The special @ 2 -Aronszajn tree property and GCH David Asper o - - PowerPoint PPT Presentation

The special @2-Aronszajn tree property and GCH

David Asper´

• University of East Anglia

Workshop on Axiomatic set theory and its applications RIMS, Kyoto, Nov 2018

Aronszajn trees and Suslin trees

Let  be a regular uncountable cardinal.

• A -tree is a tree T of height  all of whose levels are

smaller than . A -Aronszajn tree is a -tree which has no -branches.

• A -Suslin tree is a -tree which has no -branches and no

antichains of size .

• If  = +, a -Aronszajn tree T is said to be special if there

exists a function f : T ! such that f(x) 6= f(y) whenever x, y 2 T are such that x <T y.

Aronszajn trees were introduced by Kurepa, and Aronszajn (1934) proved the existence, in ZFC, of a special @1-Aronszajn

• tree. Later, Specker (1949) showed that 2< = implies the

existence of special +-Aronszajn trees for regular, and Jensen (1972) produced special +-Aronszajn trees for singular in L. Baumgartner, Malitz and Reinhardt (1970) showed that Martin’s Axiom + 2@0 > @1 implies that all @1-Aronszajn trees are

• special. In particular, under this assumption there are no

@1-Suslin trees. Later, Jensen (1974) produced a model of GCH in which there are no @1-Suslin trees, and in fact all @1-Aronszajn trees are special.

Aronszajn trees were introduced by Kurepa, and Aronszajn (1934) proved the existence, in ZFC, of a special @1-Aronszajn

• tree. Later, Specker (1949) showed that 2< = implies the

existence of special +-Aronszajn trees for regular, and Jensen (1972) produced special +-Aronszajn trees for singular in L. Baumgartner, Malitz and Reinhardt (1970) showed that Martin’s Axiom + 2@0 > @1 implies that all @1-Aronszajn trees are

• special. In particular, under this assumption there are no

@1-Suslin trees. Later, Jensen (1974) produced a model of GCH in which there are no @1-Suslin trees, and in fact all @1-Aronszajn trees are special.

@2-Suslin trees

The situation at @2 turned out to be more complicated. Jensen (1972) proved that the existence of an @2-Suslin tree follows from each of the hypotheses CH +}({↵ < !2 | cf(↵) = !1}) and ⇤!1 + }({↵ < !2 | cf(↵) = !}). The second result was improved by Gregory (1976); he proved that GCH together the existence of a non–reflecting stationary subset of {↵ < !2 | cf(↵) = !} yields the existence of an @2-Suslin tree.

Theorem

(Laver–Shelah, 1981) If there is a weakly compact cardinal , then there is a forcing extension in which  = @2, CH holds, and all @2-Aronszajn trees are special (and hence there are no @2-Suslin trees).

The proof proceeds by

• L´

evy–collapsing  to become !2, and then

• running a countable–support iteration of length + in which
• ne specializes, with countable conditions, all -Aronszajn

trees given by some book-keeping function.

• One uses the weak compactness of  in V in a crucial way

in order to show that the iteration has the -c.c. and hence everything goes as planned. In the Laver–Shelah model, 2@1 = @3, and the following remained a major open problem (s. e.g. Kanamori–Magidor 1977):

Question

Is ZFC+GCH consistent with the non–existence of @2-Suslin trees?

Forcing with symmetric systems of models as side conditions

Finite–support forcing iterations involving symmetric systems of models as side conditions are useful in situations in which, for example, we want to force

• consequences of classical forcing axioms at the level of

H(!2), together with

• 2@0 large.

Given a cardinal  and T ✓ H(), a finite N ✓ [H()]@0 is a T–symmetric system if (1) for every N 2 N, (N, 2, T) 4 (H(), 2, T), (2) given N0, N1 2 N, if N0 \ !1 = N1 \ !1, then there is a unique isomorphism ΨN0,N1 : (N0, 2, T) ! (N1, 2, T) and ΨN0,N1 is the identity on N0 \ N1. (3) Given N0, N1 2 N such that N0 \ !1 = N1 \ !1 and M 2 N0 \ N, ΨN0,N1(M) 2 N. (4) Given M, N0 2 N such that M \ !1 < N0 \ !1, there is some N1 2 N such that N1 \ !1 = N0 \ !1 and M 2 N1.

The pure side condition forcing P0 = ({N : N a T–symmetric system}, ◆) (for any fixed T ✓ H()) preserves CH: This exploits the fact that given N, N0 2 N, N a symmetric system, if N \ !1 = N0 \ !1, then ΨN,N0 is an isomorphism ΨN,N0 : (N; 2, N \ N) ! (N0; 2, N \ N0) Proof: Suppose (˙ r⇠)⇠<!2 are names for subsets of ! and N P0 ˙ r⇠ 6= ˙ r⇠0 for all ⇠ 6= ⇠0. For each ⇠, let N⇠ be a sufficiently correct model such that N, ˙ r⇠ 2 N⇠.

