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Some applications of semi–formal systems Wolfram Pohlers Abstract semi–formal systems M–Logic Applications to Logic Applications to Structure Theory Bibliography

Some applications of semi–formal systems

Wolfram Pohlers June 4, 2012

Some applications of semi–formal systems Wolfram Pohlers Abstract semi–formal systems M–Logic Applications to Logic Applications to Structure Theory Bibliography

1 Abstract semi–formal systems

Syntax The verification calculus Semantics Abstract Logical Consequences The Verification Calculus for Logical Consequences

2 M–Logic

The langugage LM Search Trees Truth Complexity Term–models Admissible fragments M–Consequences

3 Applications to Logic

First order logic Infinitary Logic Lκ,ω Further Possible Applications

Some applications of semi–formal systems Wolfram Pohlers Abstract semi–formal systems M–Logic Applications to Logic Applications to Structure Theory Bibliography

Introductory remark

By a semi–formal system I understand a proof system that includes an infinitary rule.

Some applications of semi–formal systems Wolfram Pohlers Abstract semi–formal systems M–Logic Applications to Logic Applications to Structure Theory Bibliography

Introductory remark

By a semi–formal system I understand a proof system that includes an infinitary rule. The use of infinitary rules was first suggested by David Hilbert in his paper “Die Grundlegung der elementaren Zahlentheorie” ([5]) and was later systematically used by Kurt Sch¨ utte in his work on proof theory. Of special interest are cut–free semi–formal systems.

Some applications of semi–formal systems Wolfram Pohlers Abstract semi–formal systems M–Logic Applications to Logic Applications to Structure Theory Bibliography

Introductory remark

By a semi–formal system I understand a proof system that includes an infinitary rule. The use of infinitary rules was first suggested by David Hilbert in his paper “Die Grundlegung der elementaren Zahlentheorie” ([5]) and was later systematically used by Kurt Sch¨ utte in his work on proof theory. Of special interest are cut–free semi–formal systems. Today I will not talk about applications of semi–formal systems in proof theory. I want to indicate that there are series of applications also outside proof theory. My intention is to show that semi–formal systems provide a versatile general framework to obtain many results in a very uniform way. Most of the results mentioned in this talk will not be new and many of them are due to Jon Barwise.

Some applications of semi–formal systems Wolfram Pohlers Abstract semi–formal systems M–Logic Applications to Logic Applications to Structure Theory Bibliography

This paper is therefore dedicated to Kurt Sch¨ utte at the

• ccasion of his 100 birthday and to the memory of Jon Barwise.

Some applications of semi–formal 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

Abstract logical languages

An abstract logical language L consists of individual and predicate variables and constants together with function symbols (for functions on individuals) and logical operations.

Some applications of semi–formal 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

Abstract logical languages

An abstract logical language L consists of individual and predicate variables and constants together with function symbols (for functions on individuals) and logical operations. Terms and atomic formulas are built up from individual variables and constants by functions and predicate constants or variables in the familiar way.

Some applications of semi–formal 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

Abstract logical languages

An abstract logical language L consists of individual and predicate variables and constants together with function symbols (for functions on individuals) and logical operations. Terms and atomic formulas are built up from individual variables and constants by functions and predicate constants or variables in the familiar way. Composed formulas a built up from atomic formulas by logical

• perations where some operations may bind variables.

Some applications of semi–formal 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

Dual formulas

We use the Tait formulation of L. I.e. there is no negation

• peration but for every predicate variable X and every

predicate constant P there is a dual symbol X c and Pc.

Some applications of semi–formal 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

Dual formulas

We use the Tait formulation of L. I.e. there is no negation

• peration but for every predicate variable X and every

predicate constant P there is a dual symbol X c and Pc. Moreover we require that for every logcial operation O there is a dual operation Oc.

