Mathematical Logics 15. Elements of Model theory Luciano Serafini Fondazione Bruno Kessler, Trento, Italy November 19, 2014 Luciano Serafini Mathematical Logics
What is model theory? First-order model theory. . . , . . . also known as classical model theory, is a branch of mathematics that deals with the relationships between descriptions (i.e., formulas and terms) in first-order languages and the structures that satisfy these descriptions [Stanford Encyclopedia of Philosophy] Model theory in our course Compactness theorem If an infinite set of fomrulas Γ is satisfiable iff every finite subset Σ ⊂ Γ is satisfiable Countable model theorem: A set of first-order formulas has a model if and only if it has a countable model. Herbrand’s theorem: A set of universally quantified formulas is unsatisfiable iff there is a finite grounding of them which is unsatisfiable. Luciano Serafini Mathematical Logics
What is model theory? First-order model theory. . . , . . . also known as classical model theory, is a branch of mathematics that deals with the relationships between descriptions (i.e., formulas and terms) in first-order languages and the structures that satisfy these descriptions [Stanford Encyclopedia of Philosophy] Model theory in our course Compactness theorem If an infinite set of fomrulas Γ is satisfiable iff every finite subset Σ ⊂ Γ is satisfiable Countable model theorem: A set of first-order formulas has a model if and only if it has a countable model. Herbrand’s theorem: A set of universally quantified formulas is unsatisfiable iff there is a finite grounding of them which is unsatisfiable. Luciano Serafini Mathematical Logics
Compactness Theorem Theorem An infinite set of fomrulas Γ is satisfiable iff every finite subset Σ ⊂ Γ is satisfiable. Proof. ⇒ If Γ is satisfiable, then there is an interpretation I , such that I | = γ for all γ ∈ Γ. Since Σ ⊂ Γ, then I | = σ for all σ ∈ Σ and therefore Σ is satisfiable ⇐ Suppose by contradiction that Γ is not satisfiable, then Γ | = ⊥ . By soundness we have that Γ ⊢ ⊥ , i.e., there is a deduction Π of ⊥ from Γ. Since deductions are finite, there is a finite subset of formulas Σ ⊂ Γ that appear in Π. This implies that Π is a deduction of ⊥ from Σ. By soundness, we have that Σ | = ⊥ , i.e., that Σ ⊂ Γ and finite is inconsistent. But this is a contradiction. Luciano Serafini Mathematical Logics
Compactness Theorem Theorem An infinite set of fomrulas Γ is satisfiable iff every finite subset Σ ⊂ Γ is satisfiable. Proof. ⇒ If Γ is satisfiable, then there is an interpretation I , such that I | = γ for all γ ∈ Γ. Since Σ ⊂ Γ, then I | = σ for all σ ∈ Σ and therefore Σ is satisfiable ⇐ Suppose by contradiction that Γ is not satisfiable, then Γ | = ⊥ . By soundness we have that Γ ⊢ ⊥ , i.e., there is a deduction Π of ⊥ from Γ. Since deductions are finite, there is a finite subset of formulas Σ ⊂ Γ that appear in Π. This implies that Π is a deduction of ⊥ from Σ. By soundness, we have that Σ | = ⊥ , i.e., that Σ ⊂ Γ and finite is inconsistent. But this is a contradiction. Luciano Serafini Mathematical Logics
Σ -structure A first order interpretation of the language that contains the signature Σ = { c 1 , c 2 , . . . , f 1 , f 2 . . . , R 1 , R 2 , . . . } is called a Σ-structure, to stress the fact that it is relative to a specific vocabulary. Σ -structure Given a vocabulary/signature Σ = � c 1 , c 2 , . . . , f 1 , f 2 , . . . , R 1 , R 2 , . . . � a Σ-structure is I is composed of a non empty set ∆ I and an interpretation function such that c I i ∈ ∆ I ∈ (∆ I ) arity ( f i ) − f I → ∆ I : The set of functions from n -tuples i of elements of ∆ I to ∆ I with n − arity ( f i ) i ∈ (∆ I ) arity ( R i ) the set of n -tuples of elements of ∆ I with R I n = arity ( R i ). Luciano Serafini Mathematical Logics
Substructures Substructure A Σ-structure I is a substructure of a Σ-structure J , in symbols I ⊆ J if ∆ I ⊆ ∆ J c I = c J f I is the restriction of f J to the set ∆ I , i.e., for all a 1 , . . . , a n ∈ ∆ I , f I ( a 1 , . . . , a n ) = f J ( a 1 , . . . , a n ). R I = R J ∩ (∆ I ) n where n is the arity of f and R . Example (Substructure (syntax, semantics) ) Let Σ = � zero , one , plus ( · , · ) , positive ( · ) , negative ( · ) � � ∆ I , · I � � ∆ I , · I � I = J = ∆ I = { 0 , 1 , 2 , 3 , . . . } ∆ J = { . . . , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , . . . } zero I = 0 , one I = 1 zero J = 0 , one I = 1 plus J ( x , y ) = x + y plus I ( x , y ) = x + y positive I = { 1 , 2 , . . . } positive J = { 1 , 2 , . . . } negative I = ∅ negative J = {− 1 , − 2 , . . . } Luciano Serafini Mathematical Logics