The pure side condition forcing P0 = ({N : N a T–symmetric system}, ◆) (for any fixed T ✓ H()) preserves CH: This exploits the fact that given N, N0 2 N, N a symmetric system, if N \ !1 = N0 \ !1, then ΨN,N0 is an isomorphism ΨN,N0 : (N; 2, N \ N) ! (N0; 2, N \ N0) Proof: Suppose (˙ r⇠)⇠<!2 are names for subsets of ! and N P0 ˙ r⇠ 6= ˙ r⇠0 for all ⇠ 6= ⇠0. For each ⇠, let N⇠ be a sufficiently correct model such that N, ˙ r⇠ 2 N⇠.

By CH we may find ⇠ 6= ⇠0 such that there is an isomorphism Ψ : (N⇠; 2, T ⇤, N, ˙ r⇠) ! (N⇠0; 2, T ⇤, N, ˙ r⇠0) (where T ⇤ is the satisfaction predicate for (H(); 2, T)). Then N ⇤ = N [ {N⇠, N⇠0} 2 P0. But N ⇤ is (N⇠, P0)–generic and (N⇠0, P0)–generic. Now, let n < ! and let N 0 be an extension of N ⇤. Suppose N 0 P0 n 2 ˙ r⇠. Then there is N 00 2 P0 extending both N 0 and some M 2 N⇠ \ P0 such that M P0 n 2 ˙ r⇠. By symmetry, N 00 extends also Ψ(M). But Ψ(M) P0 n 2 Ψ(˙ r⇠) = ˙ r⇠0. We have shown N ⇤ P0 ˙ r⇠ ✓ ˙ r⇠0, and similarly we can show N ⇤ P0 ˙ r⇠0 ✓ ˙ r⇠. Contradiction since N ⇤ extends N and ⇠ 6= ⇠0. ⇤

By CH we may find ⇠ 6= ⇠0 such that there is an isomorphism Ψ : (N⇠; 2, T ⇤, N, ˙ r⇠) ! (N⇠0; 2, T ⇤, N, ˙ r⇠0) (where T ⇤ is the satisfaction predicate for (H(); 2, T)). Then N ⇤ = N [ {N⇠, N⇠0} 2 P0. But N ⇤ is (N⇠, P0)–generic and (N⇠0, P0)–generic. Now, let n < ! and let N 0 be an extension of N ⇤. Suppose N 0 P0 n 2 ˙ r⇠. Then there is N 00 2 P0 extending both N 0 and some M 2 N⇠ \ P0 such that M P0 n 2 ˙ r⇠. By symmetry, N 00 extends also Ψ(M). But Ψ(M) P0 n 2 Ψ(˙ r⇠) = ˙ r⇠0. We have shown N ⇤ P0 ˙ r⇠ ✓ ˙ r⇠0, and similarly we can show N ⇤ P0 ˙ r⇠0 ✓ ˙ r⇠. Contradiction since N ⇤ extends N and ⇠ 6= ⇠0. ⇤

By CH we may find ⇠ 6= ⇠0 such that there is an isomorphism Ψ : (N⇠; 2, T ⇤, N, ˙ r⇠) ! (N⇠0; 2, T ⇤, N, ˙ r⇠0) (where T ⇤ is the satisfaction predicate for (H(); 2, T)). Then N ⇤ = N [ {N⇠, N⇠0} 2 P0. But N ⇤ is (N⇠, P0)–generic and (N⇠0, P0)–generic. Now, let n < ! and let N 0 be an extension of N ⇤. Suppose N 0 P0 n 2 ˙ r⇠. Then there is N 00 2 P0 extending both N 0 and some M 2 N⇠ \ P0 such that M P0 n 2 ˙ r⇠. By symmetry, N 00 extends also Ψ(M). But Ψ(M) P0 n 2 Ψ(˙ r⇠) = ˙ r⇠0. We have shown N ⇤ P0 ˙ r⇠ ✓ ˙ r⇠0, and similarly we can show N ⇤ P0 ˙ r⇠0 ✓ ˙ r⇠. Contradiction since N ⇤ extends N and ⇠ 6= ⇠0. ⇤

In typical forcing iterations with symmetric systems as side conditions, 2@0 is large in the final extension. Even if P0 can be seen as the first stage of these iterations, the forcing is in fact designed to add reals at (all) subsequent successor stages. Something one may want to try at this point: Extend the symmetry requirements also to the working parts in such a way that the above CH–preservation argument goes trough. Hope to be able to force something interesting this way.