Some applications of semi–formal 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

Dual formulas

We use the Tait formulation of L. I.e. there is no negation

• peration but for every predicate variable X and every

predicate constant P there is a dual symbol X c and Pc. Moreover we require that for every logcial operation O there is a dual operation Oc. For an L–formula F we define its dual F c by (Xt1, . . . , tn)c = (X ct1, . . . , tn)

Some applications of semi–formal 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

Dual formulas

We use the Tait formulation of L. I.e. there is no negation

• peration but for every predicate variable X and every

predicate constant P there is a dual symbol X c and Pc. Moreover we require that for every logcial operation O there is a dual operation Oc. For an L–formula F we define its dual F c by (Xt1, . . . , tn)c = (X ct1, . . . , tn) and (X ct1, . . . , tn)c = (Xt1, . . . , tn)

Some applications of semi–formal 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

Dual formulas

We use the Tait formulation of L. I.e. there is no negation

• peration but for every predicate variable X and every

predicate constant P there is a dual symbol X c and Pc. Moreover we require that for every logcial operation O there is a dual operation Oc. For an L–formula F we define its dual F c by (Xt1, . . . , tn)c = (X ct1, . . . , tn) and (X ct1, . . . , tn)c = (Xt1, . . . , tn) For predicate constants the definition is analogous.

Some applications of semi–formal 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

Dual formulas

We use the Tait formulation of L. I.e. there is no negation

• peration but for every predicate variable X and every

predicate constant P there is a dual symbol X c and Pc. Moreover we require that for every logcial operation O there is a dual operation Oc. For an L–formula F we define its dual F c by (Xt1, . . . , tn)c = (X ct1, . . . , tn) and (X ct1, . . . , tn)c = (Xt1, . . . , tn) For predicate constants the definition is analogous. For composed formulas we put (O

• Fι

ι ∈ I

• )c = Oc

Fιc ι ∈ I

• (Oc

Fι ι ∈ I

• )c = O
• Fιc

ι ∈ I

• .

Some applications of semi–formal 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

Types of formulas

The L formulas are sorted into three categories Formulas without type

Some applications of semi–formal 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

Types of formulas

The L formulas are sorted into three categories Formulas without type Formulas of –type

Some applications of semi–formal 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

Types of formulas

The L formulas are sorted into three categories Formulas without type Formulas of –type Formulas of –type

Some applications of semi–formal 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

Types of formulas

The L formulas are sorted into three categories Formulas without type Formulas of –type Formulas of –type satisfying all formulas without type are atomic.

Some applications of semi–formal 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

Characteristic subformula sequences

Each L–formula F possesses a characteristic subformula sequence CS(F) fulfilling (C0) CS(F c) =

• G c

G ∈ CS(F)

Some applications of semi–formal 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

Characteristic subformula sequences

Each L–formula F possesses a characteristic subformula sequence CS(F) fulfilling (C0) CS(F c) =

• G c

G ∈ CS(F)

• (C1) The complexity of every formula in CS(F) is less than the

complexity of F

Some applications of semi–formal 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

Characteristic subformula sequences

Each L–formula F possesses a characteristic subformula sequence CS(F) fulfilling (C0) CS(F c) =

• G c

G ∈ CS(F)

• (C1) The complexity of every formula in CS(F) is less than the

complexity of F (C2) The formulas in CS(F) must not contain free variables which are not also free in F.

Some applications of semi–formal 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

Characteristic subformula sequences

Each L–formula F possesses a characteristic subformula sequence CS(F) fulfilling (C0) CS(F c) =

• G c

G ∈ CS(F)

• (C1) The complexity of every formula in CS(F) is less than the

complexity of F (C2) The formulas in CS(F) must not contain free variables which are not also free in F. We call a language L countable if it contains only countably many terms and all characteristic subformula sequences are countable.

Some applications of semi–formal 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

Admissible Fragment

Let A be an admissible structure in the sense of Barwise [1] and L a logical language that is ∆1–definable in A. The admissible fragment LA of L consists of

Some applications of semi–formal 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

Admissible Fragment

Let A be an admissible structure in the sense of Barwise [1] and L a logical language that is ∆1–definable in A. The admissible fragment LA of L consists of {t ∈ L t is an L–term and t ∈ A}

Some applications of semi–formal 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

Admissible Fragment

Let A be an admissible structure in the sense of Barwise [1] and L a logical language that is ∆1–definable in A. The admissible fragment LA of L consists of {t ∈ L t is an L–term and t ∈ A} {F ∈ L F is an L–formula, F ∈ A and CS(F) ∈ A}

Some applications of semi–formal 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

The Verification Calculus

We define a verification calculus

α L ∆ for finite sets ∆ of

L–formulas which should be viewed as finite disjunctions by the following clauses

Some applications of semi–formal 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

The Verification Calculus

We define a verification calculus

α L ∆ for finite sets ∆ of

L–formulas which should be viewed as finite disjunctions by the following clauses (Ax) If ∆ is a finite set of L–formulas and F is a formula without type such that {F, F c} ⊆ ∆ then

α L ∆ holds true

for all ordinals α.