Proposition If I ⊆ J then for every ground formula φ , I | = φ iff J | = φ Proof. A ground formula is a formula that does not contain individual variables and quantifiers. So φ is ground if it is a boolean combination of atomic formulas of the form P ( t 1 , . . . , t n ) with t i ’s ground terms, i.e., terms that do not contain variables. If t is a ground term then t I = t J (proof by induction on the construction of t ) if t is the constant c , then by definition c I = c J if t is f ( t 1 , . . . , t n ), then t is ground implies that each t i is ground. By ∈ ∆ I ⊆ ∆ J . Since the definitions of f I and f J = t J induction t I i i coincide on the elements of ∆ I ∩ ∆ J , we have that f I ( t I 1 , . . . , t I n ) = f I ( t I 1 , . . . , t I n ) and therefore ( f ( t 1 , . . . , t n )) I = ( f ( t 1 , . . . , t n )) J if φ is P ( t 1 , . . . , t n ) with t i ’s ground terms, then, by induction we have that ∈ ∆ I ⊆ ∆ J for 1 ≤ i ≤ n . The fact that P I = P J ∩ (∆ I ) n implies t I = t J i i that I | = P ( t 1 , . . . , t n ) iff J | = P ( t 1 , . . . , t n ) the fact that I and J agree on all the atomic ground formulas implies that they agree also on all the boolean combinations of the ground formulas. Luciano Serafini Mathematical Logics
Proposition If I ⊆ J then for every ground formula φ , I | = φ iff J | = φ Proof. A ground formula is a formula that does not contain individual variables and quantifiers. So φ is ground if it is a boolean combination of atomic formulas of the form P ( t 1 , . . . , t n ) with t i ’s ground terms, i.e., terms that do not contain variables. If t is a ground term then t I = t J (proof by induction on the construction of t ) if t is the constant c , then by definition c I = c J if t is f ( t 1 , . . . , t n ), then t is ground implies that each t i is ground. By ∈ ∆ I ⊆ ∆ J . Since the definitions of f I and f J = t J induction t I i i coincide on the elements of ∆ I ∩ ∆ J , we have that f I ( t I 1 , . . . , t I n ) = f I ( t I 1 , . . . , t I n ) and therefore ( f ( t 1 , . . . , t n )) I = ( f ( t 1 , . . . , t n )) J if φ is P ( t 1 , . . . , t n ) with t i ’s ground terms, then, by induction we have that ∈ ∆ I ⊆ ∆ J for 1 ≤ i ≤ n . The fact that P I = P J ∩ (∆ I ) n implies t I = t J i i that I | = P ( t 1 , . . . , t n ) iff J | = P ( t 1 , . . . , t n ) the fact that I and J agree on all the atomic ground formulas implies that they agree also on all the boolean combinations of the ground formulas. Luciano Serafini Mathematical Logics
Proposition If I ⊆ J then for every ground formula φ , I | = φ iff J | = φ Proof. A ground formula is a formula that does not contain individual variables and quantifiers. So φ is ground if it is a boolean combination of atomic formulas of the form P ( t 1 , . . . , t n ) with t i ’s ground terms, i.e., terms that do not contain variables. If t is a ground term then t I = t J (proof by induction on the construction of t ) if t is the constant c , then by definition c I = c J if t is f ( t 1 , . . . , t n ), then t is ground implies that each t i is ground. By ∈ ∆ I ⊆ ∆ J . Since the definitions of f I and f J = t J induction t I i i coincide on the elements of ∆ I ∩ ∆ J , we have that f I ( t I 1 , . . . , t I n ) = f I ( t I 1 , . . . , t I n ) and therefore ( f ( t 1 , . . . , t n )) I = ( f ( t 1 , . . . , t n )) J if φ is P ( t 1 , . . . , t n ) with t i ’s ground terms, then, by induction we have that ∈ ∆ I ⊆ ∆ J for 1 ≤ i ≤ n . The fact that P I = P J ∩ (∆ I ) n implies t I = t J i i that I | = P ( t 1 , . . . , t n ) iff J | = P ( t 1 , . . . , t n ) the fact that I and J agree on all the atomic ground formulas implies that they agree also on all the boolean combinations of the ground formulas. Luciano Serafini Mathematical Logics
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