Syntax Types of formulas Some applications of semi–formal systems Wolfram Pohlers The L formulas are sorted into three categories Abstract Formulas without type semi–formal systems Syntax The verification calculus Semantics Abstract Logical Consequences The Verification Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Syntax Types of formulas Some applications of semi–formal systems Wolfram Pohlers The L formulas are sorted into three categories Abstract Formulas without type semi–formal systems Formulas of � –type Syntax The verification calculus Semantics Abstract Logical Consequences The Verification Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Syntax Types of formulas Some applications of semi–formal systems Wolfram Pohlers The L formulas are sorted into three categories Abstract Formulas without type semi–formal systems Formulas of � –type Syntax The verification calculus Formulas of � –type Semantics Abstract Logical Consequences The Verification Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Syntax Types of formulas Some applications of semi–formal systems Wolfram Pohlers The L formulas are sorted into three categories Abstract Formulas without type semi–formal systems Formulas of � –type Syntax The verification calculus Formulas of � –type Semantics Abstract Logical Consequences satisfying The Verification Calculus for Logical all formulas without type are atomic. Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Syntax Characteristic subformula sequences Some applications of semi–formal systems Each L –formula F possesses a characteristic subformula Wolfram Pohlers sequence CS ( F ) fulfilling (C0) CS ( F c ) = G c � � Abstract G ∈ CS ( F ) semi–formal systems Syntax The verification calculus Semantics Abstract Logical Consequences The Verification Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Syntax Characteristic subformula sequences Some applications of semi–formal systems Each L –formula F possesses a characteristic subformula Wolfram Pohlers sequence CS ( F ) fulfilling (C0) CS ( F c ) = G c � � Abstract G ∈ CS ( F ) semi–formal systems (C1) The complexity of every formula in CS ( F ) is less than the Syntax The verification complexity of F calculus Semantics Abstract Logical Consequences The Verification Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Syntax Characteristic subformula sequences Some applications of semi–formal systems Each L –formula F possesses a characteristic subformula Wolfram Pohlers sequence CS ( F ) fulfilling (C0) CS ( F c ) = G c � � Abstract G ∈ CS ( F ) semi–formal systems (C1) The complexity of every formula in CS ( F ) is less than the Syntax The verification complexity of F calculus Semantics (C2) The formulas in CS ( F ) must not contain free variables Abstract Logical Consequences The Verification which are not also free in F . Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Syntax Characteristic subformula sequences Some applications of semi–formal systems Each L –formula F possesses a characteristic subformula Wolfram Pohlers sequence CS ( F ) fulfilling (C0) CS ( F c ) = G c � � Abstract G ∈ CS ( F ) semi–formal systems (C1) The complexity of every formula in CS ( F ) is less than the Syntax The verification complexity of F calculus Semantics (C2) The formulas in CS ( F ) must not contain free variables Abstract Logical Consequences The Verification which are not also free in F . Calculus for Logical Consequences We call a language L countable if it contains only countably M –Logic many terms and all characteristic subformula sequences are Applications countable. to Logic Applications to Structure Theory Bibliography
Syntax Admissible Fragment Some applications of semi–formal systems Wolfram Pohlers Let A be an admissible structure in the sense of Barwise [1] Abstract semi–formal and L a logical language that is ∆ 1 –definable in A . The systems Syntax admissible fragment L A of L consists of The verification calculus Semantics Abstract Logical Consequences The Verification Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Syntax Admissible Fragment Some applications of semi–formal systems Wolfram Pohlers Let A be an admissible structure in the sense of Barwise [1] Abstract semi–formal and L a logical language that is ∆ 1 –definable in A . The systems Syntax admissible fragment L A of L consists of The verification calculus { t ∈ L t is an L –term and t ∈ A } Semantics Abstract Logical Consequences The Verification Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Syntax Admissible Fragment Some applications of semi–formal systems Wolfram Pohlers Let A be an admissible structure in the sense of Barwise [1] Abstract semi–formal and L a logical language that is ∆ 1 –definable in A . The systems Syntax admissible fragment L A of L consists of The verification calculus { t ∈ L t is an L –term and t ∈ A } Semantics Abstract Logical Consequences { F ∈ L F is an L –formula, F ∈ A and CS ( F ) ∈ A } The Verification Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Syntax The Verification Calculus Some applications of semi–formal α We define a verification calculus L ∆ for finite sets ∆ of systems L –formulas which should be viewed as finite disjunctions by the Wolfram Pohlers following clauses Abstract semi–formal systems Syntax The verification calculus Semantics Abstract Logical Consequences The Verification Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Syntax The Verification Calculus Some applications of semi–formal α We define a verification calculus L ∆ for finite sets ∆ of systems L –formulas which should be viewed as finite disjunctions by the Wolfram Pohlers following clauses Abstract semi–formal (Ax) If ∆ is a finite set of L –formulas and F is a formula systems α without type such that { F , F c } ⊆ ∆ then L ∆ holds true Syntax The verification calculus for all ordinals α . Semantics Abstract Logical Consequences The Verification Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Syntax The Verification Calculus Some applications of semi–formal α We define a verification calculus L ∆ for finite sets ∆ of systems L –formulas which should be viewed as finite disjunctions by the Wolfram Pohlers following clauses Abstract semi–formal (Ax) If ∆ is a finite set of L –formulas and F is a formula systems α without type such that { F , F c } ⊆ ∆ then L ∆ holds true Syntax The verification calculus for all ordinals α . Semantics α G Abstract Logical ( � ) If F ∈ � –type ∩ ∆ and ∆ , G as well as α G < α holds Consequences The Verification L Calculus for α Logical true for all G ∈ CS ( F ) then L ∆. Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Syntax The Verification Calculus Some applications of semi–formal α We define a verification calculus L ∆ for finite sets ∆ of systems L –formulas which should be viewed as finite disjunctions by the Wolfram Pohlers following clauses Abstract semi–formal (Ax) If ∆ is a finite set of L –formulas and F is a formula systems α without type such that { F , F c } ⊆ ∆ then L ∆ holds true Syntax The verification calculus for all ordinals α . Semantics α G Abstract Logical ( � ) If F ∈ � –type ∩ ∆ and ∆ , G as well as α G < α holds Consequences The Verification L Calculus for α Logical true for all G ∈ CS ( F ) then L ∆. Consequences M –Logic α 0 ( � ) If F ∈ � –type ∩ ∆ and ∆ , Γ as well as α 0 < α holds L Applications α to Logic true for some finite set Γ ⊆ CS ( F ) then L ∆. Applications to Structure Theory Bibliography
Syntax First observations Some Lemma applications of semi–formal α β L ∆ , α ≤ β and ∆ ⊆ Γ imply L Γ . systems Wolfram Pohlers Abstract semi–formal systems Syntax The verification calculus Semantics Abstract Logical Consequences The Verification Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Syntax First observations Some Lemma applications of semi–formal α β L ∆ , α ≤ β and ∆ ⊆ Γ imply L Γ . systems Wolfram Pohlers Lemma ( � –inversion) Abstract α α semi–formal F ∈ � –type and L ∆ , G for all G ∈ CS ( F ) L ∆ , F imply systems Syntax The verification calculus Semantics Abstract Logical Consequences The Verification Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Syntax First observations Some Lemma applications of semi–formal α β L ∆ , α ≤ β and ∆ ⊆ Γ imply L Γ . systems Wolfram Pohlers Lemma ( � –inversion) Abstract α α semi–formal F ∈ � –type and L ∆ , G for all G ∈ CS ( F ) L ∆ , F imply systems Syntax The verification calculus Lemma ( � –inversion) Semantics Abstract Logical Consequences α If ∈ � –type and CS ( F ) is finite then L ∆ , F implies The Verification Calculus for Logical α Consequences L ∆ , CS ( F ) . M –Logic Applications to Logic Applications to Structure Theory Bibliography
Syntax First observations Some Lemma applications of semi–formal α β L ∆ , α ≤ β and ∆ ⊆ Γ imply L Γ . systems Wolfram Pohlers Lemma ( � –inversion) Abstract α α semi–formal F ∈ � –type and L ∆ , G for all G ∈ CS ( F ) L ∆ , F imply systems Syntax The verification calculus Lemma ( � –inversion) Semantics Abstract Logical Consequences α If ∈ � –type and CS ( F ) is finite then L ∆ , F implies The Verification Calculus for Logical α Consequences L ∆ , CS ( F ) . M –Logic Applications Lemma to Logic Applications Let rnk ( F ) denote the complexity of F. Then to Structure 2 · rnk ( F ) Theory ∆ , F , F c holds true for all finite sets ∆ of L –formulas. Bibliography L
Semantics L –Structures and Assignments Some applications of semi–formal systems Let L be a logical language. As usual an L -Structure M Wolfram Pohlers consists of a non empty set | M | , the domain of M together Abstract with the interpretations of all the non–logical symbols of L with semi–formal the requirement that ( P c ) M for a predicate constant P has to systems Syntax be the complement of P M . The verification calculus Semantics Abstract Logical Consequences The Verification Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Semantics L –Structures and Assignments Some applications of semi–formal systems Let L be a logical language. As usual an L -Structure M Wolfram Pohlers consists of a non empty set | M | , the domain of M together Abstract with the interpretations of all the non–logical symbols of L with semi–formal the requirement that ( P c ) M for a predicate constant P has to systems Syntax be the complement of P M . The verification calculus Semantics An M –assignment Φ is a map that assigns an element Abstract Logical Consequences Φ( x ) ∈ | M | to every individual variable x and a set The Verification Calculus for Φ( X ) ⊆ | M | n to every n –ary predicate variable X . Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Semantics L –Structures and Assignments Some applications of semi–formal systems Let L be a logical language. As usual an L -Structure M Wolfram Pohlers consists of a non empty set | M | , the domain of M together Abstract with the interpretations of all the non–logical symbols of L with semi–formal the requirement that ( P c ) M for a predicate constant P has to systems Syntax be the complement of P M . The verification calculus Semantics An M –assignment Φ is a map that assigns an element Abstract Logical Consequences Φ( x ) ∈ | M | to every individual variable x and a set The Verification Calculus for Φ( X ) ⊆ | M | n to every n –ary predicate variable X . Logical Consequences Clearly we again require that Φ( X c ) is the complement of M –Logic Φ( X ). Applications to Logic Applications to Structure Theory Bibliography
