Thintall Boolean Algebras David Asper o University of East Anglia - - PowerPoint PPT Presentation

Thin–tall Boolean Algebras

David Asper´

• University of East Anglia

Workshop on Pure and Descriptive Set Theory (Poi) Torino, 25, 26 Sep. 2015

Finite–support iterations with symmetric systems as side conditions

Proper forcing is nice:

• Proper forcing notions preserve !1.
• Properness (due to Shelah) is preserved under countable

support iterations. Hence, granted the existence of a supercompact cardinal, one can build a model of PFA, the forcing axiom for proper forcings relative to collection of @1–many dense series (Baumgartner).

PFA has many consequences. One of them is 2@0 = @2. Problem: Force some consequence of PFA or, for that matter, something we can force by iterating proper forcing, together with 2@0 > @2.

Countable support iterations won’t do. In fact, at stages of uncountable cofinality we are adding generics, over all previous models, for Add(1, !1) (= adding a Cohen subset of !1); in particular we are collapsing the continuum of all those previous models to @1. Hence, in the final model necessarily 2@0  @2. Bigger support won’t work either: The preservation lemma for properness doesn’t work in this context. Finite–support iterations won’t work either; in fact, any finite–support iteration of non–c.c.c. forcings collapses !1.

A solution: Use finite supports, together with countable elementary substructures of some H(✓) as side conditions affecting the whole iteration or initial segments of the iteration in

• rder to ensure properness (the idea of using countable

structures as side conditions in order to “force” a non–proper forcing to become proper is old (Todorˇ cevi´ c, 1980’s, implicit in work of Baumgartner (adding a club of !1 by finite con- ditions)), but this was not done in the context of actual iterations). Typically we will want our iteration to have the @2–c.c. (after all we are interested in 2@0 arbitrarily large).The natural approach

• f using finite 2–chains of structures won’t work, though, since

we have too many structures and would therefore lose the @2–c.c. We will replace 2–chains of structures by “matrices” of structures with suitable symmetry properties. If we start with CH and consider only iterands with the @2–c.c., we might succeed.

Symmetric systems of elementary substructures

Given a set N, N will denote N \ !1 (the height of N).

Definition

Let ✓ be a cardinal and T ✓ H(✓) (such that S T = H(✓)). A finite set N ✓ [H(✓)]@0 is a T–symmetric system iff the following holds for all N, N0, N1 2 N: (1) (N; 2, Y) 4 (H(✓); 2, T) (2) If N0 = N1, then there is a unique isomorphism ΨN0,N1 : (N0; 2, T) ! (N1; 2, T) Furthermore, ΨN0,N1 is the identity on N0 \ N1. (3) If N0 = N1 and N 2 N0 \ N, then ΨN0,N1(N) 2 N. (4) If N0 < N1, then there is some N0

1 2 N such that N0

1 = N1

and N0 2 N0

1.

• Symmetric systems had previously been considered in (at

least) work of Todorˇ cevi´ c, Abraham–Cummings and

• Koszmider. Again, not in the context of forcing iterations.
• The def. of symmetric system guarantees that

(4)’ if N0, N1 2 N and N0 < N1, then there is some N0

0 2 N1\N such that N0

0 = N0 and N0\N1 = N0\N0

0.

(In fact, N0

0 = ΨN0

1,N1(N0), where N0

1 2 N is such that

N0

1 = N1 and N0 2 N0

1.) This property is important in many

• applications. Sometimes it is enough to keep (1)–(3) and

weaken (4) to (4)’. The resulting object is called partial T–symmetric system.

Two amalgamation lemmas 1st amalgamation lemma: If N and N 0 are T–symmetric systems, (S N) \ (S N 0) = X, and there are enumerations (Ni)i<n and (N0

i )i<n of N, N 0, resp., for which there is an

isomorphism Ψ : ( [ N; 2, Ni, T, X) ! ( [ N 0; 2, N0

i , T, X)

then N [ N 0 is a T–symmetric system. 2nd amalgamation lemma: Let N be a T–symmetric system and M 2 N. Suppose M 2 M is a T–symmetric system such that N \ M ✓ M. Let N M(M) = N [ {ΨM,M0(N) : N 2 M, M0 2 N : M0 = M} Then N M(M) is a T–symmetric system.

