Suslin’s Problem and Martin Axiom Suslin’s Problem Martin Axiom Iterated Suslin’s Problem and Martin Axiom Forcing Direct Limit Construction of the model 23 July 2014 Suslin’s Problem and Martin Axiom 1 / 14
Suslin’s Problem and Martin Axiom Suslin’s Problem Martin Axiom Suslin’s Problem Iterated Forcing Is there a linearly ordered set which satisfies the countable Direct Limit chain condition (ccc) and is not separable? Construction of the model Suslin’s Problem and Martin Axiom 2 / 14
Suslin’s Problem and Martin Axiom Suslin’s Problem Martin Axiom Suslin’s Problem Iterated Forcing Is there a linearly ordered set which satisfies the countable Direct Limit chain condition (ccc) and is not separable? Construction of the model Such a set is called a Suslin line. The existence of a Suslin line is equivalent to the existence of a normal Suslin tree. Suslin’s Problem and Martin Axiom 2 / 14
Tree Suslin’s Problem and Martin Axiom A tree is a poset ( P , < ) such that ∀ x ∈ T { y : y < x } is well ordered by < . Suslin’s Problem Martin Axiom Iterated Forcing Direct Limit Construction of the model Suslin’s Problem and Martin Axiom 3 / 14
Tree Suslin’s Problem and Martin Axiom A tree is a poset ( P , < ) such that ∀ x ∈ T { y : y < x } is well ordered by < . Suslin’s Problem Suslin Tree Martin Axiom Iterated A tree is called a Suslin tree if: Forcing 1 height ( T ) = ω 1 Direct Limit Construction 2 every branch in T is at most countable of the model 3 every antichain in T is at most countable A Suslin tree is called normal if: 1 T has a unique least point 2 each level of T is at most countable 3 x not maximal has infinitely many immediate successors 4 ∀ x ∈ T there is some z > x at each greater level 5 if o ( x ) = o ( y ) = β with β limit and { z : z < x } = { z : z < y } then x = y Suslin’s Problem and Martin Axiom 3 / 14
Suslin’s Problem and Martin Axiom Suslin’s MA k Problem Martin Axiom If a poset ( P , < ) satisfies ccc and D is a collection of at most Iterated k dense subsets of P , then there exists a D -generic filter on P. Forcing Direct Limit Construction of the model Suslin’s Problem and Martin Axiom 4 / 14
Suslin’s Problem and Martin Axiom Suslin’s MA k Problem Martin Axiom If a poset ( P , < ) satisfies ccc and D is a collection of at most Iterated k dense subsets of P , then there exists a D -generic filter on P. Forcing Direct Limit Lemma Construction of the model If MA ℵ 1 holds then there is no Suslin tree. Suslin’s Problem and Martin Axiom 4 / 14
Suslin’s Problem and Martin Axiom Suslin’s MA k Problem Martin Axiom If a poset ( P , < ) satisfies ccc and D is a collection of at most Iterated k dense subsets of P , then there exists a D -generic filter on P. Forcing Direct Limit Lemma Construction of the model If MA ℵ 1 holds then there is no Suslin tree. Solovay-Tennenbaum = MA + 2 ℵ 0 > ℵ 1 . There is a model M of ZFC such that M | Suslin’s Problem and Martin Axiom 4 / 14
Suslin’s Problem and Martin Axiom Suslin’s Problem Martin Axiom Let P be a forcing notion in M and G 1 ⊆ P a M -generic filter. Iterated Forcing Direct Limit Construction of the model Suslin’s Problem and Martin Axiom 5 / 14
Suslin’s Problem and Martin Axiom Suslin’s Problem Martin Axiom Let P be a forcing notion in M and G 1 ⊆ P a M -generic filter. Iterated Let Q be a poset in M [ G 1 ] and G 2 ⊆ Q a M [ G 1 ]-generic filter. Forcing Direct Limit Construction of the model Suslin’s Problem and Martin Axiom 5 / 14
Suslin’s Problem and Martin Axiom Suslin’s Problem Martin Axiom Let P be a forcing notion in M and G 1 ⊆ P a M -generic filter. Iterated Let Q be a poset in M [ G 1 ] and G 2 ⊆ Q a M [ G 1 ]-generic filter. Forcing Direct Limit I want to show that there exists a G M -generic filter on R such Construction that: of the model M [ G 1 ][ G 2 ] = M [ G ] We will define this filter using Boolean algebras. Suslin’s Problem and Martin Axiom 5 / 14
Suslin’s Problem and Martin Axiom Let B be a complete Boolean algebra in M . Suslin’s Problem Martin Axiom Iterated Forcing Direct Limit Construction of the model Suslin’s Problem and Martin Axiom 6 / 14
Suslin’s Problem and Martin Axiom Let B be a complete Boolean algebra in M . Let C ∈ M B such that Suslin’s Problem || C is a complete Boolean algebra || = 1. Martin Axiom Iterated Forcing Direct Limit Construction of the model Suslin’s Problem and Martin Axiom 6 / 14