Some applications of semi–formal 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

The Verification Calculus

We define a verification calculus

α L ∆ for finite sets ∆ of

L–formulas which should be viewed as finite disjunctions by the following clauses (Ax) If ∆ is a finite set of L–formulas and F is a formula without type such that {F, F c} ⊆ ∆ then

α L ∆ holds true

for all ordinals α. () If F ∈ –type ∩ ∆ and

αG L

∆, G as well as αG < α holds true for all G ∈ CS(F) then

α L ∆.

Some applications of semi–formal 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

The Verification Calculus

We define a verification calculus

α L ∆ for finite sets ∆ of

L–formulas which should be viewed as finite disjunctions by the following clauses (Ax) If ∆ is a finite set of L–formulas and F is a formula without type such that {F, F c} ⊆ ∆ then

α L ∆ holds true

for all ordinals α. () If F ∈ –type ∩ ∆ and

αG L

∆, G as well as αG < α holds true for all G ∈ CS(F) then

α L ∆.

() If F ∈ –type ∩ ∆ and

α0 L

∆, Γ as well as α0 < α holds true for some finite set Γ ⊆ CS(F) then

α L ∆.

Some applications of semi–formal 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

Lemma

α L ∆, α ≤ β and ∆ ⊆ Γ imply β L Γ.

Some applications of semi–formal 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

Lemma

α L ∆, α ≤ β and ∆ ⊆ Γ imply β L Γ.

Lemma (–inversion) F ∈ –type and

α L ∆, F imply α L ∆, G for all G ∈ CS(F)

Some applications of semi–formal 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

Lemma

α L ∆, α ≤ β and ∆ ⊆ Γ imply β L Γ.

Lemma (–inversion) F ∈ –type and

α L ∆, F imply α L ∆, G for all G ∈ CS(F)

Lemma (–inversion) If ∈ –type and CS(F) is finite then

α L ∆, F implies α L ∆, CS(F).

Some applications of semi–formal 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

Lemma

α L ∆, α ≤ β and ∆ ⊆ Γ imply β L Γ.

Lemma (–inversion) F ∈ –type and

α L ∆, F imply α L ∆, G for all G ∈ CS(F)

Lemma (–inversion) If ∈ –type and CS(F) is finite then

α L ∆, F implies α L ∆, CS(F).

Lemma Let rnk(F) denote the complexity of F. Then

2·rnk(F) L

∆, F, F c holds true for all finite sets ∆ of L–formulas.

Some applications of semi–formal 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

Semantics

L–Structures and Assignments

Let L be a logical language. As usual an L-Structure M consists of a non empty set |M|, the domain of M together with the interpretations of all the non–logical symbols of L with the requirement that (Pc)M for a predicate constant P has to be the complement of PM.

Some applications of semi–formal 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

Semantics

L–Structures and Assignments

Let L be a logical language. As usual an L-Structure M consists of a non empty set |M|, the domain of M together with the interpretations of all the non–logical symbols of L with the requirement that (Pc)M for a predicate constant P has to be the complement of PM. An M–assignment Φ is a map that assigns an element Φ(x) ∈ |M| to every individual variable x and a set Φ(X) ⊆ |M|n to every n–ary predicate variable X.

Some applications of semi–formal 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

Semantics

L–Structures and Assignments

Let L be a logical language. As usual an L-Structure M consists of a non empty set |M|, the domain of M together with the interpretations of all the non–logical symbols of L with the requirement that (Pc)M for a predicate constant P has to be the complement of PM. An M–assignment Φ is a map that assigns an element Φ(x) ∈ |M| to every individual variable x and a set Φ(X) ⊆ |M|n to every n–ary predicate variable X. Clearly we again require that Φ(X c) is the complement of Φ(X).

Some applications of semi–formal 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

Semantics

Evaluation

As usual we define the evaluation of a formula in an structure M under an assignment Φ.

