}(n) ≤_{T} x_{γ}(n). Put a point of T_{α} over each cofinal branch of T ↾ α. Choose an ω-sequence ⟨α_{n} : n < ω⟩ cofinal in α with α_{0} = γ. Now for each n let i(n) ≤ n be maximal such that x_{α_{0}}(n) ≤_{T} … ≤_{T} x_{α_{i(n)}}(n); build a cofinal branch in T ↾ α passing through x_{α_{i(n)}}(n), and such that if possible the branch also passes through a point of S_{α}. Designate the point on top of this branch as x_{α}(n).

Hypothesis 5 is satisfied because if β ∈ lim(C_{α}) then for all n we have x_{β}(n) ≤_{T} x_{γ}(n) < x_{α}(n). For Hypothesis 2 let i < ω, and observe that for all large enough n we have n ≥ i and x_{α_{0}}(n) < … < x_{α_{i}}(n), so that i(n) ≥ i and therefore x_{α_{i}}(n) ≤_{T} x_{α_{i(n)}}(n) <_{T} x_{α}(n). For all β < α there is an i such that β < α_{i} < α, and hence x_{β}(n) <_{T} x_{α_{i}}(n) <_{T} x_{α}(n) for all large n.

5. cf(α) = ℵ_{1}. For each point x of T ↾ α build a branch which passes through x, and also if possible passes through a point of S_{α}. Put a point of T_{α} on top of each such branch. For each m the sequence ⟨x_{β}(m) : β ∈ lim(C_{α})⟩ is a cofinal branch; if we did not do so already, put a point on top of this branch. Now, for each m, designate x_{α}(m) as the point over the branch ⟨x_{β}(m) : β ∈ lim(C_{α})⟩.

As in Case 3, Hypotheses 2 and 5 are easily seen to be satisfied. It remains to be seen that this works. It is clear that we have satisfied the

demands on the x_{α}, and built a normal countably closed ℵ_{2}-tree, the problem is to see that we have built an ℵ_{2}-Souslin tree. By normality it is enough to check that there are no antichains of size ℵ_{2}. Let S be a maximal antichain and let C be the club consisting of those α such that T ↾ α = α and S ∩ α is maximal in T ↾ α. Find α such that cf(α) = ℵ_{1}, C_{α} ⊆ C and S ∩ α = S_{α}. Notice that α = sup(C_{α}) ∈ C.

Now consider the construction of T_{α}; we claim that every point of T_{α} lies over some point of S_{α}. We completed branches of T ↾ α for two reasons, to satisfy normality and to define x_{α}. Those branches which we completed to get normality clearly pass through S_{α}, because S_{α} = S ∩ α is maximal in T ↾ α, so for each x it was possible to construct a branch through x and some point of S_{α}. To see that the other branches pass through S consider β a successor point in lim(C_{α}); now β ∈ C so S_{β} = S_{α} ∩ β = S ∩ β is maximal in T ↾ β, and lim(C_{β}) = lim(C_{α}) ∩ β is bounded in β, so that when we define x_{β} we will have arranged that every point x_{β}(n) lies over some point of S_{β}. It follows that S_{α} is maximal in T, and hence that S = S_{α} and |S| ≤ ℵ_{1}.

This concludes the proof of Theorem 2.

The result has an easy generalisation.

Definition 2.2. Let κ be a regular cardinal. A _{κ}-sequence is a sequence of pairs ⟨S_{α}, C_{α} : α < κ^{+}⟩ where

1. S_{α} ⊆ α and C_{α} is club in α.

2. ot(C_{α}) ≤ κ.

3. β ∈ lim(C_{α}) implies C_{β} = C_{α} ∩ β and S_{β} = S_{α} ∩ β.

4. If C is club in κ^{+} and X ⊆ κ^{+} then there exists β with cf(β) = κ such that C_{β} ⊆ C and S_{β} = X ∩ β.

Theorem 3. Let κ^{<κ} = κ and let _{κ} hold. Then there is a κ^{+}-Souslin tree with an ω-ascent path.

Proof. Exactly like the proof of Theorem 2.

