Systems of Filters Silvia Steila (joint work with Giorgio Audrito) - PowerPoint PPT Presentation

Systems of Filters Silvia Steila (joint work with Giorgio Audrito) Universit` a degli studi di Torino Winter School in Abstract Analysis: Section of Set Theory and Topology Hejnice January 30th - February 6th, 2016

How can we express properties of elementary embeddings? F ⊆ P ( X ) is a filter on X , if F is closed under supersets and finite intersections. Filters j : V → M Extenders Normal Towers C -Systems of Filters

Why Systems of Filters? Systems of Filters: ◮ generalize both extenders and normal towers; ◮ provide a common framework in which properties of extenders and towers can be expressed in a coincise way. What is a system of filters?

Indices Extenders Normal Towers E = { F a : a ∈ [ λ ] <ω } T = { F a : a ∈ V λ } C -systems of filters S = { F a : a ∈ C} A set C ∈ V is a directed set of domains iff the following holds: 1. Ideal property: C is closed under subsets and unions; 2. Transitivity: � C is transitive.

Filter Property and Compatibility In standard extenders F a is a filter on [ κ a ] | a | . π ba : [ κ b ] | b | → [ κ a ] | a | is such that given a , b ∈ [ λ ] <ω such that b = { α 0 , . . . , α n } ⊇ a = { α i 0 , . . . , α i m } and s = { s 0 , . . . , s n } , π ba ( s ) = { s i 0 , . . . , s i m } . 0 , 1 , 74 , ω, ω 3 + 1 � � For instance if a = { 1 , ω } , b = , s = { 0 , 1 , 2 , 3 , 4 } , then π ba ( s ) = { 1 , 3 } . We can see F a as a filter on a κ a . In this case π ba : b κ b → a κ a is just the restriction of functions, i.e. π ba ( f ) = f ↾ a .

Filter Property and Compatibility In standard towers F a is a filter on P ( a ). π ba : P ( b ) → P ( a ) is such that given a , b ∈ V λ , X ∈ P ( b ), π ba ( X ) = X ∩ a . We can see F a as a filter on { π M : M ⊆ a } (where π M : M → V is the Mostowski collapse of the structure ( M , ∈ )). In this case π ba is just the restriction of functions, i.e. π ba ( f ) = f ↾ a .

Filter Property and Compatibility Extenders Normal Towers E = { F a : a ∈ [ λ ] <ω } T = { F a : a ∈ V λ } F a filter on [ κ a ] | a | F a filter on P ( a ) π ba ( s ) = s b π ba ( X ) = X ∩ a a A ∈ F a iff π − 1 A ∈ F a iff π − 1 ba [ A ] ∈ F b ba [ A ] ∈ F b C -systems of filters S = { F a : a ∈ C} F a filter on O a π ba ( f ) = f ↾ a A ∈ F a iff π − 1 ba [ A ] ∈ F b O a = { π M ↾ ( a ∩ M ) : M ⊆ trcl( a ) , M ∈ V }

Filter Property and Compatibility Extenders Normal Towers E = { F a : a ∈ [ λ ] <ω } T = { F a : a ∈ V λ } F a filter on a κ a F a filter on { π M : M ⊆ a } π ba ( f ) = f ↾ a π ba ( f ) = f ↾ a A ∈ F a iff π − 1 A ∈ F a iff π − 1 ba [ A ] ∈ F b ba [ A ] ∈ F b C -systems of filters S = { F a : a ∈ C} F a filter on O a π ba ( f ) = f ↾ a A ∈ F a iff π − 1 ba [ A ] ∈ F b O a = { π M ↾ ( a ∩ M ) : M ⊆ trcl( a ) , M ∈ V }

Fineness Extenders Normal Towers For all x ∈ a , For all x ∈ a , { f ∈ a κ a : x ∈ a } ∈ F a { X ∈ P ( a ) : x ∈ X } ∈ F a C -systems of filters For all x ∈ a , { f ∈ O a : x ∈ dom( f ) } ∈ F a

Normality Extenders: ◮ u : A → V is regressive on A ⊆ a κ a iff there exists α ∈ a such that for all f ∈ A , u ( f ) ∈ f ( α ). ◮ u : A → V is guessed on B ⊆ b κ b , b ⊇ a iff there is a β ∈ b such that for all f ∈ B , u ( π ba ( f )) = f ( β ). Towers: ◮ u : A → V is regressive on A ⊆ P ( a ) iff for all X ∈ A , u ( X ) ∈ X . ◮ u : A → V is guessed on B ⊆ P ( b ), b ⊇ a iff there is a y ∈ b such that for all X ∈ B , u ( π ba ( X )) = y . C -System of Filters: define x � y as x ∈ y ∨ x = y . ◮ u : A → V is regressive on A ⊆ O a iff for all f ∈ A , u ( f ) � f ( x f ) for some x f ∈ dom( f ). ◮ u : A → V is guessed on B ⊆ O b , b ⊇ a iff there is a y ∈ b such that for all f ∈ B , u ( π ba ( f )) = f ( y ).