Some applications of semi–formal 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

Semantics

Evaluation

As usual we define the evaluation of a formula in an structure M under an assignment Φ. M | = F[Φ] for formulas F that do not belong to a type is given by the interpretation and the assignment,

Some applications of semi–formal 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

Semantics

Evaluation

As usual we define the evaluation of a formula in an structure M under an assignment Φ. M | = F[Φ] for formulas F that do not belong to a type is given by the interpretation and the assignment, M | = F[Φ] if M | = G[Φ] for all G ∈ CS(F) for formulas F in –type

Some applications of semi–formal 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

Semantics

Evaluation

As usual we define the evaluation of a formula in an structure M under an assignment Φ. M | = F[Φ] for formulas F that do not belong to a type is given by the interpretation and the assignment, M | = F[Φ] if M | = G[Φ] for all G ∈ CS(F) for formulas F in –type and M | = F[Φ] if M | = G[Φ] for some G ∈ CS(F) for formulas F in –type.

Some applications of semi–formal 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

Semantics

Correctness and completeness for sentences

Theorem (L–Correctness) Let L be a logical language, M an L-structure and ∆ a finite set of L–formulas. Then

α L ∆ implies M |

= ∆[Φ] for all M–assignments Φ.

Some applications of semi–formal 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

Semantics

Correctness and completeness for sentences

Theorem (L–Correctness) Let L be a logical language, M an L-structure and ∆ a finite set of L–formulas. Then

α L ∆ implies M |

= ∆[Φ] for all M–assignments Φ. Theorem (L–Correctness for Π1

1–sentences)

Let L be a logical language, M an L-structure and F an L–formula the free predicate variables of which belong all to the list X1, . . . , Xn.. Then

α L F implies M |

= (∀X1) . . . (∀Xn)F.

Some applications of semi–formal 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

Semantics

Correctness and completeness for sentences

Theorem (L–Correctness) Let L be a logical language, M an L-structure and ∆ a finite set of L–formulas. Then

α L ∆ implies M |

= ∆[Φ] for all M–assignments Φ. Theorem (L–Correctness for Π1

1–sentences)

Let L be a logical language, M an L-structure and F an L–formula the free predicate variables of which belong all to the list X1, . . . , Xn.. Then

α L F implies M |

= (∀X1) . . . (∀Xn)F. Theorem (L–Completeness for sentences) Let L be a logical language and M an L-structure. For every L–sentence F that is true in M there is an ordinal α ≤ rnk(F) such that

α L F.

Some applications of semi–formal 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

Abstract Logical Consequences

Definition

Definition (L-consequence) Let L be a logical language and T a set of L–formulas. We say that an L–formula F is an L–consequence of a set T of L–formulas, denoted by T | =L F iff for every L–structure M and every M–assignment Φ such that M | = T[Φ] (i.e. M | = G[Φ] for all G ∈ T) we also get M | = F[Φ].

Some applications of semi–formal 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

Logical consequences

A verification calculus for logical consequence

To extend the verification calculus also to logical consequences we keep the old rules

α L ∆ ⇒ T α L ∆.

Some applications of semi–formal 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

Logical consequences

A verification calculus for logical consequence

To extend the verification calculus also to logical consequences we keep the old rules

α L ∆ ⇒ T α L ∆.

and augment the verification calculus by a theory–rule T

α0 L

∆, F c and α0 < α ⇒ T, F

α L ∆.

Some applications of semi–formal 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

Logical consequences

A verification calculus for logical consequence

To extend the verification calculus also to logical consequences we keep the old rules

α L ∆ ⇒ T α L ∆.

and augment the verification calculus by a theory–rule T

α0 L

∆, F c and α0 < α ⇒ T, F

α L ∆.

The, somehow weird looking, formulation of the theory rule has technical reasons. The rule we would expect is, however, an permissible rule.

Some applications of semi–formal 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

Logical consequences

A verification calculus for logical consequence

To extend the verification calculus also to logical consequences we keep the old rules

α L ∆ ⇒ T α L ∆.

and augment the verification calculus by a theory–rule T

α0 L

∆, F c and α0 < α ⇒ T, F

α L ∆.

The, somehow weird looking, formulation of the theory rule has technical reasons. The rule we would expect is, however, an permissible rule. We get F ∈ ∆ ∩ T = ∅ ⇒ T

2·rnk(F)+1 L

∆.

Some applications of semi–formal 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

Logical consequences

A verification calculus for logical consequence

To extend the verification calculus also to logical consequences we keep the old rules

α L ∆ ⇒ T α L ∆.

and augment the verification calculus by a theory–rule T

α0 L

∆, F c and α0 < α ⇒ T, F

α L ∆.