Suslin’s Problem and Martin Axiom Let B be a complete Boolean algebra in M . Let C ∈ M B such that Suslin’s Problem || C is a complete Boolean algebra || = 1. Martin Axiom D is a maximal subset in M B such that: Iterated Forcing 1 || c ∈ C || = 1 ∀ c ∈ D Direct Limit 2 c 1 , c 2 ∈ D , c 1 � = c 2 ⇒ || c 1 = c 2 || < 1 Construction of the model Suslin’s Problem and Martin Axiom 6 / 14
Suslin’s Problem and Martin Axiom Let B be a complete Boolean algebra in M . Let C ∈ M B such that Suslin’s Problem || C is a complete Boolean algebra || = 1. Martin Axiom D is a maximal subset in M B such that: Iterated Forcing 1 || c ∈ C || = 1 ∀ c ∈ D Direct Limit 2 c 1 , c 2 ∈ D , c 1 � = c 2 ⇒ || c 1 = c 2 || < 1 Construction of the model I define + D : ∀ c 1 , c 2 ∈ D ∃ c ∈ D such that || c = c 1 + C c 2 || = 1 this c is unique and I define c = c 1 + D c 2 . The operations · D and − D are defined similarly. Suslin’s Problem and Martin Axiom 6 / 14
Suslin’s Problem and Martin Axiom Let B be a complete Boolean algebra in M . Let C ∈ M B such that Suslin’s Problem || C is a complete Boolean algebra || = 1. Martin Axiom D is a maximal subset in M B such that: Iterated Forcing 1 || c ∈ C || = 1 ∀ c ∈ D Direct Limit 2 c 1 , c 2 ∈ D , c 1 � = c 2 ⇒ || c 1 = c 2 || < 1 Construction of the model I define + D : ∀ c 1 , c 2 ∈ D ∃ c ∈ D such that || c = c 1 + C c 2 || = 1 this c is unique and I define c = c 1 + D c 2 . The operations · D and − D are defined similarly. With this operations D is a complete Boolean algebra ( in M ). I define B ∗ C = D . Suslin’s Problem and Martin Axiom 6 / 14
Suslin’s Problem and Martin Axiom Theorem Let B be a complete Boolean algebra in M , let C ∈ M B be Suslin’s such that || C is a complete Boolean algebra || = 1 and let Problem D = B ∗ C such that B is a complete subalgebra of D . Then Martin Axiom Iterated 1 If G 1 is an M -generic ultrafilter on B , C = i G 1 ( C ) and G 2 Forcing is an M [ G 1 ]-generic ultrafilter on C then there is an Direct Limit M -generic ultrafilter G on B ∗ C such that: Construction of the model M [ G 1 ][ G 2 ] = M [ G ] 2 If G is an M -generic ultrafilter on B ∗ C . G 1 = G ∩ B and C = i G 1 ( C ) then there is an M [ G 1 ]-generic ultrafilter G 2 on C such that: M [ G 1 ][ G 2 ] = M [ G ] Suslin’s Problem and Martin Axiom 7 / 14
Suslin’s Problem and Martin Axiom Suslin’s Problem Martin Axiom Iterated Forcing Lemma Direct Limit Construction B satisfies ccc and || C satisfies ccc || = 1 iff B ∗ C satisfies ccc. of the model Suslin’s Problem and Martin Axiom 8 / 14
Suslin’s Problem and Martin Axiom Let α be a limit ordinal. Suslin’s Let { B i } i <α a sequence such that Problem - B i is a complete Boolean algebra Martin Axiom Iterated - if i < j B i is a complete subalgebra of B j Forcing Direct Limit Construction of the model Suslin’s Problem and Martin Axiom 9 / 14
Suslin’s Problem and Martin Axiom Let α be a limit ordinal. Suslin’s Let { B i } i <α a sequence such that Problem - B i is a complete Boolean algebra Martin Axiom Iterated - if i < j B i is a complete subalgebra of B j Forcing Direct Limit Direct limit Construction of the model The completion B of � i <α B i is called direct limit of { B i } . B = limdir i ≤ α B i . Suslin’s Problem and Martin Axiom 9 / 14
Suslin’s Problem and Martin Axiom Let α be a limit ordinal. Suslin’s Let { B i } i <α a sequence such that Problem - B i is a complete Boolean algebra Martin Axiom Iterated - if i < j B i is a complete subalgebra of B j Forcing Direct Limit Direct limit Construction of the model The completion B of � i <α B i is called direct limit of { B i } . B = limdir i ≤ α B i . Lemma Then if each B i is k -saturated then B is k -saturated. In particular if each B i satisfies ccc then B satisfies ccc. Suslin’s Problem and Martin Axiom 9 / 14
Suslin’s Problem and Martin Axiom Suslin’s Problem Martin Axiom Let M be a transitive model of ZFC + GCH. Iterated Forcing We will construct a complete Boolean algebra B such that if G Direct Limit is an M -generic filter on B then Construction of the model = MA + 2 ℵ 0 ≤ ℵ 2 M [ G ] | Suslin’s Problem and Martin Axiom 10 / 14
Suslin’s Problem and Martin Axiom B α Suslin’s Let { B α } be a sequence such that: Problem 1 α < β ⇒ B α is a complete subalgebra of B β Martin Axiom Iterated 2 γ limit ⇒ B γ = limdir i ≤ γ B i Forcing 3 each B α satisfies ccc Direct Limit Construction 4 | B α | ≤ ℵ 2 of the model Suslin’s Problem and Martin Axiom 11 / 14
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