Normality Extenders Normal Towers Normal Normal C -systems of filters Normal (Normality) Every function u : A → V in V that is regressive on a set A ∈ I + a for some a ∈ C is guessed on a set B ∈ I + b for some b ∈ C such that B ⊆ π − 1 ba [ A ];

Ultrafilter Property Extenders Normal Towers E = { F a : a ∈ [ λ ] <ω } T = { F a : a ∈ V λ } F a ultrafilter on a κ a F a ultrafilter on P ( a ) C -systems of filters S = { F a : a ∈ C} F a ultrafilter on O a O a = { π M ↾ ( a ∩ M ) : M ⊆ trcl( a ) , M ∈ V }

� κ, λ � -Systems of Filters � κ, λ � -Extenders: F a is κ -complete. The support κ a is the least ξ such that [ ξ ] | a | ∈ F a . And ◮ if a ⊆ b ∈ [ λ ] <ω then ◮ κ a ≤ κ b ; ◮ if max( a ) = max( b ), then κ a = κ b ; ◮ κ { κ } = κ ; S is a � κ, λ � -system of filters if: κ a is the support of a iff it is the minimum ξ such that O a ∩ a V ξ ∈ F a . ◮ rank( C ) = λ and κ ⊆ � C , ◮ F { γ } is principal generated by id ↾ { γ } whenever γ < κ , ◮ κ a ≤ κ whenever a ∈ V κ +2 .

From a system of ultrafilters to elementary embeddings Let S be a C -system of ultrafilters, and define U S = { u : O a → V : a ∈ C} and the relations u = S v ⇔ { f ∈ O c : u ( π ca ( f )) = v ( π cb ( f )) } ∈ F c u ∈ S v ⇔ { f ∈ O c : u ( π ca ( f )) ∈ v ( π cb ( f )) } ∈ F c where O a = dom( u ), O b = dom( v ), c = a ∪ b . The ultrapower of V by S is Ult( V , S ) = � U S / = S , ∈ S � . Define j S : V → Ult( V , S ) by j S ( x ) = [ c x ] S , c x : O ∅ → { x } .

From elementary embedding to system of ultrafilters Let j : V → M ⊆ V [ G ] be a generic elementary embedding, C ∈ V be a directed set of domains such that ( j ↾ a ) − 1 ∈ M for all a ∈ C . The C -system of ultrafilters derived from j is S = � F a : a ∈ C� such that: A ⊆ O a : ( j ↾ a ) − 1 ∈ j ( A ) � � F a = .

C -systems of filters from a single j Let j : V → M ⊆ W be a generic elementary embedding definable in W , S n be the C n -system of V -ultrafilters derived from j for n = 1 , 2, C 1 ⊆ C 2 . Then Ult( V , S 2 ) can be factored into Ult( V , S 1 ), and crit( k 1 ) ≤ crit( k 2 ) where k 1 , k 2 are the corresponding factor maps. j V M Ult( V , S 2 ) j 2 k 2 j 1 k 1 k Ult( V , S 1 )

C -systems of filters from a single j non( F )= min {| A | : A ∈ F } . non( S )= sup { non( F a ) + 1 : a ∈ C} . Let j : V → M ⊆ W be a generic elementary embedding definable in W , S be the C -system of filters derived from j , E be the extender of length λ ⊇ j [non( S )] derived from j . Then Ult( V , E ) can be factored into Ult( V , S ), and crit( k S ) ≤ crit( k E ). j V M Ult( V , E ) j E k E j S k S k Ult( V , S )

Generic C -Systems of ultrafilters F be a B -name for an ultrafilter on P V ( X ). Define ◮ Let ˙ � � � � I ( ˙ Y ∈ ˙ ˇ F ) = Y ⊂ X : = 0 B F ◮ Let I be an ideal in V on P ( X ) and consider the poset B = P ( X ) / I . Let ˙ F ( I ) be the B -name defined by ˙ � ˇ � � F ( I ) = Y , [ Y ] I � : Y ⊆ X ◮ Let ˙ S be a B -name for a C -system of ultrafilters. Then we define the corresponding C -system of filters in V , I ( ˙ S ). ◮ Conversely, S be a C -system of filters in V . Then we define the corresponding name for a C -system of ultrafilters, ˙ F ( S ).

Generic C -Systems of ultrafilters Let S be a � κ, λ � - C -system of filters, C be a κ -cc cBa. Define S C = � A ⊆ ( O a ) V C � : ∃ B ∈ ˇ � F C � where F C a : a ∈ C a = F a A ⊇ B . S C is a C -system of filters, C ∗ S C is isomorphic to S ∗ j ( C ) and the following diagram commutes. j ˙ F ( S ) ⊆ V M V S ⊆ ⊆ ⊆ ⊆ = V C ∗ S C V C M j ( C ) V S ∗ j ( C ) j ˙ F ( S C )

Generic C -Systems of ultrafilters Let S be a � κ, λ � - C -system of filters, C be a κ -cc cBa. Define S C = � A ⊆ ( O a ) V C � : ∃ B ∈ ˇ � F C � where F C a : a ∈ C a = F a A ⊇ B . S C is a C -system of filters, C ∗ S C is isomorphic to S ∗ j ( C ) and the following diagram commutes. j ˙ F ( S ) ⊆ V M V S ⊆ ⊆ ⊆ ⊆ = V C ∗ S C V C M j ( C ) V S ∗ j ( C ) j ˙ F ( S C ) Thank you!