Corollaries Let SymmT = ({N : N T–symmetric system}, ◆) Using 1st amalgamation lemma: Corollary 1 (CH) SymmT is @2–Knaster. Corollary 2 (CH) SymmT adds new reals but preserves CH. Using 2nd amalgamation lemma: Corollary 3 SymmT is proper.

Iterating: General template of the constructions.

Start with CH, let  regular with 2<κ = . Fix suitable T ✓ H(). Let (Pα : ↵  ) be such that for all ↵, a condition in Pα is a pair q = (F, ∆) such that: (1) F is a finite function such that dom(F) ✓ ↵ (dom(F) is the support of q). (2) ∆ is a finite set of pairs (N, ), where N 2 [H()]@0,  ↵,  sup(N \ ), and where dom(∆) is a (partial) T–symmetric system ( is the marker associated to N). (3) For all < ↵, q|β := (F , {(N, min{, }) : (N, ) 2 ∆}) is a Pβ–condition.

(4) For every ⇠ 2 dom(F), q|ξ Pξ F(⇠) 2 Φ⇤(⇠) where Φ⇤(⇠) is a Pξ–name for a suitable forcing, and Φ⇤(⇠) = Φ(⇠) if Φ(⇠) is a Pξ–name for a suitable forcing (and where Φ is a suitable bookkeeping function on ). (5) For every ⇠ 2 dom(F) and every (N, ) 2 ∆, if ⇠  and ⇠ 2 N, then q|ξ Pξ F(⇠) is (N[ ˙ Gξ], Φ⇤(⇠))–generic Given Pα–conditions q0 = (F0, ∆0), q1 = (F1, ∆1), q1 α q0 iff (a) for every (N, ) 2 ∆0 there is some 0 such that (N, 0) 2 ∆1, (b) dom(F0) ✓ dom(F1), and (c) for every ⇠ 2 dom(F0), q0|ξ Pξ F1(⇠) Φ⇤(ξ) F0(⇠)

Typical properties

(1) Pβ is always a complete suborder of Pα whenever < ↵: Thanks to the markers in (N, ) 2 ∆. (2) Each Pβ is typically @2–c.c.: This often uses CH and standard ∆–system arguments as in the proof of Corollary 1. (3) Properness of Pα: We define a sequence (Mα)ακ of clubs

• f [H()]@0 (of increasing “richness”); e.g., by picking increasing

sequence (✓α)ακ of cardinals above  and letting Mα = {N⇤ \ H() : N⇤ 4 H(✓α), T, Φ, (✓β)β<α 2 N⇤}

Proof by induction on ↵: Let p 2 Pα \ N⇤, N⇤ 4 H(✓) countable containing everything relevant. We build q⇤ from q by essentially adding (N, min{↵, sup(N \ )}) to ∆p. We argue that q⇤ can be built and is (N⇤, Pα)–generic. For this, let A 2 N⇤, A ✓ Pα maximal antichain, and let q 2 Pα extend both a condi- tion t 2 A and q⇤. We want to find r 2 N \A, r compatible with q.

Case ↵ = 0: Corollary 3. Case ↵ = + 1: Usually easy, since, by definition, q|β Pβ Fq() is N[ ˙ Gβ], Φ⇤())–generic (if 2 dom(Fq)). Case ↵ 6= 0 limit: The case when cf(↵) 6= !1 is typically easy, since then there is 2 2 N \ ↵ bounding the support of (some condition in A extended by) q (obvious when cf(↵) = !, using that |A|  @1 if cf(↵) !2). Then we apply induction hypothesis to : Working in N[ ˙ Gσ], find r 2 A such that r|σ 2 ˙ Gσ, r compatible with everything N[ ˙ Gσ] can see, dom(Fr) \ [, ↵) = ;. By extending q|σ we can assume r 2 N (since, by induction hypothesis, q|σ is (N[ ˙ Gσ], Pσ)–generic). Now we can amalgamate q and r into a condition. In the case cf(↵) = !1, go to the blackboard.

A typical application:

Theorem

(A.–Mota, A generalization of Martin’s Axiom, Israel J. Math., to appear) (GCH) For every regular  !2, there is a proper @2–c.c. forcing notion forcing MA1.5

<κ + 2@0 = .