Semantics Evaluation Some applications of semi–formal systems Wolfram As usual we define the evaluation of a formula in an structure Pohlers M under an assignment Φ. Abstract semi–formal systems Syntax The verification calculus Semantics Abstract Logical Consequences The Verification Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Semantics Evaluation Some applications of semi–formal systems Wolfram As usual we define the evaluation of a formula in an structure Pohlers M under an assignment Φ. Abstract semi–formal M | = F [Φ] for formulas F that do not belong to a type is systems Syntax given by the interpretation and the assignment, The verification calculus Semantics Abstract Logical Consequences The Verification Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Semantics Evaluation Some applications of semi–formal systems Wolfram As usual we define the evaluation of a formula in an structure Pohlers M under an assignment Φ. Abstract semi–formal M | = F [Φ] for formulas F that do not belong to a type is systems Syntax given by the interpretation and the assignment, The verification calculus M | = F [Φ] if M | = G [Φ] for all G ∈ CS ( F ) for formulas F Semantics Abstract Logical Consequences in � –type The Verification Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Semantics Evaluation Some applications of semi–formal systems Wolfram As usual we define the evaluation of a formula in an structure Pohlers M under an assignment Φ. Abstract semi–formal M | = F [Φ] for formulas F that do not belong to a type is systems Syntax given by the interpretation and the assignment, The verification calculus M | = F [Φ] if M | = G [Φ] for all G ∈ CS ( F ) for formulas F Semantics Abstract Logical Consequences in � –type and The Verification Calculus for Logical M | = F [Φ] if M | = G [Φ] for some G ∈ CS ( F ) for formulas Consequences F in � –type. M –Logic Applications to Logic Applications to Structure Theory Bibliography
Semantics Correctness and completeness for sentences Some Theorem ( L –Correctness) applications of semi–formal systems Let L be a logical language, M an L -structure and ∆ a finite α Wolfram L ∆ implies M | = � ∆[Φ] for all set of L –formulas. Then Pohlers M –assignments Φ . Abstract semi–formal systems Syntax The verification calculus Semantics Abstract Logical Consequences The Verification Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Semantics Correctness and completeness for sentences Some Theorem ( L –Correctness) applications of semi–formal systems Let L be a logical language, M an L -structure and ∆ a finite α Wolfram L ∆ implies M | = � ∆[Φ] for all set of L –formulas. Then Pohlers M –assignments Φ . Abstract semi–formal systems Theorem ( L –Correctness for Π 1 1 –sentences) Syntax The verification calculus Let L be a logical language, M an L -structure and F an Semantics Abstract Logical L –formula the free predicate variables of which belong all to the Consequences α The Verification L F implies M | = ( ∀ X 1 ) . . . ( ∀ X n ) F. list X 1 , . . . , X n .. Then Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Semantics Correctness and completeness for sentences Some Theorem ( L –Correctness) applications of semi–formal systems Let L be a logical language, M an L -structure and ∆ a finite α Wolfram L ∆ implies M | = � ∆[Φ] for all set of L –formulas. Then Pohlers M –assignments Φ . Abstract semi–formal systems Theorem ( L –Correctness for Π 1 1 –sentences) Syntax The verification calculus Let L be a logical language, M an L -structure and F an Semantics Abstract Logical L –formula the free predicate variables of which belong all to the Consequences α The Verification L F implies M | = ( ∀ X 1 ) . . . ( ∀ X n ) F. list X 1 , . . . , X n .. Then Calculus for Logical Consequences M –Logic Theorem ( L –Completeness for sentences) Applications to Logic Let L be a logical language and M an L -structure. For every Applications L –sentence F that is true in M there is an ordinal α ≤ rnk ( F ) to Structure Theory α such that L F. Bibliography
Abstract Logical Consequences Definition Some applications of semi–formal systems Wolfram Definition ( L -consequence) Pohlers Let L be a logical language and T a set of L –formulas. We say Abstract semi–formal that an L –formula F is an L –consequence of a set T of systems Syntax L –formulas, denoted by The verification calculus Semantics T | = L F Abstract Logical Consequences The Verification Calculus for iff for every L –structure M and every M –assignment Φ such Logical Consequences that M | = T [Φ] (i.e. M | = G [Φ] for all G ∈ T ) we also get M –Logic M | = F [Φ]. Applications to Logic Applications to Structure Theory Bibliography
Logical consequences A verification calculus for logical consequence To extend the verification calculus also to logical consequences Some applications of we keep the old rules semi–formal systems α α Wolfram L ∆ ⇒ T L ∆. Pohlers Abstract semi–formal systems Syntax The verification calculus Semantics Abstract Logical Consequences The Verification Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Logical consequences A verification calculus for logical consequence To extend the verification calculus also to logical consequences Some applications of we keep the old rules semi–formal systems α α Wolfram L ∆ ⇒ T L ∆. Pohlers Abstract and augment the verification calculus by a theory–rule semi–formal α 0 α systems ∆ , F c and α 0 < α ⇒ T , F T L ∆. Syntax L The verification calculus Semantics Abstract Logical Consequences The Verification Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Logical consequences A verification calculus for logical consequence To extend the verification calculus also to logical consequences Some applications of we keep the old rules semi–formal systems α α Wolfram L ∆ ⇒ T L ∆. Pohlers Abstract and augment the verification calculus by a theory–rule semi–formal α 0 α systems ∆ , F c and α 0 < α ⇒ T , F T L ∆. Syntax L The verification calculus Semantics The, somehow weird looking, formulation of the theory rule has Abstract Logical Consequences technical reasons. The rule we would expect is, however, an The Verification Calculus for Logical permissible rule. Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Logical consequences A verification calculus for logical consequence To extend the verification calculus also to logical consequences Some applications of we keep the old rules semi–formal systems α α Wolfram L ∆ ⇒ T L ∆. Pohlers Abstract and augment the verification calculus by a theory–rule semi–formal α 0 α systems ∆ , F c and α 0 < α ⇒ T , F T L ∆. Syntax L The verification calculus Semantics The, somehow weird looking, formulation of the theory rule has Abstract Logical Consequences technical reasons. The rule we would expect is, however, an The Verification Calculus for Logical permissible rule. Consequences We get M –Logic Applications 2 · rnk ( F )+1 F ∈ ∆ ∩ T � = ∅ ⇒ T ∆. to Logic L Applications to Structure Theory Bibliography
