Many-Sorted First-Order Model Theory Lecture 11 23 rd July, 2020 1 / 22
Fra¨ ıss´ e’s limits 2 / 22
Fra¨ ıss´ e Theorem Theorem 1 (Fra¨ ıss´ e) Let Σ be an at most countable signature and let K be a nonempty at most countable set of finitely generated Σ -structures which has HP, JEP and AP. Then, there is a Σ -structure D ( called Fra¨ ıss´ e limit of K ) , unique up to isomorphism, such that 1. D is at most countable, 2. K = age ( D ) , 3. D is homogeneous. Where we are in the proof ◮ We have shown that such a D is unique – if it exists. ◮ On the way we have shown that homogeneity is equivalent to an ostensibly weaker notion: weak homogeneity. 3 / 22
Existence of Fra¨ ıss´ e limit Existence proof. ◮ Assume K is nonempty, has HP, JEP and AP, and (wlog) is closed under taking isomorphic copies. ◮ Assume also that K contains only countably many isomorphism types of structures (an assumption automatically satisfied if K is itself at most countable). ◮ Suppose we have constructed a chain ( D i : i ∈ ω ) of structures from K such that the following holds: ( ⋆ ) If A , B ∈ K and A ≤ B , and there is an embedding f : A ֒ → D i for some i ∈ ω , then there is an embedding g : B ֒ → D j extending f . ◮ Pictorially, ( ⋆ ) is: D j B D i A ◮ Put D = � i <ω D i . Since age ( D i ) ⊆ K for every i < ω , by HP we have age ( D ) ⊆ K . ◮ Moreover, if A ∈ K , then by JEP, there is a B ∈ K such that both A and D 0 are embeddable in B . ◮ As K is closed under taking isomorphic copies, we can assume A ≤ B . Then, by ( ⋆ ) we have an embedding of B into some D j . ◮ Then, we have A ≤ B ∈ age ( D j ) ⊆ age ( D ) and so A ∈ age ( D ). Therefore age ( D ) = K . 4 / 22
Existence proof Existence proof. ◮ Thus, we have shown that D satisfies (1) and (2) of Fra¨ ıss´ e Theorem. ◮ For (3) note that D is weakly homogeneous by construction. ◮ But weak homogeneity is the same as homogeneity. ◮ All that remains now is to construct the chain ( D i : i ∈ ω ). Constructing the chain. ◮ Let P be a countable set containing a representative of each isomorphism type of a pair ( A , B ) such that A ≤ B ∈ K . Fix some enumeration of ω × ω , for example 0 1 2 3 . 0 0 1 4 9 . 1 3 2 5 10 . 2 8 7 6 11 . 3 15 14 13 12 . . . . . . . ◮ Pick D 0 from K arbitrarily. ◮ Enumerate all pairs ( A , B ) from P , with A embeddable in D 0 , as ( f i 0 , A i 0 , B i 0 ), where f i 0 : A i 0 ֒ → D 0 is the embedding (this fills the first column of the table above). 5 / 22
Existence proof Existence proof. ◮ Thus, we have shown that D satisfies (1) and (2) of Fra¨ ıss´ e Theorem. ◮ For (3) note that D is weakly homogeneous by construction. ◮ But weak homogeneity is the same as homogeneity. ◮ All that remains now is to construct the chain ( D i : i ∈ ω ). Constructing the chain. ◮ Let P be a countable set containing a representative of each isomorphism type of a pair ( A , B ) such that A ≤ B ∈ K . Fix some enumeration of ω × ω , for example 0 1 2 3 . 0 0 1 4 9 . 1 3 2 5 10 . 2 8 7 6 11 . 3 15 14 13 12 . . . . . . . ◮ Pick D 0 from K arbitrarily. ◮ Enumerate all pairs ( A , B ) from P , with A embeddable in D 0 , as ( f i 0 , A i 0 , B i 0 ), where f i 0 : A i 0 ֒ → D 0 is the embedding (this fills the first column of the table above). 6 / 22
Existence proof Existence proof. ◮ Thus, we have shown that D satisfies (1) and (2) of Fra¨ ıss´ e Theorem. ◮ For (3) note that D is weakly homogeneous by construction. ◮ But weak homogeneity is the same as homogeneity. ◮ All that remains now is to construct the chain ( D i : i ∈ ω ). Constructing the chain. ◮ Let P be a countable set containing a representative of each isomorphism type of a pair ( A , B ) such that A ≤ B ∈ K . Fix some enumeration of ω × ω , for example 0 1 2 3 . 0 0 1 4 9 . 1 3 2 5 10 . 2 8 7 6 11 . 3 15 14 13 12 . . . . . . . ◮ Pick D 0 from K arbitrarily. ◮ Enumerate all pairs ( A , B ) from P , with A embeddable in D 0 , as ( f i 0 , A i 0 , B i 0 ), where f i 0 : A i 0 ֒ → D 0 is the embedding (this fills the first column of the table above). 7 / 22