3. Successors of singulars

We now turn to λ^{+}-trees with λ singular. It is hopeless to demand that such a tree be < λ-closed because λ^{cf(λ)} > λ. We will prove that under the right hypotheses there can exist λ^{+}-Souslin trees with κ-ascent paths for some κ ≠ cf(λ). Our construction is very similar to Baumgartner’s construction described in [2]. We start by treating the case when cf(λ) = ω.

Theorem 4. Let λ be a cardinal such that 1. cf(λ) = ω.

2. γ < λ ⟹ γ^{ℵ_{1}} < λ.

3. 2^{λ} = λ^{+}.

4. □_{λ} holds. Then there exists an λ^{+}-Souslin tree with an ℵ_{1}-ascent path.

Proof. We begin by deriving some consequences of the assumptions. The following is well-known.

Claim . There exists E a stationary subset of { α < λ^{+} : cf(α) = ℵ_{1} } and a sequence ⟨C_{α} : α < λ^{+}⟩ such that C⃗ is a □_{λ}(E)-sequence, that is a □_{λ}-sequence with the additional property that ∀α lim(C_{α}) ∩ E = ∅.

Proof. Let ⟨D_{α} : α < λ^{+}⟩ be a □_{λ}-sequence. By Fodor’s lemma find E such that E is a stationary set of cofinality ℵ_{1} ordinals, and α ∈ E ⟹ ot(D_{α}) = β for some fixed β. Now let C_{α} = D_{α} if ot(D_{α}) ≤ β, and C_{α} = D_{α} − (γ + 1) if ot(D_{α}) > β and γ is the β^{th} element of D_{α}. Check that C⃗ is a □_{λ}(E)-sequence.

Now fix some E, C⃗ as in this Claim.

Claim . There exists ⟨S_{α} : α ∈ E⟩ such that S⃗ is a ♢(E)-sequence.

Proof. Apply Theorem 32 from [5].

Fix S⃗ as in this claim. We will construct the tree T along with the functions x_{α} : ℵ_{1} ⟶ T_{α} by induction on the levels, maintaining the following hypotheses.

1. The elements of T are the members of λ^{+}, and |T_{α}| ≤ λ for all α. The ordinals appearing in T ↾ α form an initial segment of λ^{+}.

2. If ρ < σ then there exists i < ℵ_{1} such that j > i implies x_{ρ}(j) <_{T} x_{σ}(j).

3. T ↾ α is normal.

4. For every limit α and every x ∈ T ↾ α there exists a “canonical branch” b^{x,α} = { b_{γ}^{x,α} : γ ∈ C_{α}, γ ≥ ht_{T} (x) } such that

(a) b^{x,α} is cofinal in T ↾ α.

(b) b_{γ}^{x,α} ∈ T_{γ}

(c) x ≤_{T} b_{min(C_{α}−ht(x))}^{x,α}.

(d) For all γ, b_{γ}^{x,α} is the least element of the level T_{γ} which is above b_{δ}^{x,α} for

all δ ∈ [ht(x), γ) ∩ C_{α}.

5. For all β ∉ E, if α ∈ lim(C_{β}) then x_{α}(i) < x_{β}(i) for all i < ℵ_{1}.

A few remarks on the Hypotheses are in order. As usual in this type of construction, we will maintain the truth of Hypothesis 4 by putting a point of T_{α} over each canonical branch of T ↾ α whenever α ∉ E. Since for all α we have lim(C_{α})∩E = ∅, the limit stages in the inductive construction of a canonical branch will then be unproblematic, so canonical branches will always exist.

The key point is that Hypothesis 5 is only to apply when β ∉ E. This will be convenient, in that Hypothesis 5 would present difficulties at a stage where we are trying to seal off an antichain. Observe that if γ, δ ∈ lim(C_{α}) and γ < δ then δ ∉ E and γ ∈ lim(C_{δ}), so that if Hypothesis 5 holds we have x_{γ}(i) < x_{δ}(i) for all i.

We describe the inductive construction of T and x⃗.

1. β = 0. T_{0} has a single point and x_{0} is constant with value that point.

2. β = α+1. Put two points of T_{β} over each point of T_{α} and for each n designate one of the successors of x_{α}(n) to be x_{β}(n).