Logical consequences A verification calculus for logical consequence To extend the verification calculus also to logical consequences Some applications of we keep the old rules semi–formal systems α α Wolfram L ∆ ⇒ T L ∆. Pohlers Abstract and augment the verification calculus by a theory–rule semi–formal α 0 α systems ∆ , F c and α 0 < α ⇒ T , F T L ∆. Syntax L The verification calculus Semantics The, somehow weird looking, formulation of the theory rule has Abstract Logical Consequences technical reasons. The rule we would expect is, however, an The Verification Calculus for Logical permissible rule. Consequences We get M –Logic Applications 2 · rnk ( F )+1 F ∈ ∆ ∩ T � = ∅ ⇒ T ∆. to Logic L Applications Another simple observation is to Structure Theory α β T L ∆ implies S L Γ for T ⊆ S , α ≤ β and ∆ ⊆ Γ. Bibliography
Logical consequences Correctness Some applications of semi–formal systems Wolfram Lemma (Correctness) Pohlers α T L ∆ implies T | = L � ∆ . Abstract semi–formal systems Syntax The verification calculus Semantics Abstract Logical Consequences The Verification Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Logical consequences Correctness Some applications of semi–formal systems Wolfram Lemma (Correctness) Pohlers α T L ∆ implies T | = L � ∆ . Abstract semi–formal systems Syntax Proof By a simple induction on α we show that for every The verification calculus L –structure M and M –assignment Φ satisfying all formulas in Semantics Abstract Logical T we also get M | = � ∆[Φ]. Consequences The Verification Calculus for Logical Consequences M –Logic Applications to Logic Applications to Structure Theory Bibliography
Logical consequences Correctness Some applications of semi–formal systems Wolfram Lemma (Correctness) Pohlers α T L ∆ implies T | = L � ∆ . Abstract semi–formal systems Syntax Proof By a simple induction on α we show that for every The verification calculus L –structure M and M –assignment Φ satisfying all formulas in Semantics Abstract Logical T we also get M | = � ∆[Φ]. Consequences The Verification Calculus for Logical Consequences We just remark the triviality M –Logic α α ∅ L ∆ ⇔ L ∆. Applications to Logic Applications to Structure Theory Bibliography
M –Logic The language L M Some applications of semi–formal systems Wolfram Pohlers We now assume that the logical language L comprises first Abstract semi–formal order logic. Let M be an L –structure. We expand the language systems L to the language L M by adding a name m for every m ∈ | M | . M –Logic The langugage This allows us to dispense with free first order variables. L M Search Trees Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
M –Logic The language L M Some applications of semi–formal systems Wolfram Pohlers We now assume that the logical language L comprises first Abstract semi–formal order logic. Let M be an L –structure. We expand the language systems L to the language L M by adding a name m for every m ∈ | M | . M –Logic The langugage This allows us to dispense with free first order variables. L M Search Trees Therefore there are only closed terms in L M and we identify Truth Complexity terms that obtain the same value in M . Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
M –Logic Semantics Some applications of semi–formal systems To fix the semantics for the first order part of L M we define Wolfram Pohlers Abstract semi–formal systems M –Logic The langugage L M Search Trees Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
M –Logic Semantics Some applications of semi–formal systems To fix the semantics for the first order part of L M we define Wolfram Pohlers The � –type comprises the diagram of M and all formulas Abstract the outmost logical symbol of which is ∧ or ∀ . semi–formal systems M –Logic The langugage L M Search Trees Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
M –Logic Semantics Some applications of semi–formal systems To fix the semantics for the first order part of L M we define Wolfram Pohlers The � –type comprises the diagram of M and all formulas Abstract the outmost logical symbol of which is ∧ or ∀ . semi–formal systems The � –type comprises all atomic sentences that are false M –Logic in M and all formulas whose outmost logical symbol is ∨ The langugage L M or ∃ . Search Trees Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
M –Logic Semantics Some applications of semi–formal systems To fix the semantics for the first order part of L M we define Wolfram Pohlers The � –type comprises the diagram of M and all formulas Abstract the outmost logical symbol of which is ∧ or ∀ . semi–formal systems The � –type comprises all atomic sentences that are false M –Logic in M and all formulas whose outmost logical symbol is ∨ The langugage L M or ∃ . Search Trees Truth Complexity If F is atomic then CS ( F ) = ∅ . Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
M –Logic Semantics Some applications of semi–formal systems To fix the semantics for the first order part of L M we define Wolfram Pohlers The � –type comprises the diagram of M and all formulas Abstract the outmost logical symbol of which is ∧ or ∀ . semi–formal systems The � –type comprises all atomic sentences that are false M –Logic in M and all formulas whose outmost logical symbol is ∨ The langugage L M or ∃ . Search Trees Truth Complexity If F is atomic then CS ( F ) = ∅ . Term–models Admissible fragments CS ( F 1 ◦ · · · ◦ F n ) = � F 1 , . . . , F n � where ◦ = ∧ or ◦ = ∨ . M – Consequences Applications to Logic Applications to Structure Theory Bibliography
