Scott Ranks of Models of a Theory Matthew Harrison-Trainor University of California, Berkeley Notre Dame, September 2015 Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 1 / 38
Overview The Scott rank of a countable structure is a measure of the complexity of describing that structure. 2 theory have a model of Scott rank ≤ α ? Must a Π in Answer: No, it may have only models of high Scott rank. What are the possible Scott spectra of theories? Answer: Certain Σ 1 1 classes of ordinals. Can every computable structure of high Scott rank be approximated by structures of lower Scott rank? Answer: No, there is a computable structure of high Scott rank which cannot be approximated. What is the Scott height of L ω 1 ω ? Answer: δ 1 2 . We will answer these questions as applications of a general construction. Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 2 / 38
Countable structure theory All of our languages and structures will be countable. Some of the results are about computable structures. A structure is computable if its domain is ω and its atomic diagram is computable. Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 3 / 38
Infinitary logic L ω 1 ω is the infinitary logic which allows countable conjunctions and disjunctions. By a “theory” we mean a sentence of L ω 1 ω . A formula is Σ in α if it has α -many alternations of quantifiers and begins with a disjunction / existential quantifier. A formula is Π in α if it has α -many alternations of quantifiers and begins with a conjunction / universal quantifier. Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 4 / 38
Infinitary logic L ω 1 ω is the infinitary logic which allows countable conjunctions and disjunctions. By a “theory” we mean a sentence of L ω 1 ω . A formula is Σ in α if it has α -many alternations of quantifiers and begins with a disjunction / existential quantifier. A formula is Π in α if it has α -many alternations of quantifiers and begins with a conjunction / universal quantifier. Example There is a Π in 2 formula which describes the class of torsion groups. It consists of the group axioms together with: (∀ x ) ⩔ nx = 0 . n ∈ N Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 4 / 38
Back-and-forth relations Theorem (Scott) Let A be a countable structure. There is an L ω 1 ω -sentence ϕ , the Scott sentence of A , such that B ⊧ ϕ if and only if B ≅ A . Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 5 / 38
Back-and-forth relations Theorem (Scott) Let A be a countable structure. There is an L ω 1 ω -sentence ϕ , the Scott sentence of A , such that B ⊧ ϕ if and only if B ≅ A . Definition The standard (non-symmetric) back-and-forth relations ≤ α on A , for α < ω 1 , are defined by: a ≤ 0 ¯ ¯ b if for each quantifier-free formula ψ ( ¯ x ) with G¨ odel number less a ) then A ⊧ ψ ( ¯ than the length of ¯ a , if A ⊧ ψ ( ¯ b ) . a ≤ α ¯ b if for each β < α and ¯ For α > 0, ¯ d there is ¯ c such that b ¯ ¯ d ≤ β ¯ a ¯ c . a ≡ α ¯ a ≤ α ¯ b and ¯ Let ¯ b if ¯ b ≤ α ¯ a , a ≡ α ¯ a and ¯ ¯ b if and only if ¯ b satisfy the same Σ α formulas. Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 5 / 38
Scott rank, version 1 Let A be a structure. Definition (Scott rank, version 1) a ≡ α ¯ a and ¯ SR ( ¯ a ) is the least α such that: if ¯ b , then ¯ b are in the same automorphism orbit of A . Then SR ( A ) = sup ( SR ( ¯ a ∈ A ) . a ) + 1 ∶ ¯ Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 6 / 38
Scott rank, version 2 Theorem (Montalb´ an) Let A be a countable structure, and α a countable ordinal. The following are equivalent: A has a Π in α + 1 Scott sentence. Every automorphism orbit in A is Σ in α -definable without parameters. A is uniformly (boldface) ∆ 0 α -categorical without parameters. α type realized in A is implied by a Σ in Every Π in α formula. No tuple in A is α -free. Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 7 / 38
Scott rank, version 2 Theorem (Montalb´ an) Let A be a countable structure, and α a countable ordinal. The following are equivalent: A has a Π in α + 1 Scott sentence. Every automorphism orbit in A is Σ in α -definable without parameters. A is uniformly (boldface) ∆ 0 α -categorical without parameters. α type realized in A is implied by a Σ in Every Π in α formula. No tuple in A is α -free. Definition (Scott rank, version 2) SR ( A ) is the least ordinal α such that A has a Π in α + 1 Scott sentence. Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 7 / 38
