Many-Sorted First-Order Model Theory Lecture 11 23 rd July, 2020 1 - - PowerPoint PPT Presentation

Many-Sorted First-Order Model Theory

Lecture 11 23rd July, 2020

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Fra¨ ıss´ e’s limits

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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

• stensibly weaker notion: weak homogeneity.

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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 (Di : 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 ֒ → Di for some i ∈ ω, then there is an embedding g : B ֒ → Dj extending f . ◮ Pictorially, (⋆) is: A B Di Dj ◮ Put D =

i<ω Di. Since age(Di) ⊆ 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 D0 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 Dj. ◮ Then, we have A ≤ B ∈ age(Dj) ⊆ age(D) and so A ∈ age(D). Therefore age(D) = K.

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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 (Di : 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

1 2 3 . 1 4 9 . 1 3 2 5 10 . 2 8 7 6 11 . 3 15 14 13 12 . . . . . . .

◮ Pick D0 from K arbitrarily. ◮ Enumerate all pairs (A, B) from P, with A embeddable in D0, as (fi0, Ai0, Bi0), where fi0 : Ai0 ֒ → D0 is the embedding (this fills the first column of the table above).

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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 (Di : 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

1 2 3 . 1 4 9 . 1 3 2 5 10 . 2 8 7 6 11 . 3 15 14 13 12 . . . . . . .

◮ Pick D0 from K arbitrarily. ◮ Enumerate all pairs (A, B) from P, with A embeddable in D0, as (fi0, Ai0, Bi0), where fi0 : Ai0 ֒ → D0 is the embedding (this fills the first column of the table above).

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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 (Di : 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

1 2 3 . 1 4 9 . 1 3 2 5 10 . 2 8 7 6 11 . 3 15 14 13 12 . . . . . . .

◮ Pick D0 from K arbitrarily. ◮ Enumerate all pairs (A, B) from P, with A embeddable in D0, as (fi0, Ai0, Bi0), where fi0 : Ai0 ֒ → D0 is the embedding (this fills the first column of the table above).

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Existence proof: chain construction

Constructing the chain.

◮ Consider f00 : A00 ֒ → D0. By AP, we can pick a D1 ∈ K as below A00 D0 B00 D1 ◮ Now that D1 has been picked, we enumerate as (fi1, Ai1, Bi1) all pairs (A, B) from P, with A embeddable in D1 (second column of the table). ◮ The first entry that has not yet been considered is (f01, A01, B01) with f01 : A01 ֒ → D1. Use AP again to obtain D2. ◮ Continuing inductively we arrive at the following picture A00 D0 B00 D1 A01 B01 D2 A11 B11 D3 A10 B10 D4

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Existence proof: chain construction

A00 D0 B00 D1 A01 B01 D2 A11 B11 D3 A10 B10 D4

Now observe:

◮ All pairs (A, B) with A, B ∈ K eventually appear in the enumeration. ◮ By construction of the chain (Di : i ∈ ω), we can strengthen the previous statement to: all pairs (A, B) with A, B ∈ K and with A embeddable in some Di appear in the enumeration. ◮ By construction of the chain again, for each such pair (A, B) and embedding f : A ֒ → Di, we can extend f to an embedding g : B ֒ → Dj, for some Dj appearing “later” than Di. ◮ Thus, (Di : i ∈ ω) indeed satisfies (⋆) as we needed. This completes the proof.

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Fra¨ ıss´ e’s limits of uniformly locally finite classes are ω-categorical

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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(x1, . . . , xn, xn+1). Only finitely many of them are non-equivalent

• ver T. Pick some representatives of equivalence classes, and list them as θ1, . . . , θm.

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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(x1, . . . , xn, xn+1). Only finitely many of them are non-equivalent

• ver T. Pick some representatives of equivalence classes, and list them as θ1, . . . , θm.

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A characterisation of ω-categoricity

Proof sketch.

◮ Pick an arbitrary c from A. For each i we have either A | = θi(a, c), or A | = ¬θi(a, c). Choose θi or ¬θi 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 EFω(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. ◮ Thus, C | = Ψ(c), but D | = Ψ(d) for every d, so C ∼ = D. ◮ By L¨

• wenheim-Skolem, C and D can be taken countable, contradicting (1).