M –Logic Semantics Some applications of semi–formal systems To fix the semantics for the first order part of L M we define Wolfram Pohlers The � –type comprises the diagram of M and all formulas Abstract the outmost logical symbol of which is ∧ or ∀ . semi–formal systems The � –type comprises all atomic sentences that are false M –Logic in M and all formulas whose outmost logical symbol is ∨ The langugage L M or ∃ . Search Trees Truth Complexity If F is atomic then CS ( F ) = ∅ . Term–models Admissible fragments CS ( F 1 ◦ · · · ◦ F n ) = � F 1 , . . . , F n � where ◦ = ∧ or ◦ = ∨ . M – Consequences � � s ∈ | M | for Q ∈ {∀ , ∃} . CS ((Q x ) F ( x )) = F ( s ) Applications to Logic Applications to Structure Theory Bibliography
M –Logic Search Trees Some applications of semi–formal Definition (Search tree) systems Wolfram Let L be a countable language, M a countable L –structure, T Pohlers a countable and ∆ a finite sequence of L M –formulas. We Abstract define the search tree S L M T , ∆ together with a label function semi–formal systems → L <ω by the following clauses. ∆: S L M T , ∆ − M –Logic The langugage � � ∈ S L M L M T , ∆ and ∆ � � := ∆. Search Trees Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
M –Logic Search Trees Some applications of semi–formal Definition (Search tree) systems Wolfram Let L be a countable language, M a countable L –structure, T Pohlers a countable and ∆ a finite sequence of L M –formulas. We Abstract define the search tree S L M T , ∆ together with a label function semi–formal systems → L <ω by the following clauses. ∆: S L M T , ∆ − M –Logic The langugage � � ∈ S L M L M T , ∆ and ∆ � � := ∆. Search Trees Truth Complexity For the following clauses we assume s ∈ S L M T , ∆ such that ∆ s is Term–models Admissible not a logical axiom according to (Ax). fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
M –Logic Search Trees Some applications of semi–formal Definition (Search tree) systems Wolfram Let L be a countable language, M a countable L –structure, T Pohlers a countable and ∆ a finite sequence of L M –formulas. We Abstract define the search tree S L M T , ∆ together with a label function semi–formal systems → L <ω by the following clauses. ∆: S L M T , ∆ − M –Logic The langugage � � ∈ S L M L M T , ∆ and ∆ � � := ∆. Search Trees Truth Complexity For the following clauses we assume s ∈ S L M T , ∆ such that ∆ s is Term–models Admissible not a logical axiom according to (Ax). fragments M – Consequences The redex of a finite sequence ∆ s of L M –formulas is the Applications leftmost formula that possesses a type. We obtain the reduced to Logic sequence ∆ r s by discharging the redex in ∆ s . Applications to Structure Theory Bibliography
M –Logic Search trees (continued) Some applications of semi–formal Definition (Search tree continued) systems Wolfram Pohlers If ∆ s has no redex then s ⌢ � 0 � ∈ S L M T , ∆ and ∆ s ⌢ � 0 � := ∆ s , F ι c where F ι is the first formula in T that Abstract semi–formal does not occur in � t ⊑ s ∆ t . systems M –Logic The langugage L M Search Trees Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
M –Logic Search trees (continued) Some applications of semi–formal Definition (Search tree continued) systems Wolfram Pohlers If ∆ s has no redex then s ⌢ � 0 � ∈ S L M T , ∆ and ∆ s ⌢ � 0 � := ∆ s , F ι c where F ι is the first formula in T that Abstract semi–formal does not occur in � t ⊑ s ∆ t . systems M –Logic If F is the redex of ∆ s and F ≃ �� � G ι ι ∈ I then The langugage L M s ⌢ � ι � ∈ S L M T , ∆ for all ι ∈ I and ∆ s ⌢ � ι � := ∆ r Search Trees s , G ι . Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
M –Logic Search trees (continued) Some applications of semi–formal Definition (Search tree continued) systems Wolfram Pohlers If ∆ s has no redex then s ⌢ � 0 � ∈ S L M T , ∆ and ∆ s ⌢ � 0 � := ∆ s , F ι c where F ι is the first formula in T that Abstract semi–formal does not occur in � t ⊑ s ∆ t . systems M –Logic If F is the redex of ∆ s and F ≃ �� � G ι ι ∈ I then The langugage L M s ⌢ � ι � ∈ S L M T , ∆ for all ι ∈ I and ∆ s ⌢ � ι � := ∆ r Search Trees s , G ι . Truth Complexity If F is the redex of ∆ s and F ≃ �� � Term–models G ι ι ∈ I then Admissible s , G , F , F ι c where G is the fragments s ⌢ � 0 � ∈ S L M T , ∆ and ∆ s ⌢ � 0 � := ∆ r M – Consequences first formula in CS ( F ) and F ι the first formula in T that Applications to Logic does not occur in � t ⊑ s ∆ t . Applications to Structure Theory Bibliography
M –Logic The Syntactical Main Lemma Some applications of Lemma (Syntactial Main Lemma) semi–formal systems Let L be a countable language, M a countable L –structure, T Wolfram a countable and ∆ a finite set of L M –formulas. If S L T , ∆ is Pohlers well–founded of ordertype α then there is a subset T 0 of T Abstract α semi–formal such that T 0 L ∆ . systems M –Logic The langugage L M Search Trees Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
M –Logic The Syntactical Main Lemma Some applications of Lemma (Syntactial Main Lemma) semi–formal systems Let L be a countable language, M a countable L –structure, T Wolfram a countable and ∆ a finite set of L M –formulas. If S L T , ∆ is Pohlers well–founded of ordertype α then there is a subset T 0 of T Abstract α semi–formal such that T 0 L ∆ . systems In case that L is an admissible fragment L A and T is M –Logic The langugage Σ –definable in A then T 0 can be chosen A –finite. L M Search Trees Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
M –Logic The Syntactical Main Lemma Some applications of Lemma (Syntactial Main Lemma) semi–formal systems Let L be a countable language, M a countable L –structure, T Wolfram a countable and ∆ a finite set of L M –formulas. If S L T , ∆ is Pohlers well–founded of ordertype α then there is a subset T 0 of T Abstract α semi–formal such that T 0 L ∆ . systems In case that L is an admissible fragment L A and T is M –Logic The langugage Σ –definable in A then T 0 can be chosen A –finite. L M Search Trees Truth Proof Straight forward by induction on α . Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