In typical forcing iterations with symmetric systems as side conditions, 2@0 is large in the final extension. Even if P0 can be seen as the first stage of these iterations, the forcing is in fact designed to add reals at (all) subsequent successor stages. Something one may want to try at this point: Extend the symmetry requirements also to the working parts in such a way that the above CH–preservation argument goes trough. Hope to be able to force something interesting this way.

While Visiting Mohammad Golshani in Tehran in December 2017, we thought about implementing these ideas (with 2@1 = @2 instead of 2@0 = @1 and @1-sized models instead if countable models) for the Laver–Shelah construction, in order to build a model of GCH with no @2-Suslin trees. We eventually succeeded:

The result

Theorem (A.–Golshani) Suppose  is a weakly compact

• cardinal. Then there exists a set–generic extension of the

universe in which (1) GCH holds, (2)  = @2, and (3) All @2-Aronszajn trees are special (and hence there are no @2-Suslin trees).

Proof sketch

Let  be weakly compact. W.l.o.g. we may assume 2µ = µ+ for all µ . Let Φ : + ! H(+) be such that for each x 2 H(+), Φ1(x) is an unbounded subset of +. Φ exists by 2 = +. Let also (Φ↵)↵<+ be the following sequence of subsets of H(+).

• Φ0 = Φ
• If ↵ > 0, then Φ↵ codes, in some fixed canonical way, the

satisfaction predicate for the structure hH(+), 2, ~ Φ↵i, where ~ Φ↵ = (Φ↵0)↵0<↵.

Let F be the weak compactness filter on , i.e., the filter on  generated by the sets { <  | (V, 2, B \ V) | = }, where B ✓ V and where is a Π1

1 sentence for the structure

(V, 2, B) such that (V, 2, B) | = F is a proper normal filter on . Let also S be the collection of F-positive subsets of , i.e., S = {X ✓  | X \ C 6= ; for all C 2 F}

The main ingredient: Revisionism (copying information from the future into the past).

Let us call h(N0, 0), (N1, 1)i an edge below if (0) For all i 2 {0, 1}, Ni ✓ H(+), Ni := N \  2 , |Ni| = |Ni|, and <|Ni|Ni ✓ Ni. (1) For all i 2 {0, 1}, i is an ordinal in the closure of Ni \ {⇠ + 1 : ⇠ < } and (Ni, 2, Φ↵) 4 (H(+), 2, Φ↵) for all ↵ 2 Ni \ i. (2) N0 ⇠ = N1 via an isomorphism ΨN0,N1 : N0 ! N1 such that

(i) (N0, 2, Φ↵) ⇠ = (N1, 2, ΦΨN0,N1(↵)) for all ↵ < 0 such that ΨN0,N1(↵) < 1, (ii) ΨN0,N1 is the identity on N0 \ N1, and (iii) ΨN0,N1(⇠)  ⇠ for every ordinal ⇠ 2 N0.

Given  +, we will build Q as a forcing with side conditions consisting of sets of edges below . Given an edge h(N0, 0), (N1, 1)i in the side condition, we will copy information in N0 attached to ↵ < 0 via ΨN0,N1 into N1 if ΨN0,N1(↵) < 1. We do not require that information in N1 attached to ΨN0,N1(↵) be copied into N0.

Given models with markers (N, ), (N0, 0) and (N1, 1), if (N, ) 2 N0 and (N0, 2) ⇠ = (N1, 2), then we let ⇡,N

N0,0,N1,1 denote

the supremum of the set of ordinals ⇠ + 1 2 N1 such that

• ⇠ < ΨN0,N1(),
• ΨN1,N0(⇠) < 0, and
• ⇠ < 1

Assuming ↵ < +, Q↵ defined and Q↵ CH + = !2, we let T ⇠↵ 2 H(+) be a canonically chosen Q↵-name for -Aronszajn tree such that T ⇠↵ = Φ↵ if Φ(↵) is a Q↵-name for a -Aronszajn tree. We assume that for each ⇢ < , the ⇢-th level of T ⇠↵ is {⇢} ⇥ !1.

Definition of the forcing

Let  + and suppose Q↵ defined for all ↵ < . A condition in Q is an ordered pair of the form q = (fq, ⌧q) with the following properties. (1) fq is a countable function such that dom(fq) ✓ + \ and such that the following holds for every ↵ 2 dom(fq).

(a) If ↵ = 0, then fq(↵) 2 Col(!1, <). (b) If ↵ > 0, then fq(↵) :  ⇥ !1 ! !1 is a countable function.

(2) ⌧q is a countable set of edges below .

(3) The following holds for every edge h(N0, 0), (N1, 1)i 2 ⌧q.

