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Forcing with matrices ofcountable elementary submodels
Borisa Kuzeljevic
IMSJSPS 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 2c.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 2c.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 2c.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 theth 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 theth 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 theth 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 2c.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 2c.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