3. cf(β) = ω, lim(C_{β}) unbounded in β. Put a point of T_{β} over each canonical branch b^{x,β}, and also over the cofinal branches ⟨x_{γ}(i) : γ ∈ lim(C_{β})⟩ for each i < ℵ_{1}. Designate the point which is put over the cofinal branch ⟨x_{γ}(i) : γ ∈ lim(C_{β})⟩ as x_{β}(i). As in Case 3 from the construction for Theorem 2, it is easy to see that Hypotheses 2 and 5 are satisfied.

4. cf(β) = ω, lim(C_{β}) bounded in β. Put a point of T_{β} over each canonical branch b^{x,β}. Let γ = max(lim(C_{β})), and choose ⟨β_{n} : n < ω⟩ increasing and cofinal in β with β_{0} = γ.

Since there are only countably many pairs (β_{m}, β_{n}), we may find i < ℵ_{1} so large that if j > i then x_{β_{m}}(j) <_{T} x_{β_{n}}(j) whenever m < n. Now for all j > i, put a point over the cofinal branch ⟨x_{β_{m}}(j) : m < ω⟩ and designate it as x_{β}(j). If j < i then designate the point that was put over b^{x_{γ}(j),β} as x_{β}(j). Hypotheses 2 and 5 are satisfied as in Case 4 from the construction for Theorem 2.

5. cf(β) ≥ ℵ_{1}, and β ∉ E or S_{β} is not a maximal antichain of T ↾ β. Put a point on top of each canonical branch, and on top of the branch ⟨x_{γ}(i) : γ ∈ lim(C_{β})⟩ for each i < ℵ_{1}. Designate the point that was put over ⟨x_{γ}(i) : γ ∈ lim(C_{β})⟩ as x_{β}(i). It is clear that Hypotheses 2 and 5 are satisfied.

6. cf(β) = ℵ_{1}, β ∈ E, S_{β} is a maximal antichain of T ↾ β. For each z ∈ T ↾ β choose w(z) ≥_{T} z such that w(z) is also above some point of S_{β}. Put a point u(z) over the branch b^{w(z),β}.

Choose ⟨β_{i} : i < ℵ_{1}⟩ increasing and cofinal in β. Given i < ℵ_{1}, choose j(i) to be the maximal j ≤ i such that x_{β_{0}}(i) <_{T} … <_{T} x_{β_{j(i)}}(i). Define x_{β}(i) = u(x_{β_{j(i)}}(i)). Hypothesis 2 is satisfied as in Case 4 from the construction for Theorem 2. Hypothesis 5 is satisfied vacuously because β ∈ E.

It is routine to check that we have satisfied the Hypotheses, so in particular T has an ℵ_{1}-ascent path. As usual, if S is maximal in T then for some β we have that S ∩ β = S_{β} and S_{β} is maximal in T ↾ β, so that |S_{β}| ≤ λ. This concludes the proof of Theorem 4.

The proof of the following result is very similar (and in fact is exactly parallel to Baumgartner’s construction from [2]).

Theorem 5. Let λ be a cardinal such that 1. cf(λ) > ω.

2. γ < λ ⟹ γ^{ℵ_{0}} < λ.

3. 2^{λ} = λ^{+}.

4. □_{λ} holds. Then there exists an λ^{+}-Souslin tree with an ω-ascent path.

References

[1] J. Baumgartner, J. Malitz and W. Reinhardt, Embedding trees in the rationals, Proceedings of the National Academy of Sciences 67 (1970), pp 1748–1753.

[2] K. J. Devlin, Reduced products of ℵ_{2}-trees, Fundamenta Mathematicae 118 (1983), pp 129– 134.

[3] C. Gray, Iterated forcing from the strategic point of view, PhD thesis, University of California, Berkeley, 1980.

[4] A. Levy, Basic set theory, Springer-Verlag, Berlin, 1979.

[5] S. Shelah, On successors of singular cardinals, in Logic Colloquium ’78 (ed: M. Boffa, D. van Dalen and K. McAloon), North-Holland, Amsterdam, pp 357–380.

[6] S. Shelah and L. Stanley, Weakly compact cardinals and non-special Aronszajn trees, Proceedings of the American Mathematical Society 104 (1988), pp 887–897.

Mathematics Institute, Hebrew University, Givat Ram, 91904 Jerusalem, ISRAEL

E-mail address: cummings@@math.huji.ac.il