(a) If h(N0

0, 0 0), (N0 1, 0 1)i 2 N0 \ ⌧q, then

h(ΨN0,N1(N0

0), ⇤ 0), (ΨN0,N1(N0 1), ⇤ 1)i 2 ⌧q

for some ⇤

0 ⇡0

0,N0

N0,0,N1,1 and ⇤ 1 ⇡0

1,N0 1

N0,0,N1,1.

(b) The following holds for each nonzero ordinal ↵ 2 dom(fq) \ N0 \ 0 such that ΨN0,N1(↵) < 1.

(i) ΨN0,N1(α) 2 dom(fq) (ii) fq(α) δN0 ⇥ ω1 ✓ fq(ΨN0,N1(α))

(4) For all ↵ < , q ↵ 2 Q↵, where q ↵ = (fq ↵, ⌧q ↵)

(5) The following holds for every nonzero ↵ < .

(a) If ↵ 2 dom(fq), then q ↵ forces that fq(↵) is a partial specializing function for T ⇠↵. (b) For every edge h(N0, 0), (N1, 1)i 2 ⌧q, if ↵ 2 N0 \ 0, then Q↵+1 \ N0 l QN0

↵+1, where QN0 ↵+1 is the partial order whose

conditions are ordered pairs p = (fp, ⌧p) such that

(i) fp is a function such that dom(fp) ✓ α + 1, (ii) if α 2 dom(fp), then fp(α) : κ ⇥ ω1 ! ω1 is a countable function, (iii) τp is a set of edges below α + 1, (iv) γ0, γ1  α for every h(N0

0, γ0), (N0 1, γ1)i 2 τp \ N0,

(v) p α 2 Q↵, (vi) p N0 2 Q↵+1, and (vii) if α 2 dom(fp), then p α forces that fp(α) is a partial specializing function for T ⇠↵,

• rdered by setting p1 Q

N0 ↵+1 p0 if

• p1 α Q↵ p0 α and
• fp0(α) ✓ fp1(α) in case α 2 dom(fp0).

The extension relation: Given q1, q0 2 Q, q1  q0 (q1 is an extension of q0) if and

• nly if the following holds.

(A) dom(fq0) ✓ dom(fq1) (B) for every ↵ 2 dom(fq0), fq0(↵) ✓ fq1(↵). (C) For every h(N0, 0), (N1, 1)i 2 ⌧q0 there are 0

0 0 and 1 1 such that h(N0, 0 0), (N1, 0 1)i 2 ⌧q1.

Main facts

(0) For every < +, Q is definable over the structure (H(+), 2, Φ+1) without parameters. Moreover, this definition can be taken to be uniform in . (1) Q1 forces  = !2. (2) For every  +,

(i) Q↵ ✓ Q for all ↵ < , and (ii) if cf() , then Q = S

↵< Q↵.

(3) Thanks to the fact that we are only copying information ‘from the future into the past’, (Q)+ is a forcing iteration (i.e., Q↵ l Q for all ↵ < ): Given q 2 Q and r 2 Q↵, if r ↵ q ↵, then (fr [ fq [↵, ), ⌧q [ ⌧r) is a common extension of q and r in Q.

(4) Q is -closed for every  +. In fact, every decreasing !-sequence (fn)n<! of Q-conditions has a greatest lower bound q⇤ in Q, q⇤ = (f, S

n ⌧qn), where

dom(f) = S

n dom(fqn), and

f(↵) = [ {fqm(↵) : m n} for all n and ↵ 2 dom(fqn). In particular, forcing with Q does not add new !-sequences of ordinals, and therefore it preserves both !1 and CH. (5) Q+ has the -c.c. (6) Q+ adds -many new subsets of !1, but not more than that; in particular, Q+ preserves 2@1 = @2 [essentially the same argument we saw a few slides back]. (7) Q+ forces that all @2-Aronszajn are special.

Proof of (6): Q+ adds less than +-many new subsets of !1. Suppose, towards a contradiction, that q 2 Q+ and there is a sequence ( r ⇠i)i<+ of names for subsets of !1 such that q Q+ r ⇠i 6= r ⇠i0 for all i < i0 < + By -c.c. we may assume, for each i, that r ⇠i 2 H(+) and r ⇠i is a Qi-name for some i < +. Let ✓ be a large enough. For each i < + let N⇤

i H(✓) be

such that (1) |N⇤

i | = |N⇤ i \ |,

(2) N⇤

i is closed under sequences of length less than |N⇤ i |,

(3) q, r ⇠i, i, (Φ↵)↵<+, (Q↵)↵<+ 2 N⇤

i , and

(4) Q↵ \ N⇤

i l Q↵ for every ↵ 2 + \ N⇤ i .