Here MA1.5

λ

is the forcing axiom for the class of @1.5–c.c. partial

• rders relative to collections of  –many dense sets, where P

has the @1.5–c.c. iff for every ✓ > |TC(P)| there is a club D ✓ [H(✓)]@0 such that for any finitely many N0, . . . , Nm 2 D and every p 2 P, if p 2 Ni for all i with Ni minimal among N0, . . . , Nm, then there is q P p such that q is (Nk, P)–generic for all k  m. MA1.5

λ

• f course extends MAλ but also has, for example, many

consequences at the level of strong failures of Club Guessing at !1.

Some nice spin–offs: For every , there is a homogeneous @2–c.c. proper forcing AddB() adding –many Baumgartner clubs to !1 (CH not needed!). AddB() Adding a Baumgartner club = Add(, !) Cohen forcing In particular, AddB() has applications in the context of cardinal characteristics for ω1!1 and [!1]@1. AddB() also figures prominently in the construction, in ZFC, of a forcing notion collapsing @3 but preserving all other cadinals (the existence of this forcing answers a 1983 question of Abraham, who built in ZFC a forcing collapsing @2 and preserving all other cardinals).

Changing the side conditions (I): Larger structures.

The theory of T–symmetric systems goes through unchanged if we re- place |N| = @0 with |N| = (and we can also ask that <λN ✓ N). We would then expect to be able to iterate <–closed forcings which are suitably –proper (i.e., proper with respect to sufficiently many elem. substructures N such that |N| = , perhaps in the strong sense of the def. of @1.5–c.c.). This doesn’t work: In the above inductive proof of properness it is essential that supports be finite.In fact, this cannot work. Otherwise we should be able to build models falsifying instances of Club–Guessing on !2 which hold true in ZFC! Specifically we would be able to destroy the Club–Guessing in the following slide.

Changing the side conditions (I): Larger structures.

The theory of T–symmetric systems goes through unchanged if we re- place |N| = @0 with |N| = (and we can also ask that <λN ✓ N). We would then expect to be able to iterate <–closed forcings which are suitably –proper (i.e., proper with respect to sufficiently many elem. substructures N such that |N| = , perhaps in the strong sense of the def. of @1.5–c.c.). This doesn’t work: In the above inductive proof of properness it is essential that supports be finite.In fact, this cannot work. Otherwise we should be able to build models falsifying instances of Club–Guessing on !2 which hold true in ZFC! Specifically we would be able to destroy the Club–Guessing in the following slide.

Theorem

(Shelah, Claim 3.3 in Colouring and non-productivity of @2–c.c.,

• Ann. Pure and Applied Logic, vol. 84, 2 (1997), 153–174)

Let  > !1 be a regular cardinal. Then for every stationary S ✓ Sκ+

κ

there is a club–sequence hCδ : 2 Si such that for all 2 S,

• ot(Cδ) = , and
• cf(Cδ(↵ + 1)) =  for all ↵ < ,

and such that for every club D ✓ + there is some 2 S (equivalently, stationary many 2 S) such that {↵ <  : Cδ(↵ + 1) 2 D} is stationary (where Cδ() is the –th member of Cδ).

See also D. Soukup and L. Soukup, Club guessing for dummies for a nicely written proof of the above. Question (Shelah, Question 5.4 in On what I do not understand (and have something to say): Part I, Fundamenta Math., vol. 166, 1–2 (2000), 1–82.) Is it true in ZFC that for every regular cardinal  !1 there is a club–sequence ~ C = hCδ : 2 Sκ+

κ i with ot(Cδ) =  for all

such that for every club D ✓ + there is some such that {↵ <  : {Cδ(↵ + 1), Cδ(↵ + 2)} ✓ D} is stationary?

According to Shelah in the above paper, if there is a club–sequence as in the above question on Sκ+

κ

and GCH holds, then there is a +–Souslin tree. In particular, an affirmative answer to above question would yield an affirmative answer to the following well–known open question. Question: Does GCH imply that there is an !2–Souslin tree?

The following (from A., The consistency of a club–guessing failure at the successor of a regular cardinal, in “Infinity, computability, and metamathematics: Festschrift celebrating the 60th birthdays of Peter Koepke and Philip Welch,” 2014, 5–27) answers the above question:

Theorem

(GCH) For every regular cardinal  !1 there is a cardinal–preserving poset forcing that there is no club–sequence ~ C = hCδ : 2 Sκ+

κ i with ot(Cδ) =  for all and

such that for every club D ✓ + there is no such that {↵ <  : {Cδ(↵ + 1), Cδ(↵ + 2)} ✓ D} is stationary. Proof by a <–support iteration of length ++ using, as side conditions, symmetric systems of size < of models N such that |N| =  and <κN ✓ N. Proof of –properness is direct, not by induction.

