Forcing with matrices ofcountable elementary submodels

Borisa Kuzeljevic

IMS-JSPS Joint Workshop - IMS Singapore, January 2016

Borisa Kuzeljevic (NUS) Matrices of elementary submodels January 2016 1 / 14

Joint work with Stevo Todorcevic

Borisa Kuzeljevic (NUS) Matrices of elementary submodels January 2016 2 / 14

Matrices

Definition

Let 2 be a regular cardinal. By H we denote the collection of allsets whose transitive closure has cardinality < . We consider it as a modelof the form (H,,

Matrices

Theorem (Todorcevic, 1984)

PFA implies that fails for every uncountable cardinal .

Theorem (Todorcevic, 1985)

It is consistent with ZF that every directed set of cardinality 1 is cofinallyequivalent to one of the following five: 1, , 1, 1, [1] 2.

Borisa Kuzeljevic (NUS) Matrices of elementary submodels January 2016 4 / 14

Matrices

Theorem (Todorcevic, 1984)

PFA implies that fails for every uncountable cardinal .

Theorem (Todorcevic, 1985)

It is consistent with ZF that every directed set of cardinality 1 is cofinallyequivalent to one of the following five: 1, , 1, 1, [1] 2.

Borisa Kuzeljevic (NUS) Matrices of elementary submodels January 2016 4 / 14

Matrices

Theorem (Todorcevic, 1984)

PFA implies that fails for every uncountable cardinal .

Theorem (Todorcevic, 1985)

It is consistent with ZF that every directed set of cardinality 1 is cofinallyequivalent to one of the following five: 1, , 1, 1, [1] 2.

Borisa Kuzeljevic (NUS) Matrices of elementary submodels January 2016 4 / 14

Matrices

By M H we denote that M is a countable elementary submodel of H.

If G P is a filter in P generic over V , then we define G : 1 H as thefunction satisfying

G () = {M H : p G such that M p()} .

For p, q P we will define their join p q as the function from 1 to Hsatisfying (p q)() = p() q() for < 1.

Borisa Kuzeljevic (NUS) Matrices of elementary submodels January 2016 5 / 14

Matrices

By M H we denote that M is a countable elementary submodel of H.

If G P is a filter in P generic over V , then we define G : 1 H as thefunction satisfying

G () = {M H : p G such that M p()} .

For p, q P we will define their join p q as the function from 1 to Hsatisfying (p q)() = p() q() for < 1.

Borisa Kuzeljevic (NUS) Matrices of elementary submodels January 2016 5 / 14

Matrices

By M H we denote that M is a countable elementary submodel of H.

If G P is a filter in P generic over V , then we define G : 1 H as thefunction satisfying

G () = {M H : p G such that M p()} .

For p, q P we will define their join p q as the function from 1 to Hsatisfying (p q)() = p() q() for < 1.

Borisa Kuzeljevic (NUS) Matrices of elementary submodels January 2016 5 / 14

Matrices

If a condition q P and M H (M = M 1) for are given, it isclear what intersection q M represents.

We define the restriction of q to M as a function with finite supportq | M : 1 H satisfying supp(q | M) = supp(q) M and for < Mwe let (q | M)() to be the set of all M(N) where M q(M),M : M

= M H and N q() M .

Note that the function q | M is in P.

Borisa Kuzeljevic (NUS) Matrices of elementary submodels January 2016 6 / 14

Matrices

If a condition q P and M H (M = M 1) for are given, it isclear what intersection q M represents.

We define the restriction of q to M as a function with finite supportq | M : 1 H satisfying supp(q | M) = supp(q) M and for < Mwe let (q | M)() to be the set of all M(N) where M q(M),M : M

= M H and N q() M .

Note that the function q | M is in P.

Borisa Kuzeljevic (NUS) Matrices of elementary submodels January 2016 6 / 14

Matrices

If a condition q P and M H (M = M 1) for are given, it isclear what intersection q M represents.

We define the restriction of q to M as a function with finite supportq | M : 1 H satisfying supp(q | M) = supp(q) M and for < Mwe let (q | M)() to be the set of all M(N) where M q(M),M : M

= M H and N q() M .

Note that the function q | M is in P.

Borisa Kuzeljevic (NUS) Matrices of elementary submodels January 2016 6 / 14

Matrices

We will also need the following notion which we call the closure of pbelow .

Let p P and supp(p). Then cl(p) : 1 H is a function such thatsupp(cl(p)) = supp(p) and cl(p)() = p() for , while for < we define cl(p)() to be the set of all N1,N2(M) where M p() N1,N1,N2 p() and N1,N2 : N1

= N2.

Borisa Kuzeljevic (NUS) Matrices of elementary submodels January 2016 7 / 14

Matrices

We will also need the following notion which we call the closure of pbelow .

Let p P and supp(p). Then cl(p) : 1 H is a function such thatsupp(cl(p)) = supp(p) and cl(p)() = p() for , while for < we define cl(p)() to be the set of all N1,N2(M) where M p() N1,N1,N2 p() and N1,N2 : N1

= N2.

Borisa Kuzeljevic (NUS) Matrices of elementary submodels January 2016 7 / 14

Basic properties

Lemma

P is strongly proper.

Lemma

Let G be a filter generic in P over V , let M,M H(2)+ andp,P M M . If : M

= M then for = M 1 = M 1 thecondition pMM = p {, {M H,M H}} satisfies:

pMM [G M] = G M .