The, somehow weird looking, formulation of the theory rule has technical reasons. The rule we would expect is, however, an permissible rule. We get F ∈ ∆ ∩ T = ∅ ⇒ T

2·rnk(F)+1 L

∆. Another simple observation is T

α L ∆ implies S β L Γ for T ⊆ S, α ≤ β and ∆ ⊆ Γ.

Some applications of semi–formal 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

Logical consequences

Correctness

Lemma (Correctness) T

α L ∆ implies T |

=L ∆.

Some applications of semi–formal 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

Logical consequences

Correctness

Lemma (Correctness) T

α L ∆ implies T |

=L ∆. Proof By a simple induction on α we show that for every L–structure M and M–assignment Φ satisfying all formulas in T we also get M | = ∆[Φ].

Some applications of semi–formal 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

Logical consequences

Correctness

Lemma (Correctness) T

α L ∆ implies T |

=L ∆. Proof By a simple induction on α we show that for every L–structure M and M–assignment Φ satisfying all formulas in T we also get M | = ∆[Φ]. We just remark the triviality ∅

α L ∆ ⇔ α L ∆.

Some applications of semi–formal systems Wolfram Pohlers Abstract semi–formal systems M–Logic

The langugage LM Search Trees Truth Complexity Term–models Admissible fragments

M–

Consequences

Applications to Logic Applications to Structure Theory Bibliography

M–Logic

The language LM

We now assume that the logical language L comprises first

• rder logic. Let M be an L–structure. We expand the language

L to the language LM by adding a name m for every m ∈ |M|. This allows us to dispense with free first order variables.

Some applications of semi–formal systems Wolfram Pohlers Abstract semi–formal systems M–Logic

The langugage LM Search Trees Truth Complexity Term–models Admissible fragments

M–

Consequences

Applications to Logic Applications to Structure Theory Bibliography

M–Logic

The language LM

We now assume that the logical language L comprises first

• rder logic. Let M be an L–structure. We expand the language

L to the language LM by adding a name m for every m ∈ |M|. This allows us to dispense with free first order variables. Therefore there are only closed terms in LM and we identify terms that obtain the same value in M.

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M–Logic

Semantics

To fix the semantics for the first order part of LM we define

Some applications of semi–formal systems Wolfram Pohlers Abstract semi–formal systems M–Logic

The langugage LM Search Trees Truth Complexity Term–models Admissible fragments

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M–Logic

Semantics

To fix the semantics for the first order part of LM we define The –type comprises the diagram of M and all formulas the outmost logical symbol of which is ∧ or ∀.

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M–Logic

Semantics

To fix the semantics for the first order part of LM we define The –type comprises the diagram of M and all formulas the outmost logical symbol of which is ∧ or ∀. The –type comprises all atomic sentences that are false in M and all formulas whose outmost logical symbol is ∨

• r ∃.

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M–Logic

Semantics

To fix the semantics for the first order part of LM we define The –type comprises the diagram of M and all formulas the outmost logical symbol of which is ∧ or ∀. The –type comprises all atomic sentences that are false in M and all formulas whose outmost logical symbol is ∨

• r ∃.

If F is atomic then CS(F) = ∅.

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M–Logic

Semantics

To fix the semantics for the first order part of LM we define The –type comprises the diagram of M and all formulas the outmost logical symbol of which is ∧ or ∀. The –type comprises all atomic sentences that are false in M and all formulas whose outmost logical symbol is ∨

• r ∃.

If F is atomic then CS(F) = ∅. CS(F1 ◦ · · · ◦ Fn) = F1, . . . , Fn where ◦ = ∧ or ◦ = ∨.

Some applications of semi–formal systems Wolfram Pohlers Abstract semi–formal systems M–Logic

The langugage LM Search Trees Truth Complexity Term–models Admissible fragments

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M–Logic

Semantics

To fix the semantics for the first order part of LM we define The –type comprises the diagram of M and all formulas the outmost logical symbol of which is ∧ or ∀. The –type comprises all atomic sentences that are false in M and all formulas whose outmost logical symbol is ∨

• r ∃.

If F is atomic then CS(F) = ∅. CS(F1 ◦ · · · ◦ Fn) = F1, . . . , Fn where ◦ = ∧ or ◦ = ∨. CS((Qx)F(x)) =

• F(s)

s ∈ |M|

• for Q ∈ {∀, ∃}.