N⇤

i can be found by a Π1 1-reflection argument, using the weak

compactness of  and the -c.c. of all Q↵: Let M⇤ H(✓) be such that

• q, r

⇠i, i, (Φ↵)↵<+, (Q↵)↵<+ 2 M⇤,

• |M⇤| = , and
• <M⇤ ✓ M⇤.

By -c.c. of all Q↵, Q↵ l Q⇤

↵ := Q↵ \ M⇤ for each ↵ 2 M⇤ \ +.

Let B ✓ V code (Q⇤

↵)↵2M⇤\+ and the sequence of maximal

antichains of Q⇤

↵ (for ↵ 2 M⇤ \ +). But now, for a suitable Π1 1

sentence with B as parameter, we may find a set C 2 F of <  such that (V, 2) | = (B \ V). With a sensible choice of it follows that we may find 2 C such that N⇤

i = SkM⇤()

satisfies (1)–(4). Let Ni = N⇤

i \ H(+) for each i.

Let P be the satisfaction predicate for the structure hH(+), 2, ~ Φi, where ~ Φ ✓ H(+) codes (Φ↵)↵<+ in some canonical way, and let M be an elementary submodel of H(✓) containing q, r ⇠i, (i)i<+, (Q↵)↵+, (N⇤

i )i<+ and P, and such that |M| =  and <M ✓ M.

Let i0 2 + \ M. By a standard reflection argument we may find i1 2 + \ M for which there exists an isomorphism Ψ : (Ni0, 2, P, r ⇠i0, i0, q) ⇠ = (Ni1, 2, P, r ⇠i1, i1, q), such that Ψ(⇠)  ⇠ for every ordinal in Ni0. Indeed, the existence of such an i1 follows from the correctness of M in H(✓) about a suitable statement with parameters (Ni)i<+, q, P, (i)i<+, ( r ⇠i)i<+, and Ni0 \ M, all of which are in M.

Let ¯ q = (fq, ⌧¯

q), where

⌧¯

q = ⌧q [ {h(Ni0, i0 + 1), (Ni1, i1 + 1)i}

But Q

Ni0 ↵+1 ✓ Q↵+1 for each ↵. Hence, Q↵+1 \ Ni0 l Q Ni0 ↵+1 for

every ↵ 2 N⇤

• i0. It follows that ¯

q 2 Q+ thanks to the choice of N⇤

i0

and N⇤

• i1. We show that ¯

q Q+ r ⇠i0 = r ⇠i1. Suppose not, and we will derive a contradiction. Thus we can find ⌫ < !1 and q0 + ¯ q such that q0 Q+ “⌫ 2 r ⇠i0 $ ⌫ / 2 r ⇠i1”. Let us assume, for concreteness, that q0 Q+ “⌫ 2 r ⇠i0 and ⌫ / 2 r ⇠i1” (the proof in the case that q0 Q+ “⌫ 2 r ⇠i1 and ⌫ / 2 r ⇠i0” is exactly the same).

By correctness of N⇤

i0 we have that this model contains a

maximal antichain A of conditions in Qi0 deciding the statement “⌫ 2 r ⇠i0”. |A| <  by -c.c. Hence, since N⇤

i0 \  2 ,

A ✓ N⇤

i0 \ H(+) = Ni0. Hence, we may find a common

extension q00 of q0 and some r 2 Ni0 \ A such that r Q+“⌫ 2 r ⇠i0”. Also, note that, since Ψ is an isomorphism between the structures (Ni0, 2, P, r ⇠i0, i0, q) and (Ni1, 2, P, r ⇠i1, i1, q), and by the choice of P, we have that Ψ(r) Qi1 “⌫ 2 Ψ( r ⇠i0) = r ⇠i1 ” But then, by clause (3) in the definition of condition, we have that q00  Ψ(r). We thus obtain that q00 Q+“⌫ 2 r ⇠i1”, which is impossible as q0 Q+“⌫ / 2 r ⇠i1” and q00  q0. Contradiction. ⇤

The κ-chain condition

Some technical facts first. Given conditions q0, q1 2 Q+, we denote by q0 q1 the natural amalgamation of q0 and q1; i.e., q0 q1 is the

• rdered pair (f, ⌧) resulting from closing q0 and q1 under

relevant isomorphisms ΨN0,N1 so that clause (3) in the definition

• f condition holds in the end.

Lemma 1 Let  +, and suppose q0, q1 2 Q are such that for every ↵ < , if (q0 ↵) (q1 ↵) 2 Q↵, then (q0 ↵ + 1) (q1 ↵ + 1) 2 Q↵+1 Then q0 q1 2 Q.