Borisa Kuzeljevic (NUS) Matrices of elementary submodels January 2016 8 / 14

Basic properties

Lemma

P is strongly proper.

Lemma

Let G be a filter generic in P over V , let M,M H(2)+ andp,P M M . If : M

= M then for = M 1 = M 1 thecondition pMM = p {, {M H,M H}} satisfies:

pMM [G M] = G M .

Borisa Kuzeljevic (NUS) Matrices of elementary submodels January 2016 8 / 14

Basic properties

For p P, by p we define p : 1 [H1 ] as a function with the samesupport as p which maps supp(p) to the transitive collapse of somemodel from p(), while for 1 \ supp(p) take p() = .

Lemma

Let p, q P. If p = q, then p and q are compatible conditions.

Lemma (CH)

P satisfies 2-c.c.

Lemma

P preserves CH.

Borisa Kuzeljevic (NUS) Matrices of elementary submodels January 2016 9 / 14

Basic properties

For p P, by p we define p : 1 [H1 ] as a function with the samesupport as p which maps supp(p) to the transitive collapse of somemodel from p(), while for 1 \ supp(p) take p() = .

Lemma

Let p, q P. If p = q, then p and q are compatible conditions.

Lemma (CH)

P satisfies 2-c.c.

Lemma

P preserves CH.

Borisa Kuzeljevic (NUS) Matrices of elementary submodels January 2016 9 / 14

Basic properties

For p P, by p we define p : 1 [H1 ] as a function with the samesupport as p which maps supp(p) to the transitive collapse of somemodel from p(), while for 1 \ supp(p) take p() = .

Lemma

Let p, q P. If p = q, then p and q are compatible conditions.

Lemma (CH)

P satisfies 2-c.c.

Lemma

P preserves CH.

Borisa Kuzeljevic (NUS) Matrices of elementary submodels January 2016 9 / 14

Kurepa tree I

Definition

A tree T ,

Kurepa tree I

Definition

A tree T ,

Kurepa tree II

Theorem

P adds a Kurepa tree T .

Sketch of the proof:Consider the family of functions f : 1 1 ( < 2) defined by

f() =

{, if there is M G () such that M and M() = ,0, otherwise.

Denote this family by F = {f : < 2} and for a fixed < 2 let the-th branch of our tree be given by F = {f : < 1}.

So the Kurepa tree will be T =

Kurepa tree II

Theorem

P adds a Kurepa tree T .

Sketch of the proof:Consider the family of functions f : 1 1 ( < 2) defined by

f() =

{, if there is M G () such that M and M() = ,0, otherwise.

Denote this family by F = {f : < 2} and for a fixed < 2 let the-th branch of our tree be given by F = {f : < 1}.

So the Kurepa tree will be T =

Kurepa tree II

Theorem

P adds a Kurepa tree T .

Sketch of the proof:Consider the family of functions f : 1 1 ( < 2) defined by

f() =

{, if there is M G () such that M and M() = ,0, otherwise.

Denote this family by F = {f : < 2} and for a fixed < 2 let the-th branch of our tree be given by F = {f : < 1}.

So the Kurepa tree will be T =

Continuous matrices

Definition

Let Pc be the suborder of P containing all the conditions satisfying: for every p Pc there is a continuous -chain M : < 1 (i.e. if

is a limit ordinal, then M =

Almost Souslin Kurepa tree I

Definition

A tree S of height 1 is an almost Souslin tree if for every antichainX S , the set (S is the -th level of S)

L(X ) = { < 1 : X S 6= }

is not stationary in 1.

The existence of an almost Souslin Kurepa tree was asked byZakrzewski (1986).

Todorcevic showed that the existence of an almost Souslin Kurepatree is consistent (1987).

Golshani showed there is an almost Souslin Kurepa tree in L (2013).

Borisa Kuzeljevic (NUS) Matrices of elementary submodels January 2016 13 / 14

Almost Souslin Kurepa tree I

Definition

A tree S of height 1 is an almost Souslin tree if for every antichainX S , the set (S is the -th level of S)

L(X ) = { < 1 : X S 6= }

is not stationary in 1.

The existence of an almost Souslin Kurepa tree was asked byZakrzewski (1986).

Todorcevic showed that the existence of an almost Souslin Kurepatree is consistent (1987).

Golshani showed there is an almost Souslin Kurepa tree in L (2013).

Borisa Kuzeljevic (NUS) Matrices of elementary submodels January 2016 13 / 14

Almost Souslin Kurepa tree II

Theorem (CH)

The tree T is an almost Souslin Kurepa tree.

Sketch of the proof: Let H be a Pc -name. Then the set

= { < 1 : M Gc () M & M[Gc ] 1 = M 1 = } .

is closed and unbounded in 1.

Now, let be a Pc -name for an antichain X in T .

Because CH holds in V , Pc is 2-c.c. so there is a Pc -name for X whichis in H.

Then L(X ) = .

Borisa Kuzeljevic (NUS) Matrices of elementary submodels January 2016 14 / 14

Almost Souslin Kurepa tree II

Theorem (CH)

The tree T is an almost Souslin Kurepa tree.

Sketch of the proof: Let H be a Pc -name. Then the set

= { < 1 : M Gc () M & M[Gc ] 1 = M 1 = } .

is closed and unbounded in 1.

Now, let be a Pc -name for an antichain X in T .

Because CH holds in V , Pc is 2-c.c. so there is a Pc -name for X whichis in H.

Then L(X ) = .

Borisa Kuzeljevic (NUS) Matrices of elementary submodels January 2016 14 / 14