Also using this type of side conditions:

Theorem

(Mota–Weiss) (GCH) For every regular  there is a cardinal–preserving poset forcing the existence of a superatomic Boolean algebra of width  and height ++. Baumgartner–Shelah (1987) did the case  = !; it was open whether  > ! is possible.

Changing the side conditions (II): Structures of two types

Neeman considers side conditions consisting of structures of two cardinalities types (small and large) {Q0, . . . , Qm} such that for all i,

1 Qi 2 Qi+1 2 If Qi is large and Qi+1 is small, then there is some j < i

such that Qi \ Qi+1 = Qj. In his (main) applications:

• small = countable, large = transitive (some H()). Uses

these to build a model of PFA using finite supports.

• small = countable, large = cardinality @1. Uses these to

add objects on !2 (various types of ⇤ω1–sequences).

Now we want to solve ?

• Lin. ordered side conds. 2 types =

Symmetric systems 2–chains structures same card.

Definition Let ✓ be a cardinal and T ✓ H(✓) (such that S T = H(✓)). Let  be an infinite cardinal. [Given a set N, N will now denote sup(N \ ++).] A set N ✓ H(✓) with |N| <  is a T–symmetric system of type {, +} iff the following holds for all Q0, Q1, Q 2 N: (1) (Q; 2, T) 4 (H(✓); 2, T), |Q| 2 {, +}, and <|Q|Q ✓ Q. (2) If Q0 = Q1, then the following holds.

(a) Q0 \ ++ = Q1 \ ++ (b) There is a (unique) isomorphism ΨQ0,Q1 : (Q0; 2, T) ! (Q1; 2, T) Furthermore, ΨQ0,Q1 is the identity on Q0 \ Q1.

(3) If Q0 = Q1 and Q 2 Q0, then ΨQ0,Q1(Q) 2 N.

(4) If Q0 < Q1, then there is some Q0

1 2 N such that

Q0

1 = Q1 and such that the following holds.

(a) If Q0 2 Q0

1, then Q0 \ Q0 1 2 N.

(b) If Q0 / 2 Q0

1, then there is some N 2 Q0 1 \ N such that

|N| = +, Q0 2 N, Q0

1 \ N 2 N and Q0

1\N < N0.

Two amalgamation lemmas 1st amalgamation lemma: If N and N 0 are T–symmetric systems of type {, +}, (S N) \ (S N 0) = X, and there are enumerations (Ni)i<n and (N0

i )i<n of N, N 0, resp., for which

there is an isomorphism Ψ : ( [ N; 2, Ni, T, X) ! ( [ N 0; 2, N0

i , T, X)

then N [ N 0 is a T–symmetric system of type {, +}. 2nd amalgamation lemma: Let N be a T–symmetric system and M 2 N of type {, +}. Suppose M 2 M is a T–symmetric system of type {, +} such that N \ M ✓ M. Let N M(M) = N [ {ΨM,M0(N) : N 2 M, M0 2 N : M0 = M} Then N M(M) is a T–symmetric system of type {, +}.

Corollaries Let SymmT = ({N : N T–symmetric system of type {, +}}, ◆) SymmT is clearly <–closed. Using 1st amalgamation lemma: Corollary 1 (2κ+ = ++) SymmT is +++–Knaster. Corollary 2

• (2κ+ = ++) SymmT adds new subsets of + but

preserves 2κ+ = ++.

• (2κ = +) SymmT adds new subsets of  but preserves

2κ = +.

Using 2nd amalgamation lemma: Corollary 3 SymmT is proper for structures Q with |Q| 2 {, +} such that <|Q|Q ✓ Q.

Can we hope to iterate suitably proper and @1–proper forcing, with finite supports, using symmetric systems of type {@0, @1} with markers to ensure preservation of !1 and !2? (Remember that having to rely on infinite supports was a problem for proving properness in our context.) No: Go to the board. In fact, if there were a reasonable iteration theory here, starting from GCH we would be able to add clubs of !2 by finite conditions in length, say, !3, and in the end we would have killed Club–Guessing on !2 (which is a ZFC–theorem)!

An application of symmetric systems of type {@0, @1}: Adding a sBa of width ω and height ω3

The following question remained open since the work of Baumgartner–Shelah 1987:

Question

Is the existence of a superatomic Boolean algebra of width ! and height !3 consistent?