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M–Logic

Search Trees

Definition (Search tree) Let L be a countable language, M a countable L–structure, T a countable and ∆ a finite sequence of LM–formulas. We define the search tree SLM

T,∆ together with a label function

∆: SLM

T,∆ −

→ L<ω by the following clauses. ∈ SLM

T,∆ and ∆ := ∆.

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M–Logic

Search Trees

Definition (Search tree) Let L be a countable language, M a countable L–structure, T a countable and ∆ a finite sequence of LM–formulas. We define the search tree SLM

T,∆ together with a label function

∆: SLM

T,∆ −

→ L<ω by the following clauses. ∈ SLM

T,∆ and ∆ := ∆.

For the following clauses we assume s ∈ SLM

T,∆ such that ∆s is

not a logical axiom according to (Ax).

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M–Logic

Search Trees

Definition (Search tree) Let L be a countable language, M a countable L–structure, T a countable and ∆ a finite sequence of LM–formulas. We define the search tree SLM

T,∆ together with a label function

∆: SLM

T,∆ −

→ L<ω by the following clauses. ∈ SLM

T,∆ and ∆ := ∆.

For the following clauses we assume s ∈ SLM

T,∆ such that ∆s is

not a logical axiom according to (Ax). The redex of a finite sequence ∆s of LM–formulas is the leftmost formula that possesses a type. We obtain the reduced sequence ∆r

s by discharging the redex in ∆s.

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M–Logic

Search trees (continued)

Definition (Search tree continued) If ∆s has no redex then s⌢0 ∈ SLM

T,∆ and

∆s⌢0 := ∆s, Fιc where Fι is the first formula in T that does not occur in

t⊑s ∆t.

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M–Logic

Search trees (continued)

Definition (Search tree continued) If ∆s has no redex then s⌢0 ∈ SLM

T,∆ and

∆s⌢0 := ∆s, Fιc where Fι is the first formula in T that does not occur in

t⊑s ∆t.

If F is the redex of ∆s and F ≃ Gι ι ∈ I

• then

s⌢ι ∈ SLM

T,∆ for all ι ∈ I and ∆s⌢ι := ∆r s, Gι.

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M–Logic

Search trees (continued)

Definition (Search tree continued) If ∆s has no redex then s⌢0 ∈ SLM

T,∆ and

∆s⌢0 := ∆s, Fιc where Fι is the first formula in T that does not occur in

t⊑s ∆t.

If F is the redex of ∆s and F ≃ Gι ι ∈ I

• then

s⌢ι ∈ SLM

T,∆ for all ι ∈ I and ∆s⌢ι := ∆r s, Gι.

If F is the redex of ∆s and F ≃ Gι ι ∈ I

• then

s⌢0 ∈ SLM

T,∆ and ∆s⌢0 := ∆r s, G, F, Fιc where G is the

first formula in CS(F) and Fι the first formula in T that does not occur in

t⊑s ∆t.

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M–Logic

The Syntactical Main Lemma

Lemma (Syntactial Main Lemma) Let L be a countable language, M a countable L–structure, T a countable and ∆ a finite set of LM–formulas. If SL

T,∆ is

well–founded of ordertype α then there is a subset T0 of T such that T0

α L ∆.

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M–Logic

The Syntactical Main Lemma

Lemma (Syntactial Main Lemma) Let L be a countable language, M a countable L–structure, T a countable and ∆ a finite set of LM–formulas. If SL

T,∆ is

well–founded of ordertype α then there is a subset T0 of T such that T0

α L ∆.

In case that L is an admissible fragment LA and T is Σ–definable in A then T0 can be chosen A–finite.

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M–Logic

The Syntactical Main Lemma

Lemma (Syntactial Main Lemma) Let L be a countable language, M a countable L–structure, T a countable and ∆ a finite set of LM–formulas. If SL

T,∆ is

well–founded of ordertype α then there is a subset T0 of T such that T0

α L ∆.

In case that L is an admissible fragment LA and T is Σ–definable in A then T0 can be chosen A–finite. Proof

Straight forward by induction on α.