Given an ordinal ↵ and a set ⌧ of edges, we will call a finite sequence (↵i)i<n of ordinals a ⌧-orbit of ↵ if there is a sequence ~ E = (h(Ni

0, i 0), (Ni 1, i 1)i)i<n of edges in ⌧ and a

sequence (✏i)i<n of ordinals in {0, 1} such that

• ↵ 2 N0

✏0 \ 0 ✏0, and

• for each i < n,

↵i = (ΨNi

✏i ,Ni 1✏i ΨNi1 ✏i1,Ni1 1✏i1

. . . ΨN0

✏0,N0 1✏0)(↵)

is such that ↵i < i

1✏i and such that ↵i 2 Ni+1 ✏i+1 \ i+1 ✏i+1 if

i + 1 < n. We will call (↵i)i<n a descending orbit if ↵i+1  ↵i whenever i + 1 < n.

Lemma 2 For all  +, q0, q1 2 Q and ↵ 2 dom(fq0q1) there is some ↵⇤ 2 dom(fq0) [ dom(fq1) such that ↵ is on some ⌧q0 [ ⌧q1–orbit of ↵⇤. In fact, for any ⌧q0q1-orbit (↵i)i<n of an

• rdinal ↵⇤ and any ordinal ↵ on (↵i)i<n there is in fact a

(possibly longer) ⌧q0 [ ⌧q1-orbit of ↵⇤ on which ↵ is.

Given ↵ < + and given nodes x, y 2  ⇥ !1, if Q↵ is -c.c., then we denote by A↵

x,y the first, in some well–order of H(+)

canonically definable from Φ, maximal antichain of Q↵ consisting of conditions deciding whether or not x and y are comparable in T ⇠↵. Given q 2 Q+, we will say that q is adequate in case: (1) For all nonzero ↵, ↵0 in dom(fq), if x 2 dom(fq(↵)), y 2 dom(fq(↵0)), and Q↵ is -c.c., then q ↵ extends a condition in A↵

x,y.

(2) For every edge h(N0, 0), (N1, 1)i 2 ⌧q and every ↵ 2 dom(fq) \ N1 \ 1, if ΨN1,N0(↵) < 0, then ΨN1,N0(↵) 2 dom(fq) and fq(ΨN1,N0(↵)) N1 ⇥ !1 = fq(↵) N1 ⇥ !1 Let us call a condition weakly adequate if it satisfies clause (1) in the above definition.

The set of weakly adequate conditions is trivially dense. Also: Lemma 3 Suppose q is a weakly adequate Q+-condition, ↵ 2 dom(fq), and ↵0 < + is on a descending ⌧q-orbit of ↵ as witnessed by a sequence ~ E = (h(Ni

0, i 0), (Ni 1, i 1)i)in of edges

in ⌧q. Suppose x0 = (⇢0, ⇣0) and x1 = (⇢1, ⇣1) are two nodes such that (1) ⇢0, ⇢1 < min{Ni

0 | i  n}, and

(2) q ↵ forces that x and y are incomparable in T ⇠↵. Then q ↵0 forces that x0 and x1 are incomparable in T ⇠↵0.

Lemma 4 For every  +, the set of adequate Q-conditions is dense in Q. We call a model Q suitable if Q is an elementary submodel of cardinality  of some high enough H(✓), closed under <-sequences, and such that hQ↵ | ↵ < +i 2 Q. Given a suitable model Q, a bijection ' :  ! Q, and an ordinal < , we will denote '“ by M'

.

Lemma 4 For every  +, the set of adequate Q-conditions is dense in Q. We call a model Q suitable if Q is an elementary submodel of cardinality  of some high enough H(✓), closed under <-sequences, and such that hQ↵ | ↵ < +i 2 Q. Given a suitable model Q, a bijection ' :  ! Q, and an ordinal < , we will denote '“ by M'

.

Given  +, we will say that Q has the strong -chain condition if for every X 2 S, every suitable model Q such that , X 2 Q, every bijection ' :  ! Q, and every two sequences (q0

| 2 X) 2 Q

and (q1

| 2 X) 2 Q

• f Q-conditions, if q0

M' = q1 M' for every 2 X, then

there is some Y 2 S, Y ✓ X, together with sequences (q00

| 2 Y)

and (q11

| 2 Y)

• f Q-conditions with the following properties.

(1) q00

Q q0 and q11 Q q1 for every 2 Y.

(2) For all < ⇤ in Y, q00

q11 ⇤ is a common extension of q00

• and q11

⇤.

Given a suitable model Q such that 2 Q, a bijection ' :  ! Q, a Q-condition q 2 Q, and < , let us say that q is

• compatible with respect to ' and if, letting Q⇤

= Q \ Q, we

have that

• Q⇤

\ M' l Q⇤ ,

• q M'

2 Q⇤ , and

• q M'

forces in Q⇤ \ M' that q is in the quotient forcing

Q⇤

/ ˙

GQ⇤

\M' ; equivalently, for every r Q⇤ \M' q M'

, r is

compatible with q.