M –Logic The Syntactical Main Lemma Some applications of Lemma (Syntactial Main Lemma) semi–formal systems Let L be a countable language, M a countable L –structure, T Wolfram a countable and ∆ a finite set of L M –formulas. If S L T , ∆ is Pohlers well–founded of ordertype α then there is a subset T 0 of T Abstract α semi–formal such that T 0 L ∆ . systems In case that L is an admissible fragment L A and T is M –Logic The langugage Σ –definable in A then T 0 can be chosen A –finite. L M Search Trees Truth Proof Straight forward by induction on α . Complexity Term–models Only the case that the redex is a formula �� � G ι ι ∈ I needs some Admissible fragments care in the proof of the addendum. Then s ⌢ � ι � is a node in S L M – T , ∆ Consequences whose order–type may be α ι . By the induction hypothesis we have Applications αι ∆ r , G ι . Since to Logic for all ι ∈ I a set T ι ∈ A and T ι ⊆ T such that T ι L Applications α ι ∆ r , G ι I ∈ A there is by Σ–collection a set T 0 ∈ A such that T 0 to Structure L Theory α and we obtain T 0 L ∆ by an inference � . Bibliography
M –Logic The Semantical Main Lemma Some applications of semi–formal Lemma (Semantical Main Lemma) systems Wolfram Let L be a countable language, M a countable L –structure, T Pohlers a countable and ∆ a finite sequence of L –formulas. If the Abstract search tree S L M T , ∆ is not well–founded then there is an semi–formal systems M –assignment Φ that satisfies all formulas in T but falsifies M –Logic the formulas in ∆ . The langugage L M Search Trees Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
M –Logic The Semantical Main Lemma Some applications of semi–formal Lemma (Semantical Main Lemma) systems Wolfram Let L be a countable language, M a countable L –structure, T Pohlers a countable and ∆ a finite sequence of L –formulas. If the Abstract search tree S L M T , ∆ is not well–founded then there is an semi–formal systems M –assignment Φ that satisfies all formulas in T but falsifies M –Logic the formulas in ∆ . The langugage L M Search Trees (Sketch) Let f be an infinite path in S L M Proof T , ∆ . Then ∆ f Truth Complexity Term–models “contains” all the formulas that are dual to the formulas in T . Admissible fragments We define an assignment M – Consequences ( X c t 1 , . . . , t n ) occurs in ∆ f } Φ( X ) = { ( t 1 , . . . , t n ) Applications to Logic and prove by induction on the complexity of F ∈ ∆ f that Applications to Structure M �| = F [Φ]. Theory Bibliography
M –Logic Correctness and Completeness for Pi 1 1 –sentences Some applications of semi–formal systems Theorem ( M –Correctness and –Completeness) Wolfram Let L be a countable language, M a countable L –structure and Pohlers F an L M –formula that may contain predicate variables Abstract semi–formal X 1 , . . . , X n . Then M | = ( ∀ X 1 ) . . . ( ∀ X n ) F iff there is a systems α countable ordinal α such that L M F. M –Logic The langugage L M Search Trees Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
M –Logic Correctness and Completeness for Pi 1 1 –sentences Some applications of semi–formal systems Theorem ( M –Correctness and –Completeness) Wolfram Let L be a countable language, M a countable L –structure and Pohlers F an L M –formula that may contain predicate variables Abstract semi–formal X 1 , . . . , X n . Then M | = ( ∀ X 1 ) . . . ( ∀ X n ) F iff there is a systems α countable ordinal α such that L M F. M –Logic The langugage L M α Search Trees Proof From L M F we get M | = F [Φ] for all M –assignments Truth Complexity Φ, hence M | = ( ∀ X 1 ) . . . ( ∀ X n ) F . Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
M –Logic Correctness and Completeness for Pi 1 1 –sentences Some applications of semi–formal systems Theorem ( M –Correctness and –Completeness) Wolfram Let L be a countable language, M a countable L –structure and Pohlers F an L M –formula that may contain predicate variables Abstract semi–formal X 1 , . . . , X n . Then M | = ( ∀ X 1 ) . . . ( ∀ X n ) F iff there is a systems α countable ordinal α such that L M F. M –Logic The langugage L M α Search Trees Proof From L M F we get M | = F [Φ] for all M –assignments Truth Complexity Φ, hence M | = ( ∀ X 1 ) . . . ( ∀ X n ) F . Term–models Admissible α L M F for all countable ordinals α then the search tree S L M fragments If � M – ∆ Consequences cannot be well–founded. By the Semantical Main Lemma there Applications to Logic is an M –assignment Φ such that M �| = F [Φ]. Hence Applications M �| = ( ∀ X 1 ) . . . ( ∀ X n ) F . to Structure Theory Bibliography
M –logic Truth complexity in M Some applications of semi–formal systems Definition Wolfram Pohlers We call Abstract α � min { α L M F } if this exists semi–formal tc L M ( F ) := systems ℵ 1 otherwise M –Logic The langugage the truth complexity of ( ∀ X 1 ) . . . ( ∀ X n ) F in the structure M . L M Search Trees Truth Complexity The M –Correctness and –Completeness theorem then says that Term–models Admissible all Π 1 1 –sentences that are valid in a countable structure for a fragments M – Consequences countable logical language L possess a countable truth Applications complexity. to Logic Applications to Structure Theory Bibliography
M –Logic Π 1 1 –relations definable in admissible languages Some applications of semi–formal To improve this result if we need more information about the systems language and the structure. The following theorem is a first Wolfram Pohlers example. Abstract semi–formal systems M –Logic The langugage L M Search Trees Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