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Decent Fra¨ ıss´ e limits are ω-categorical

Theorem 3

Let Σ be a finite signature, and K a countable, uniformly locally finite set

• f finitely generated Σ-structures with HP, JEP and AP. Let M be the

Fra¨ ıss´ e limit of K. The following hold:

• 1. Th(M) is ω-categorical.
• 2. Th(M) has quantifier elimination.

Proof sketch.

◮ By uniform local finiteness, for each n there are only finitely many (isomorphism types) of n-generated structures in K. All of them are finite, of course. ◮ Since Σ is finite, for each finite Σ-structure A generated by an n-tuple a, the set of unnested formulas in the diagram diag(A, a) is finite. ◮ Let ψA,a(x) be the conjunction of all unnested formulas in diag(A, a) with constants a replaced by variables x. ◮ Then, for every B and every tuple b we have B | = ψA,a(b) iff there is an isomorphism from A to bB which takes a to b.

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Decent Fra¨ ıss´ e limits are ω-categorical

Theorem 3

Let Σ be a finite signature, and K a countable, uniformly locally finite set

• f finitely generated Σ-structures with HP, JEP and AP. Let M be the

Fra¨ ıss´ e limit of K. The following hold:

• 1. Th(M) is ω-categorical.
• 2. Th(M) has quantifier elimination.

Proof sketch.

◮ By uniform local finiteness, for each n there are only finitely many (isomorphism types) of n-generated structures in K. All of them are finite, of course. ◮ Since Σ is finite, for each finite Σ-structure A generated by an n-tuple a, the set of unnested formulas in the diagram diag(A, a) is finite. ◮ Let ψA,a(x) be the conjunction of all unnested formulas in diag(A, a) with constants a replaced by variables x. ◮ Then, for every B and every tuple b we have B | = ψA,a(b) iff there is an isomorphism from A to bB which takes a to b.

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Decent Fra¨ ıss´ e limits are ω-categorical

Proof sketch.

◮ Let U0 be the set of all sentences ∀x ·

• ψA,a(x) → ∃y · ψB,ab(x, y)
• , where B ∈ K is

generated by (a, b) and A ≤ B is generated by a. ◮ We include empty tuples a, in which case the sentence takes the form ∃y · ψB,b(y). ◮ Let U1 be the set of sentences ∀x ·

A,a ψA,a(x), taken over all pairs A, a such that

A ∈ K and a generates A and is of the same length as x. ◮ By uniform local finiteness, this disjunction exists (i.e., is finite up to logical equivalence). ◮ Let U = U0 ∪ U1. Then M | = U. ◮ Let C be a countable model of U. We observe the following facts. ◮ For empty a the sentences in U0 say that every one-generated structure in K is embeddable in C. ◮ For non-empty a, the sentences in U0 say that if A ≤ B are finitely generated substructures of C, and B extends A by adding one more generator, then for any embedding f : A ֒ → C there exists and embedding g : B ֒ → C that extends f . ◮ By induction on the number of generators, this implies that every structure in K is embeddable in C. ◮ Now, the sentences in U1 say that every n-generated substructure of C is one of the As from K. Hence age(C) = K.

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Decent Fra¨ ıss´ e limits are ω-categorical

Proof sketch.

◮ Note that the sentencs in U0 also express a weak version of weak homogeneity: for extensions by a single generator. ◮ By induction on the size of |B| \ |A| we then obtain that C is weakly homogeneous. ◮ Therefore, by Lemma 19 of Lecture 11, C is homogeneous and isomorphic to M. ◮ So U is ω-categorical, and the closure of U under logical consequence is precisely Th(M). ◮ That is, U axiomatises Th(M), and Th(M) is ω-categorical, as claimed. ◮ It remains to prove quantifier elimination. Since Th(M) is complete, sentences are trivially equivalent to quantifier free formulas. ◮ Let ϕ(x) be a formula with free variables x. Let X be the set of all tuples a from M such that M | = ϕ(x). ◮ By homogeneity, any isomorphism i : aM → bM extends to an automorphism of M, so a ∈ X implies b ∈ X. ◮ This means that ϕ(x) is equivalent over Th(M) to the “description” of the structures generated by the tuples in X. ◮ There are, in general, infinitely many tuples in X, but by uniform local finiteness there are

• nly finitely many isomorphism types of structures generated by them, so there are finitely

many formulas ψA,a(x), describing these structures. ◮ Thus, ϕ(x) is equivalent to the (finite) disjunction of all ψA,a(x) over all a ∈ X. ◮ Formulas ψA,a(x) are quantifier free, so the proof is complete.

