Type omission Averageable Classes Torsion Modules Some compactness in some nonelementary classes Will Boney University of Illinois at Chicago January 9, 2015 Beyond First Order Model Theory Miniconference University of Texas-San Antonio
Type omission Averageable Classes Torsion Modules Goal The plan is to develop a framework that gives rise to compactness in some nonelementary contexts. This allows us to develop some nonforking notions, and we specialize to the example of torsion modules over PIDs.
Type omission Averageable Classes Torsion Modules Prototype I: Abelian torsion groups Have a nice elementary (model) theory of abelian groups Torsion groups (of infinite exponent) are not first order: � ∀ x n · x = 0 n <ω
Type omission Averageable Classes Torsion Modules Prototype I: Abelian torsion groups Have a nice elementary (model) theory of abelian groups Torsion groups (of infinite exponent) are not first order: � ∀ x n · x = 0 n <ω However, there is an easy way to pick out the torsion elements from G : tor ( G ) := { g ∈ G : ∃ n < ω. n · g = 0 }
Type omission Averageable Classes Torsion Modules Prototype I: Abelian torsion groups Have a nice elementary (model) theory of abelian groups Torsion groups (of infinite exponent) are not first order: � ∀ x n · x = 0 n <ω However, there is an easy way to pick out the torsion elements from G : tor ( G ) := { g ∈ G : ∃ n < ω. n · g = 0 } Moreover, tor ( G ) is an abelian group Key fact: given g , h and their orders, I have a bound on the order of g + h
Type omission Averageable Classes Torsion Modules Prototype II: Archimedean fields Have a nice elementary (model) theory of ordered fields of characteristic 0 Archimedean fields are not first order: � ∀ x 1 + · · · + 1 > n > − 1 − · · · − 1 n <ω
Type omission Averageable Classes Torsion Modules Prototype II: Archimedean fields Have a nice elementary (model) theory of ordered fields of characteristic 0 Archimedean fields are not first order: � ∀ x 1 + · · · + 1 > n > − 1 − · · · − 1 n <ω However, there is an easy way to pick out the standard, finite elements of a field: arch ( F ) := { f ∈ F : st ( f ) = f } Moreover, arch ( F ) is an ordered field of characteristic 0 Key fact: given standard f , g , − f , we have 1 g , f + g , and fg are standard
Type omission Averageable Classes Torsion Modules Similarities There are two key similarities here that will guide us in abstracting these situations: The types were unary Where elements omit types lets me figure out where functions of them omit types
Type omission Averageable Classes Torsion Modules Similarities There are two key similarities here that will guide us in abstracting these situations: The types were unary Where elements omit types lets me figure out where functions of them omit types I’m probably going to often use phrases like “where type omission happens.” Each type is going to have a natural index (as we’ve seen) and the “location” of type omission is that index.
Type omission Averageable Classes Torsion Modules Outline Discuss the type omitting hull and properties that lead to it being well-behaved Compactness results and ultraproducts Averageable classes Examples Torsion modules
Type omission Averageable Classes Torsion Modules Framework We will be in the following situation: M is an L -structure Γ is a set of unary L -types For ease we enumerate Γ as follows: Γ = { p j ( x ) : j < α } p j ( x ) = { φ j k ( x ) : k < β j }
Type omission Averageable Classes Torsion Modules Main definition M is an L -structure Γ is a set of unary L -types Definition Γ( M ) := { m ∈ M : ∀ j < α, ∃ k j < β j . M � ¬ φ j k j ( m ) } Γ( M ) contains all elements of M that omit all types of Γ according to M .
Type omission Averageable Classes Torsion Modules Main definition M is an L -structure Γ is a set of unary L -types Definition Γ( M ) := { m ∈ M : ∀ j < α, ∃ k j < β j . M � ¬ φ j k j ( m ) } Γ( M ) contains all elements of M that omit all types of Γ according to M . Each element has a (possibly many) witnesses to its inclusion. Namely m ∈ Γ( M ) iff there is some k ( m ) ∈ Π β j such that M � ¬ φ j k ( m )( j ) ( m ).
Type omission Averageable Classes Torsion Modules The main use Definition Γ( M ) := { m ∈ M : ∀ p ∈ Γ , ∃ φ p ∈ p . M � ¬ φ p ( m ) } Suppose { M i : i ∈ I } is a collection of L -structures that already omit Γ They probably also model a common theory T U is an ultrafilter on I Then we can form Γ(Π M i / U ) , which will omit all of the types of Γ and give enough averaging to get some compactness results...