Rather than proving that every Q has the -c.c., we prove the following more informative lemma.

Lemma

The following holds for every  +. (1) Q has the strong -chain condition. (2) Suppose D 2 F, Q is a suitable model, , D 2 Q, ' :  ! Q is a bijection, and (q0

| 2 D) 2 Q and

(q1

| 2 D) 2 Q are sequences of adequate

Q-conditions. Then there is some D0 2 F such that D0 ✓ D and such that for every 2 D0, if q0

M' = q1 M' , then there are conditions q

00

Q q0

• and q

01

Q q1 such that

(a) q

00

M' = q

01

M' and

(b) q

00

and q

01

are both -compatible with respect to ' and .

The proof is by induction on . Let  + and suppose (1)↵ and (2)↵ holds for all ↵ < . We will show that (1) and (2) also hold. There is nothing to prove for = 0, the case = 1 is trivial using the inaccessibility of , and the case = + follows form Q+ = S

<+ Q. Hence, let us assume 1 < < +.

Suppose next that < +. We start with the proof of (1). Let Q, ', X 2 S and (q0

| 2 X) and (q1 | 2 X) be as in the

definition of strong -c.c. In what follows, we will write M instead of M'

.

The proof is by induction on . Let  + and suppose (1)↵ and (2)↵ holds for all ↵ < . We will show that (1) and (2) also hold. There is nothing to prove for = 0, the case = 1 is trivial using the inaccessibility of , and the case = + follows form Q+ = S

<+ Q. Hence, let us assume 1 < < +.

Suppose next that < +. We start with the proof of (1). Let Q, ', X 2 S and (q0

| 2 X) and (q1 | 2 X) be as in the

definition of strong -c.c. In what follows, we will write M instead of M'

.

Let Q⇤

↵ = Q↵ \ Q for every ↵ 2 Q \ ( + 1). By the induction

hypothesis, Q↵ has the -c.c. for every ↵ 2 Q \ . Hence, since

<Q ✓ Q, we have that Q⇤ ↵ l Q↵ for every such ↵; in particular,

we have that for every ↵ 2 Q \ , Q⇤

↵ forces over V that T

⇠↵ does not have -branches. Given a nonzero ↵ 2 , a node x = (⇢, ⇣) and an ordinal ¯ ⇢ < ⇢, let B↵

x,¯ ⇢ denote the least, in some well–order of H(+)

canonically defined from Φ, maximal antichain of Q↵ consisting

• f conditions deciding some ¯

⇣ < !1 such that the node ¯ x = (¯ ⇢, ¯ ⇣) is below x in T ⇠↵.

Given

• conditions q0, q1,
• ↵ 2 dom(fq0) and ↵0 2 dom(fq1),
• nodes x = (⇢0, ⇣0) and y = (⇢1, ⇣1) such that

x 2 dom(fq0(↵)) and y 2 dom(fq1(↵0)),1 and

• < ,

we will say that x and y are separated below at stages ↵ and ↵0 by q1 ↵ and q0 ↵0 (via ¯ x, ¯ y) if there are ¯ ⇢ < and ⇣ 6= ⇣0 in !1 such that ¯ x = (¯ ⇢, ⇣), ¯ y = (¯ ⇢, ⇣0), and such that (1) q0 ↵ extends a condition in B↵

x,¯ ⇢ forcing ¯

x to be below x in T ⇠↵ and (2) q1 ↵0 extends a condition in B↵0

y,¯ ⇢ forcing ¯

y to be below y in T ⇠↵0.

1α and α0 may or may not be equal, and the same applies to x and y.

Given Y 2 S such that Y ✓ X and such that M Q, M \  = , and <M ✓ M for all 2 Y, and given two sequences 00 = (q00

• | 2 Y), 11 = (q11
• | 2 Y) of

adequate Q⇤

• conditions, we say that 00, 11 is a separating

pair for 0 and 1 if the following holds. (1) q00

Q q0 and q11 Q q1 for all 2 Y.

(2) For all 2 Y, all nonzero ↵ 2 dom(fq00

) \ M and

↵0 2 dom(fq11

) such that ↵0  ↵, and all

x 2 dom(fq00

(↵)) \ ( ⇥ !1) and

y 2 dom(fq11

(↵0)) \ ( ⇥ !1), x and y are separated below

at stages ↵ and ↵0 by q00

↵ and q11 ↵0 via some pair

0(x, y, ↵, ↵0, ), 1(x, y, ↵, ↵0, ).