Existence proof: chain construction Constructing the chain. ◮ Consider f 00 : A 00 ֒ → D 0 . By AP, we can pick a D 1 ∈ K as below D 0 A 00 D 1 B 00 ◮ Now that D 1 has been picked, we enumerate as ( f i 1 , A i 1 , B i 1 ) all pairs ( A , B ) from P , with A embeddable in D 1 (second column of the table). ◮ The first entry that has not yet been considered is ( f 01 , A 01 , B 01 ) with f 01 : A 01 ֒ → D 1 . Use AP again to obtain D 2 . ◮ Continuing inductively we arrive at the following picture D 0 A 00 D 1 B 00 D 2 A 01 D 3 B 01 A 11 D 4 A 10 B 11 B 10 8 / 22
Existence proof: chain construction D 0 A 00 D 1 B 00 D 2 A 01 D 3 B 01 A 11 D 4 A 10 B 11 B 10 Now observe: ◮ All pairs ( A , B ) with A , B ∈ K eventually appear in the enumeration. ◮ By construction of the chain ( D i : i ∈ ω ), we can strengthen the previous statement to: all pairs ( A , B ) with A , B ∈ K and with A embeddable in some D i appear in the enumeration. ◮ By construction of the chain again, for each such pair ( A , B ) and embedding f : A ֒ → D i , we can extend f to an embedding g : B ֒ → D j , for some D j appearing “later” than D i . ◮ Thus, ( D i : i ∈ ω ) indeed satisfies ( ⋆ ) as we needed. This completes the proof. � 9 / 22
Fra¨ ıss´ e’s limits of uniformly locally finite classes are ω -categorical 10 / 22
A characterisation of ω -categoricity Theorem 2 (Engeler, Ryll-Nardzewski, Svenonius) Let T a complete theory in a countable signature Σ . Assume T has infinite models. Then the following are equivalent: 1. T is ω -categorical. 2. For each tuple x of variables, there are only finitely many formulas ϕ ( x ) which are pairwise non-equivalent over T. Proof sketch. ◮ Assume (2) holds. Let A and B be countable models of T . We will show the following. ◮ For any n -tuples a from A and b from B such that ( A , a ) ≡ ( B , b ), we have: ( ⋆ ) for every c from A there exists d from B (and vice versa) such that ( A , a , c ) ≡ ( B , b , d ). ◮ Let a and b be n -tuples such that ( A , a ) ≡ ( B , b ). ◮ Consider all formulas ϕ ( x 1 , . . . , x n , x n +1 ). Only finitely many of them are non-equivalent over T . Pick some representatives of equivalence classes, and list them as θ 1 , . . . , θ m . 11 / 22
A characterisation of ω -categoricity Theorem 2 (Engeler, Ryll-Nardzewski, Svenonius) Let T a complete theory in a countable signature Σ . Assume T has infinite models. Then the following are equivalent: 1. T is ω -categorical. 2. For each tuple x of variables, there are only finitely many formulas ϕ ( x ) which are pairwise non-equivalent over T. Proof sketch. ◮ Assume (2) holds. Let A and B be countable models of T . We will show the following. ◮ For any n -tuples a from A and b from B such that ( A , a ) ≡ ( B , b ), we have: ( ⋆ ) for every c from A there exists d from B (and vice versa) such that ( A , a , c ) ≡ ( B , b , d ). ◮ Let a and b be n -tuples such that ( A , a ) ≡ ( B , b ). ◮ Consider all formulas ϕ ( x 1 , . . . , x n , x n +1 ). Only finitely many of them are non-equivalent over T . Pick some representatives of equivalence classes, and list them as θ 1 , . . . , θ m . 12 / 22
A characterisation of ω -categoricity Proof sketch. ◮ Pick an arbitrary c from A . For each i we have either A | = θ i ( a , c ), or A | = ¬ θ i ( a , c ). Choose θ i or ¬ θ i accordingly, and let their conjunction be Θ. Then A | = Θ( a , c ). ◮ Thus, ( A , a ) | = ∃ y · Θ( a , y ). By assumption ( B , b ) | = ∃ y · Θ( b , y ). ◮ So there exists d in B such that B | = Θ( b , d ). ◮ By construction of Θ, this implies ( A , a , c ) ≡ ( B , b , d ), proving the claim. ◮ Now we can show that ∃ has a winning strategy in the game EF ω ( A , B ). ◮ For the 0th round, we have A ≡ B because T is complete. ◮ Then, at each round ∃ has a response to ∀ by ( ⋆ ). ◮ This shows that A ∼ ω B . Hence, A ∼ = B as they are both countable. ◮ For converse, assume (1) holds, but (2) fails. ◮ From now on it is really a rough sketch: ideas only. ◮ So, for some n there are infinitely many non-equivalent n -variable formulas. For a model C and an n -tuple c let Ψ( c ) be the set of all the formulas that are true in C on c . ◮ Such a Ψ, as a set of formulas, is called an n -type. ◮ By Omitting Types Theorem ( which we omitted! ) there is a model D of T which omits Ψ, that is, such that for every d from D at least one formula from Ψ fails on d . = Ψ( d ) for every d , so C �∼ ◮ Thus, C | = Ψ( c ), but D �| = D . ◮ By L¨ owenheim-Skolem, C and D can be taken countable, contradicting (1). 13 / 22
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