M –Logic Π 1 1 –relations definable in admissible languages Some applications of semi–formal To improve this result if we need more information about the systems language and the structure. The following theorem is a first Wolfram Pohlers example. Abstract semi–formal Theorem systems Let L A be a countable admissible fragment and M a countable M –Logic The langugage L A –structure such that A is admissible above M . Then every L M Search Trees Π 1 1 –relation that is valid in M has a truth complexity less than Truth Complexity Term–models o ( A ) . Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
M –Logic Π 1 1 –relations definable in admissible languages Some applications of semi–formal To improve this result if we need more information about the systems language and the structure. The following theorem is a first Wolfram Pohlers example. Abstract semi–formal Theorem systems Let L A be a countable admissible fragment and M a countable M –Logic The langugage L A –structure such that A is admissible above M . Then every L M Search Trees Π 1 1 –relation that is valid in M has a truth complexity less than Truth Complexity Term–models o ( A ) . Admissible fragments M – Proof Since A is admissible above M the language (L A ) M is Consequences Applications A admissible. Therefore the ordertypes of search trees for this to Logic language are ordinals in A . Applications to Structure Theory Bibliography
Term–models Some applications of semi–formal systems Wolfram Pohlers Let L be a logical language comprising first order predicate logic with identity. It is not completely obvious how to define Abstract semi–formal the characteristic subformula sequences for formulas whose systems outmost logical symbol is a first order quantifier without M –Logic The langugage referring to an L –structure. L M Search Trees Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
Term–models Some applications of semi–formal systems Wolfram Pohlers Let L be a logical language comprising first order predicate logic with identity. It is not completely obvious how to define Abstract semi–formal the characteristic subformula sequences for formulas whose systems outmost logical symbol is a first order quantifier without M –Logic The langugage referring to an L –structure. Defining CS ((Q x ) F ) = � F ( x ) � L M Search Trees would violate the condition that the characteristic subformula Truth Complexity Term–models sequence of F must not contain free variables that are not free Admissible fragments in F . To overcome this problem we introduce term–models. M – Consequences Applications to Logic Applications to Structure Theory Bibliography
Term–models Some applications of semi–formal Definition (Term–models) systems Wolfram We define a term–model T L for L by the following clauses Pohlers Abstract semi–formal systems M –Logic The langugage L M Search Trees Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
Term–models Some applications of semi–formal Definition (Term–models) systems Wolfram We define a term–model T L for L by the following clauses Pohlers The domain of T L is the set { t t is an L –term } . Abstract semi–formal systems M –Logic The langugage L M Search Trees Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
Term–models Some applications of semi–formal Definition (Term–models) systems Wolfram We define a term–model T L for L by the following clauses Pohlers The domain of T L is the set { t t is an L –term } . Abstract semi–formal For every individual constant c we put c T L := c . systems M –Logic The langugage L M Search Trees Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
Term–models Some applications of semi–formal Definition (Term–models) systems Wolfram We define a term–model T L for L by the following clauses Pohlers The domain of T L is the set { t t is an L –term } . Abstract semi–formal For every individual constant c we put c T L := c . systems M –Logic For a function symbol f let f T L ( t 1 , . . . , t n ) := ( ft 1 , . . . , t n ). The langugage L M Search Trees Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
Term–models Some applications of semi–formal Definition (Term–models) systems Wolfram We define a term–model T L for L by the following clauses Pohlers The domain of T L is the set { t t is an L –term } . Abstract semi–formal For every individual constant c we put c T L := c . systems M –Logic For a function symbol f let f T L ( t 1 , . . . , t n ) := ( ft 1 , . . . , t n ). The langugage L M Let ≡ T L := { ( s , s ) Search Trees s ∈ | T L |} . Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
Term–models Some applications of semi–formal Definition (Term–models) systems Wolfram We define a term–model T L for L by the following clauses Pohlers The domain of T L is the set { t t is an L –term } . Abstract semi–formal For every individual constant c we put c T L := c . systems M –Logic For a function symbol f let f T L ( t 1 , . . . , t n ) := ( ft 1 , . . . , t n ). The langugage L M Let ≡ T L := { ( s , s ) Search Trees s ∈ | T L |} . Truth Complexity Predicate symbols are treated as predicate variables. Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
Term–models Some applications of semi–formal Definition (Term–models) systems Wolfram We define a term–model T L for L by the following clauses Pohlers The domain of T L is the set { t t is an L –term } . Abstract semi–formal For every individual constant c we put c T L := c . systems M –Logic For a function symbol f let f T L ( t 1 , . . . , t n ) := ( ft 1 , . . . , t n ). The langugage L M Let ≡ T L := { ( s , s ) Search Trees s ∈ | T L |} . Truth Complexity Predicate symbols are treated as predicate variables. Term–models Admissible fragments M – We thus have, strictly speaking, not just one term–model Consequences Applications but a variation of term–models according to the to Logic interpretation of the predicate constants. Applications to Structure Theory Bibliography
Term–models The language L T Some applications of semi–formal systems Wolfram Let L T be the language of the term model. The semantics for Pohlers the term models is then a special case of the semantics for the Abstract language L M . semi–formal systems M –Logic The langugage L M Search Trees Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