Theorem

(about a week ago, 85% true) (GCH) There is a proper, @1–proper, and @3–Knaster forcing notion adding a superatomic Boolean algebra of width ! and height !3.

An application of symmetric systems of type {@0, @1}: Adding a sBa of width ω and height ω3

The following question remained open since the work of Baumgartner–Shelah 1987:

Question

Is the existence of a superatomic Boolean algebra of width ! and height !3 consistent?

Theorem

(about a week ago, 85% true) (GCH) There is a proper, @1–proper, and @3–Knaster forcing notion adding a superatomic Boolean algebra of width ! and height !3.

Given a Boolean algebra B let I(B) be the ideal of B generated by its atoms. For an ordinal ↵, define the ↵–th Cantor–Bendixson ideal J α(B) on B:

• J 0(B) = {0}
• letting ⇡α : B

! B/J α(B) the canonical projection, J α+1(B) = ⇡1

α (I(B/Iα(B))).

• If ↵ 6= 0 is limit, J α(B) = S

β<α J β(B).

B is superatomic iff there is some ↵ such that B = J α+1(B). The least such ↵ is the height of B (denoted ht(B)). B has width  iff |B/J β(B)| =  for every < ht(B).

Definition

(essentially due to Baumgartner) Let , be infinite cardinals. An LCS( ⇥ )–structure is a pair (, b) where

•  is a partial order on  ⇥ .
• If (↵, ) < (↵0, 0), then < 0.
• For every (↵, ) 2  ⇥ and every < ,

{ : (, ) < (↵, )} is infinite.

• b : [ ⇥ ]2

! [ ⇥ ]<ω

• For all ⌫0, ⌫1 2  ⇥ ,
• for all ⌫ 2 b({⌫0, ⌫1}), ⌫  ⌫0 and ⌫  ⌫1, and
• for every ⌫, if ⌫  ⌫0 and ⌫  ⌫1, then there is some

⌫0 2 b({⌫0, ⌫1}) such that ⌫  ⌫0.

(b is called a barrier function for ).

Proposition

(Baumgartner) Let , be infinite cardinals. If there is an LCS( ⇥ )–structure, then there is a superatomic Boolean algebra of width  and height .

Proof of theorem: We build a proper @1–proper @3–Knaster forcing P adding an LCS(! ⇥ !3)–stucture as follows. Let T ✓ H(!3) code (eβ)β<ω3 where eβ : || ! bijection for all . Conditions are quadruples (N, A, , b), where:

(1) N is a T–symmetric system of type {@0, @1}. (2) A ✓ N (3)  is a partial order such that dom() ✓ ! ⇥ !3 and |  | < @0. (4) For all (↵, ), (↵0, 0) 2 dom(), if (↵, )  (↵0, 0) and (↵, ) 6= (↵0, 0), then 2 0. (5) b : [dom()]2 ! [dom()]<ω is a barrier function for . (6) For all ⌫0, ⌫1 2 dom() and all Q 2 A, if {⌫0, ⌫1} 2 Q, then b({⌫0, ⌫1}) 2 Q. (7) For all ⌫0, ⌫1 2 dom() and all M 2 A, if |M| = @0 and {⌫0, ⌫1} ✓ S{X 2 M : |X| = @1}, then b({⌫0, ⌫1}) ✓ S{X 2 M : |X| = @1}. (8) For all ⌫0, ⌫1 2 dom(), M 2 A and N 2 N \ M, if |M| = @0, |N| = @1, ⌫0 2 M, ⌫1 2 N, and ⌫0  ⌫1, then there is some ⌫ 2 M \ N such that ⌫0  ⌫  ⌫1.

Given P–conditions p0 = (N0, A0, 0, b0), p1 = (N1, A1, 1, b1), we say that p1 extends p0 iff

• N0 ✓ N1,
• A0 ✓ A1,
• dom(0) ✓ dom(1) and 1 | dom(0) =0, and
• b1 [dom(0)]2 = b0.

Conjecture: One can force a sBa of width  and height +++ for any given regular  !1. Should be doable combining these ideas with the ideas of Mota–Weiss one con prove the following. Question: Can

• ne force existence of sBa of width  and height for > +++?

Conjecture: The answer should again be yes, at least for = +4, using symmetric systems of Neeman’s side conditions

• f three types.