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M–Logic

The Syntactical Main Lemma

Lemma (Syntactial Main Lemma) Let L be a countable language, M a countable L–structure, T a countable and ∆ a finite set of LM–formulas. If SL

T,∆ is

well–founded of ordertype α then there is a subset T0 of T such that T0

α L ∆.

In case that L is an admissible fragment LA and T is Σ–definable in A then T0 can be chosen A–finite. Proof

Straight forward by induction on α. Only the case that the redex is a formula Gι ι ∈ I

• needs some

care in the proof of the addendum. Then s⌢ι is a node in SL

T,∆

whose order–type may be αι. By the induction hypothesis we have for all ι ∈ I a set Tι ∈ A and Tι ⊆ T such that Tι

αι L

∆r, Gι. Since I ∈ A there is by Σ–collection a set T0 ∈ A such that T0

αι L

∆r, Gι and we obtain T0

α L ∆ by an inference .

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M–Logic

The Semantical Main Lemma

Lemma (Semantical Main Lemma) Let L be a countable language, M a countable L–structure, T a countable and ∆ a finite sequence of L–formulas. If the search tree SLM

T,∆ is not well–founded then there is an

M–assignment Φ that satisfies all formulas in T but falsifies the formulas in ∆.

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M–Logic

The Semantical Main Lemma

Lemma (Semantical Main Lemma) Let L be a countable language, M a countable L–structure, T a countable and ∆ a finite sequence of L–formulas. If the search tree SLM

T,∆ is not well–founded then there is an

M–assignment Φ that satisfies all formulas in T but falsifies the formulas in ∆. Proof (Sketch) Let f be an infinite path in SLM

T,∆. Then ∆f

“contains” all the formulas that are dual to the formulas in T. We define an assignment Φ(X) = {(t1, . . . , tn) (X ct1, . . . , tn) occurs in ∆f } and prove by induction on the complexity of F ∈ ∆f that M | = F[Φ].

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M–Logic

Correctness and Completeness for Pi1

1 –sentences

Theorem (M–Correctness and –Completeness) Let L be a countable language, M a countable L–structure and F an LM–formula that may contain predicate variables X1, . . . , Xn. Then M | = (∀X1) . . . (∀Xn)F iff there is a countable ordinal α such that

α LM F.

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M–Logic

Correctness and Completeness for Pi1

1 –sentences

Theorem (M–Correctness and –Completeness) Let L be a countable language, M a countable L–structure and F an LM–formula that may contain predicate variables X1, . . . , Xn. Then M | = (∀X1) . . . (∀Xn)F iff there is a countable ordinal α such that

α LM F.

Proof From

α LM F we get M |

= F[Φ] for all M–assignments Φ, hence M | = (∀X1) . . . (∀Xn)F.

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M–Logic

Correctness and Completeness for Pi1

1 –sentences

Theorem (M–Correctness and –Completeness) Let L be a countable language, M a countable L–structure and F an LM–formula that may contain predicate variables X1, . . . , Xn. Then M | = (∀X1) . . . (∀Xn)F iff there is a countable ordinal α such that

α LM F.

Proof From

α LM F we get M |

= F[Φ] for all M–assignments Φ, hence M | = (∀X1) . . . (∀Xn)F. If

α LM F for all countable ordinals α then the search tree SLM ∆

cannot be well–founded. By the Semantical Main Lemma there is an M–assignment Φ such that M | = F[Φ]. Hence M | = (∀X1) . . . (∀Xn)F.

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M–logic

Truth complexity in M

Definition We call tcLM(F) :=

• min {α

α LM F}

if this exists ℵ1

• therwise

the truth complexity of (∀X1) . . . (∀Xn)F in the structure M. The M–Correctness and –Completeness theorem then says that all Π1

1–sentences that are valid in a countable structure for a

countable logical language L possess a countable truth complexity.

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Π1

1–relations definable in admissible languages

To improve this result if we need more information about the language and the structure. The following theorem is a first example.

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Π1

1–relations definable in admissible languages

To improve this result if we need more information about the language and the structure. The following theorem is a first example. Theorem Let LA be a countable admissible fragment and M a countable LA–structure such that A is admissible above M. Then every Π1

1–relation that is valid in M has a truth complexity less than

• (A).

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Π1

1–relations definable in admissible languages

To improve this result if we need more information about the language and the structure. The following theorem is a first example. Theorem Let LA be a countable admissible fragment and M a countable LA–structure such that A is admissible above M. Then every Π1

1–relation that is valid in M has a truth complexity less than

• (A).