Term–models The language L T Some applications of semi–formal systems Wolfram Let L T be the language of the term model. The semantics for Pohlers the term models is then a special case of the semantics for the Abstract language L M . semi–formal systems Observe that there is a (in fact significant) difference between M –Logic the language L and L T . There are no free first order variables The langugage L M in L T . We may, however, often identify L and L T by reading an Search Trees Truth L –formula F ( t ) as the the L T –formula F ( t ) and vice versa. Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
Term–models The language L T Some applications of semi–formal systems Wolfram Let L T be the language of the term model. The semantics for Pohlers the term models is then a special case of the semantics for the Abstract language L M . semi–formal systems Observe that there is a (in fact significant) difference between M –Logic the language L and L T . There are no free first order variables The langugage L M in L T . We may, however, often identify L and L T by reading an Search Trees Truth L –formula F ( t ) as the the L T –formula F ( t ) and vice versa. Complexity Term–models Observe moreover, that the definition of the search tree S L T Admissible fragments T , ∆ M – does not depend upon T . We therefore write briefly S L Consequences T , ∆ Applications to Logic Applications to Structure Theory Bibliography
Term–models Semantical main lemma Some applications of semi–formal systems Wolfram Pohlers The Semantical Main Lemma transfered to term–models now reads as Abstract semi–formal systems Lemma (Semantical Main Lemma) M –Logic Let L be a countable language, T a countable and ∆ a finite The langugage L M Search Trees sequence of L –formulas. If the search tree S L T , ∆ is not Truth Complexity well–founded then there is a term–model T and a T –assignment Term–models Admissible Φ that satisfies all formulas in T but falsifies the formulas in ∆ . fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
Completeness for L –consequence Main theorem Some applications of semi–formal Theorem (Correctness and Completeness) systems Wolfram Let L be a countable logical language and T a countable set of Pohlers L –formulas. Then T | = L F iff there is a countable ordinal α Abstract α and a countable subset T 0 ⊆ T such that T 0 L F. semi–formal systems M –Logic The langugage L M Search Trees Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
Completeness for L –consequence Main theorem Some applications of semi–formal Theorem (Correctness and Completeness) systems Wolfram Let L be a countable logical language and T a countable set of Pohlers L –formulas. Then T | = L F iff there is a countable ordinal α Abstract α and a countable subset T 0 ⊆ T such that T 0 L F. semi–formal systems M –Logic Proof The direction from right to left is the Correctness The langugage L M Lemma. Search Trees Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
Completeness for L –consequence Main theorem Some applications of semi–formal Theorem (Correctness and Completeness) systems Wolfram Let L be a countable logical language and T a countable set of Pohlers L –formulas. Then T | = L F iff there is a countable ordinal α Abstract α and a countable subset T 0 ⊆ T such that T 0 L F. semi–formal systems M –Logic Proof The direction from right to left is the Correctness The langugage L M Lemma. Search Trees Truth α For the opposite direction assume T 0 � L F for all countable Complexity Term–models subsets T 0 ⊆ T and all countable ordinals α . Then by the Admissible fragments M – Syntactial Main Lemma S L T , F cannot be well–founded. By the Consequences Applications Semantical Main Lemma there is a term model T L and a to Logic T L –assignment Φ that verifies all the formulas in T but falsifies Applications to Structure F [Φ]. Hence T �| = L F . Theory Bibliography
L –Validity We call an L –formula F valid in an L –structure M iff Some applications of M | = F [Φ] holds true for all M –assignments Φ. semi–formal systems Wolfram Pohlers Abstract semi–formal systems M –Logic The langugage L M Search Trees Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
L –Validity We call an L –formula F valid in an L –structure M iff Some applications of M | = F [Φ] holds true for all M –assignments Φ. semi–formal systems F is valid if F is valid in all L –structures, i.e. if ∅ | = L F . Wolfram Pohlers Abstract semi–formal systems M –Logic The langugage L M Search Trees Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
L –Validity We call an L –formula F valid in an L –structure M iff Some applications of M | = F [Φ] holds true for all M –assignments Φ. semi–formal systems F is valid if F is valid in all L –structures, i.e. if ∅ | = L F . Wolfram As a corollary to the Correctness and Completeness Theorem Pohlers we get that the term–models for L are distinguished in the Abstract semi–formal following sense. systems M –Logic The langugage L M Search Trees Truth Complexity Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
L –Validity We call an L –formula F valid in an L –structure M iff Some applications of M | = F [Φ] holds true for all M –assignments Φ. semi–formal systems F is valid if F is valid in all L –structures, i.e. if ∅ | = L F . Wolfram As a corollary to the Correctness and Completeness Theorem Pohlers we get that the term–models for L are distinguished in the Abstract semi–formal following sense. systems M –Logic Corollary The langugage L M Let L be a countable logical language. An L –formula is valid iff Search Trees Truth Complexity it is valid in all term–models of L . Term–models Admissible fragments M – Consequences Applications to Logic Applications to Structure Theory Bibliography
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