Scott spectra Let T be an L ω 1 ω -sentence. Definition The Scott spectrum of T is the set SS ( T ) = { α ∈ ω 1 ∶ α is the Scott rank of a countable model of T } . Main Question What do we know about SS ( T ) ? Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 8 / 38
Simple theories with no simple models Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 9 / 38
Simple theories with no simple models Question (Montalb´ an) If T is a Π in 2 sentence, must T have a model of Scott rank 1? Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 10 / 38
Simple theories with no simple models Question (Montalb´ an) If T is a Π in 2 sentence, must T have a model of Scott rank 1? Theorem Fix α < ω 1 . There is a Π in 2 sentence T whose models all have Scott rank α . The construction for this theorem contains many of the ideas required for our other results. Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 10 / 38
The most basic construction Fix α an ordinal. The models of T will be ranked trees. The root node has rank α . If a node has rank β , then it has infinitely many children of rank γ for each γ < β . Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 11 / 38
The most basic construction Fix α an ordinal. The models of T will be ranked trees. The root node has rank α . If a node has rank β , then it has infinitely many children of rank γ for each γ < β . We have relations ( ≡ β ) β < α on pairs of elements at the same level of the tree and with the same rank. Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 11 / 38
The most basic construction Fix α an ordinal. The models of T will be ranked trees. The root node has rank α . If a node has rank β , then it has infinitely many children of rank γ for each γ < β . We have relations ( ≡ β ) β < α on pairs of elements at the same level of the tree and with the same rank. T says that x ≡ β y are back-and-forth relations, i.e., x ≡ 0 y if and only if x and y satisfy the same unary atomic relations A i . for β > 0, x ≡ β y if and only if ▸ for all children x ′ of x and γ < β , there is a child y ′ of y with x ′ ≡ γ y ′ . ▸ for all children y ′ of y and γ < β , there is a child x ′ of x with x ′ ≡ γ y ′ . Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 11 / 38
The most basic construction Fix α an ordinal. The models of T will be ranked trees. The root node has rank α . If a node has rank β , then it has infinitely many children of rank γ for each γ < β . We have relations ( ≡ β ) β < α on pairs of elements at the same level of the tree and with the same rank. T says that x ≡ β y are back-and-forth relations, i.e., x ≡ 0 y if and only if x and y satisfy the same unary atomic relations A i . for β > 0, x ≡ β y if and only if ▸ for all children x ′ of x and γ < β , there is a child y ′ of y with x ′ ≡ γ y ′ . ▸ for all children y ′ of y and γ < β , there is a child x ′ of x with x ′ ≡ γ y ′ . We make sure that x and y are in the same orbit if and only if they are at the same level in the tree, have the same tree rank β < α , and x ≡ β y . Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 11 / 38
The most basic construction Fix α an ordinal. The models of T will be ranked trees. The root node has rank α . If a node has rank β , then it has infinitely many children of rank γ for each γ < β . We have relations ( ≡ β ) β < α on pairs of elements at the same level of the tree and with the same rank. T says that x ≡ β y are back-and-forth relations, i.e., x ≡ 0 y if and only if x and y satisfy the same unary atomic relations A i . for β > 0, x ≡ β y if and only if ▸ for all children x ′ of x and γ < β , there is a child y ′ of y with x ′ ≡ γ y ′ . ▸ for all children y ′ of y and γ < β , there is a child x ′ of x with x ′ ≡ γ y ′ . We make sure that x and y are in the same orbit if and only if they are at the same level in the tree, have the same tree rank β < α , and x ≡ β y . For each x , SR ( x ) is the tree rank of x . So if A ⊧ T , SR ( A ) = α . Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 11 / 38
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