(3) The following holds for all 0 < 1 in Y.

(a) q00

0 M0 = q11 1 M1

(b) dom(fq00

0) \ M0 = dom(fq11 1 ) \ M1

(c) q00

0 2 M1

(d) For some ordinal &, sup(R0 [ ∆0) = sup(R1 [ ∆1) = & where, for every ✏ 2 {0, 1}, R✏ = {⇢ < ✏ | ↵ 2 dom(fq✏✏

✏ ), (⇢, ⇣) 2 dom(fq✏✏ ✏ (↵))}

and ∆✏ = {N0 < ✏ | h(N0, 0), (N1, 1)i 2 ⌧q✏✏

✏ }

(e) For every edge h(N0, 0), (N1, 1)i 2 ⌧q11

1 , if N0 < 1, then

(N0 [ N1) \ (dom(fq00

0 ) [ ⌧q00 0) ✓ M0.

(4) For all 0 < 1 in X, all nonzero ↵ 2 dom(fq00

0) \ M0 and

↵0 2 dom(fq11

1) such that ↵0  ↵, and all nodes

x 2 dom(fq00

0(↵)) \ (0 ⇥ !1)

and y0 2 dom(fq11

1(↵0)) \ (1 ⇥ !1)

there are

• a node x0 2 dom(fq00

1(↵)) \ (1 ⇥ !1),

• a stage ↵† 2 dom(fq11

0) such that ↵†  ↵, and

• a node y 2 dom(fq11

0(↵†)) \ (0 ⇥ !1)

such that 0(x, y, ↵, ↵†, 0) = 0(x0, y0, ↵, ↵0, 1) and 1(x, y, ↵, ↵†, 0) = 1(x0, y0, ↵, ↵0, 1)

Claim 1: Let Y 2 S be such that M \  = for all 2 Y and suppose 00 = (q00

• | 2 Y), 11 = (q11
• | 2 Y) is a

separating pair for 0 and 1. Then for all 0 < < 1 in Y, q00

0 q11 1 is a common extension of q00 0 and q11 1 in Q.

The following is essentially due to Laver–Shelah.

Claim 1: Let Y 2 S be such that M \  = for all 2 Y and suppose 00 = (q00

• | 2 Y), 11 = (q11
• | 2 Y) is a

separating pair for 0 and 1. Then for all 0 < < 1 in Y, q00

0 q11 1 is a common extension of q00 0 and q11 1 in Q.

The following is essentially due to Laver–Shelah.

Claim 2: Suppose Z 2 S, (p0

| 2 Z) 2 Q and

(p1

| 2 Z) 2 Q are sequences of conditions in Q⇤ such that

p0

M and p1 M are compatible conditions in Q⇤ \ M for

every 2 Z, and suppose that for every 2 Z,

• p0

and p1 are -compatible with respect to ' and ↵ for all

↵ 2 \ Q,

• ↵ 2 dom(fp0

) \ M,

• ↵0

2 dom(fp1

) \ M is a nonzero ordinal such that ↵0

 ↵,

and

• x = (⇢0

, ⇣0 ) and y = (⇢1 , ⇣1 ) are nodes in ( \ ) ⇥ !1

such that x 2 dom(fp0

(↵)) and y 2 dom(fp1 (↵0

)).

Then there is D 2 F, together with two sequences (p2

| 2 Z \ D), (p3 | 2 Z \ D) of conditions in Q⇤ such that

(1) for each 2 Z \ D, p2

 q0 and p3  p1 ,

(2) for each 2 Z \ D, p2

M and p3 M are compatible in

Q⇤

\ M, and

(3) for each 2 Z \ D, x and y are separated below at stages ↵ and ↵0

by p2 ↵ and p3 ↵0 .

By Claim 1, in order to conclude the proof of current instance of (1), it suffices to prove: Claim 3: There is a separating pair for 0 and 1. Claim 3 is proved by a construction in countably many steps using Claim 2, and a pressing–down argument using the normality of F. We are left with proving (2). But this is established with essentially the same argument as in the corresponding proof in the Laver–Shelah paper. This concludes the proof of the -chain condition lemma and the proof of the theorem.

By Claim 1, in order to conclude the proof of current instance of (1), it suffices to prove: Claim 3: There is a separating pair for 0 and 1. Claim 3 is proved by a construction in countably many steps using Claim 2, and a pressing–down argument using the normality of F. We are left with proving (2). But this is established with essentially the same argument as in the corresponding proof in the Laver–Shelah paper. This concludes the proof of the -chain condition lemma and the proof of the theorem.

An open question

Question (Shelah): Is it consistent to have GCH together with a successor cardinal  !1 such that all -Aronszajn and all +-Aronszajn trees are special?