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Putting two and two together

Theorem 4

Let Σ be a finite signature, and M a countable Σ-structure. Then the following are equivalent:

• 1. M is homogeneous and uniformly locally finite.
• 2. Th(M) is ω-categorical and has quantifier elimination.

Proof sketch.

◮ (1) ⇒ (2) follows from Theorem 3 by taking K = {M}. ◮ For the converse, first note that if in a structure A generated by a we have c = d, then there is a formula ϕ such that A | = ϕ(a, c) and A | = ϕ(a, d). (Intuitively, ϕ says how c is generated, but not how d is generated. Such a formula must exist, as otherwise c and d would not be distinct.) ◮ By Theorem 2, for any n, there are only finitely many such formulas ϕ(x1, . . . , xn) non-equivalent over Th(M). Their number, say #n does not depend on the tuple a. ◮ Then #n gives a uniform bound on the number of distinct elements that can be generated from n generators. Uniform local finiteness follows. ◮ It remains to prove homogeneity.

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Putting two and two together

Theorem 4

Let Σ be a finite signature, and M a countable Σ-structure. Then the following are equivalent:

• 1. M is homogeneous and uniformly locally finite.
• 2. Th(M) is ω-categorical and has quantifier elimination.

Proof sketch.

◮ (1) ⇒ (2) follows from Theorem 3 by taking K = {M}. ◮ For the converse, first note that if in a structure A generated by a we have c = d, then there is a formula ϕ such that A | = ϕ(a, c) and A | = ϕ(a, d). (Intuitively, ϕ says how c is generated, but not how d is generated. Such a formula must exist, as otherwise c and d would not be distinct.) ◮ By Theorem 2, for any n, there are only finitely many such formulas ϕ(x1, . . . , xn) non-equivalent over Th(M). Their number, say #n does not depend on the tuple a. ◮ Then #n gives a uniform bound on the number of distinct elements that can be generated from n generators. Uniform local finiteness follows. ◮ It remains to prove homogeneity.

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Examples of Fra¨ ıss´ e classes

Proof sketch.

◮ To that end, assume aM and bM are isomorphic, via a map f . This means a and b satisfy the same quantifier free formulas. ◮ Since Th(M) has quantifier elimination, a and b satisfy precisely the same formulas. ◮ Using this, we can extend f in a game-like fashion to an automorphism of M.

Examples of Fra¨ ıss´ e classes

◮ Finite chains

◮ Fra¨ ıss´ e limit is (Q, <) (ω-categorical).

◮ Finite graphs

◮ Fra¨ ıss´ e limit is the random graph (ω-categorical).

◮ Finite Boolean algebras

◮ ... limit is the countable atomless Boolean algebra (ω-categorical).

◮ Finite lattices

◮ Fra¨ ıss´ e limit is a countable lattice into which every finite lattice embeds, but not a free lattice (not ω-categorical).

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Examples of Fra¨ ıss´ e classes

Proof sketch.

◮ To that end, assume aM and bM are isomorphic, via a map f . This means a and b satisfy the same quantifier free formulas. ◮ Since Th(M) has quantifier elimination, a and b satisfy precisely the same formulas. ◮ Using this, we can extend f in a game-like fashion to an automorphism of M.

Examples of Fra¨ ıss´ e classes

◮ Finite chains

◮ Fra¨ ıss´ e limit is (Q, <) (ω-categorical).

◮ Finite graphs

◮ Fra¨ ıss´ e limit is the random graph (ω-categorical).

◮ Finite Boolean algebras

◮ ... limit is the countable atomless Boolean algebra (ω-categorical).

◮ Finite lattices

◮ Fra¨ ıss´ e limit is a countable lattice into which every finite lattice embeds, but not a free lattice (not ω-categorical).

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Thank you!

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