Type omission Averageable Classes Torsion Modules The main use Definition Γ( M ) := { m ∈ M : ∀ p ∈ Γ , ∃ φ p ∈ p . M � ¬ φ p ( m ) } Suppose { M i : i ∈ I } is a collection of L -structures that already omit Γ They probably also model a common theory T U is an ultrafilter on I Then we can form Γ(Π M i / U ) , which will omit all of the types of Γ and give enough averaging to get some compactness results...sometimes.
Type omission Averageable Classes Torsion Modules Problems with Γ( M ) This construction turns out to be very fragile
Type omission Averageable Classes Torsion Modules Problems with Γ( M ) This construction turns out to be very fragile The types of Γ must be “honestly” unary (coding, classification over a predicate, etc.) Γ( M ) might fail to be a structure If Γ( M ) is a structure, it might still fail to be an elementary substructure This means, depending on the types, it might not even omit all of the types of Γ
Type omission Averageable Classes Torsion Modules Problems with Γ( M ) This construction turns out to be very fragile The types of Γ must be “honestly” unary (coding, classification over a predicate, etc.) Γ( M ) might fail to be a structure If Γ( M ) is a structure, it might still fail to be an elementary substructure This means, depending on the types, it might not even omit all of the types of Γ The plan is to give examples of where this can go wrong, and then give some sufficient conditions on when things work
Type omission Averageable Classes Torsion Modules Bad example I: p -adics M = � ω, + , | , 2 � I = ω , U is any non principle ultrafilter p ( x ) = { (2 k | x ) ∧ ( x � = 0) : k < ω }
Type omission Averageable Classes Torsion Modules Bad example I: p -adics M = � ω, + , | , 2 � I = ω , U is any non principle ultrafilter p ( x ) = { (2 k | x ) ∧ ( x � = 0) : k < ω } [ n �→ 1] U ∈ Γ(Π M / U ) [ n �→ 2 n − 1] U ∈ Γ(Π M / U ) [ n �→ 2 n ] U �∈ Γ(Π M / U ) Thus p -adicly valued fields don’t get mapped to substructures
Type omission Averageable Classes Torsion Modules Bad example II: some pathology with standard natural numbers M = � N , N ′ ; + , × , 1; + ′ , × ′ , 1 ′ ; × ∗ � I = ω , U is any non principle ultrafilter p ( x ) = { N 2 ( x ) ∧ (1 + · · · + 1 � = x ) : n < ω } Two copies of N linked by multiplication
Type omission Averageable Classes Torsion Modules Bad example II: some pathology with standard natural numbers M = � N , N ′ ; + , × , 1; + ′ , × ′ , 1 ′ ; × ∗ � I = ω , U is any non principle ultrafilter p ( x ) = { N 2 ( x ) ∧ (1 + · · · + 1 � = x ) : n < ω } Two copies of N linked by multiplication Two failures of � Los’ Theorem ψ ( x ) ≡ ∃ y ∈ N 2 (1 1 × ∗ y = x ) True of each n ∈ N 1 , but is not true of [ n �→ n ] U ∈ Γ(Π M / U ) φ ≡ ∀ x ∈ N 1 ∃ y ∈ N 2 (1 1 × ∗ y = x ) True in M but not in Γ(Π M / U )
Type omission Averageable Classes Torsion Modules Bad example III: Archimedean fields The construction is very fragile and does not respond well to “implicit” type omission Archimedean fields are often defined as ordered fields omitting the type of an infinite element Then, the type of infinitesimal elements and two elements infinitely close to each other are omitted by the field axioms However, using the Γ( F ) construction, we would not get a substructure if Γ is just the type of an infinite element Instead, Γ has to list each type of a nonstandard element around a standard real This example shows that the unary part is crucial and can’t be avoided through simple coding
Type omission Averageable Classes Torsion Modules When Γ( M ) is a structure There’s a straightforward condition on this: For any F ∈ L and m 0 , . . . , m n − 1 ∈ M that omit the types of Γ, there is some k j < β j for each j < α such that M � ¬ φ j k j ( F ( m 0 , . . . , m n − 1 ))
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