Proof Since A is admissible above M the language (LA)M is A admissible. Therefore the ordertypes of search trees for this language are ordinals in A.

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Term–models

Let L be a logical language comprising first order predicate logic with identity. It is not completely obvious how to define the characteristic subformula sequences for formulas whose

• utmost logical symbol is a first order quantifier without

referring to an L–structure.

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Term–models

Let L be a logical language comprising first order predicate logic with identity. It is not completely obvious how to define the characteristic subformula sequences for formulas whose

• utmost logical symbol is a first order quantifier without

referring to an L–structure. Defining CS((Qx)F) = F(x) would violate the condition that the characteristic subformula sequence of F must not contain free variables that are not free in F. To overcome this problem we introduce term–models.

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Term–models

Definition (Term–models) We define a term–model TL for L by the following clauses

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Term–models

Definition (Term–models) We define a term–model TL for L by the following clauses The domain of TL is the set {t t is an L–term }.

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Term–models

Definition (Term–models) We define a term–model TL for L by the following clauses The domain of TL is the set {t t is an L–term }. For every individual constant c we put cTL := c.

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Term–models

Definition (Term–models) We define a term–model TL for L by the following clauses The domain of TL is the set {t t is an L–term }. For every individual constant c we put cTL := c. For a function symbol f let f TL(t1, . . . , tn) := (ft1, . . . , tn).

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Term–models

Definition (Term–models) We define a term–model TL for L by the following clauses The domain of TL is the set {t t is an L–term }. For every individual constant c we put cTL := c. For a function symbol f let f TL(t1, . . . , tn) := (ft1, . . . , tn). Let ≡TL:= {(s, s) s ∈ |TL|}.

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Term–models

Definition (Term–models) We define a term–model TL for L by the following clauses The domain of TL is the set {t t is an L–term }. For every individual constant c we put cTL := c. For a function symbol f let f TL(t1, . . . , tn) := (ft1, . . . , tn). Let ≡TL:= {(s, s) s ∈ |TL|}. Predicate symbols are treated as predicate variables.

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Term–models

Definition (Term–models) We define a term–model TL for L by the following clauses The domain of TL is the set {t t is an L–term }. For every individual constant c we put cTL := c. For a function symbol f let f TL(t1, . . . , tn) := (ft1, . . . , tn). Let ≡TL:= {(s, s) s ∈ |TL|}. Predicate symbols are treated as predicate variables. We thus have, strictly speaking, not just one term–model but a variation of term–models according to the interpretation of the predicate constants.

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Term–models

The language LT

Let LT be the language of the term model. The semantics for the term models is then a special case of the semantics for the language LM.

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Term–models

The language LT

Let LT be the language of the term model. The semantics for the term models is then a special case of the semantics for the language LM. Observe that there is a (in fact significant) difference between the language L and LT. There are no free first order variables in LT. We may, however, often identify L and LT by reading an L–formula F(t) as the the LT–formula F(t) and vice versa.

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Term–models

The language LT

Let LT be the language of the term model. The semantics for the term models is then a special case of the semantics for the language LM. Observe that there is a (in fact significant) difference between the language L and LT. There are no free first order variables in LT. We may, however, often identify L and LT by reading an L–formula F(t) as the the LT–formula F(t) and vice versa. Observe moreover, that the definition of the search tree SLT

T,∆

does not depend upon T. We therefore write briefly SL

T,∆

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Term–models

Semantical main lemma

The Semantical Main Lemma transfered to term–models now reads as Lemma (Semantical Main Lemma) Let L be a countable language, T a countable and ∆ a finite sequence of L–formulas. If the search tree SL

T,∆ is not

well–founded then there is a term–model T and a T–assignment Φ that satisfies all formulas in T but falsifies the formulas in ∆.

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Completeness for L–consequence

Main theorem

Theorem (Correctness and Completeness) Let L be a countable logical language and T a countable set of L–formulas. Then T | =L F iff there is a countable ordinal α and a countable subset T0 ⊆ T such that T0

α L F.

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Completeness for L–consequence

Main theorem

Theorem (Correctness and Completeness) Let L be a countable logical language and T a countable set of L–formulas. Then T | =L F iff there is a countable ordinal α and a countable subset T0 ⊆ T such that T0

α L F.

Proof The direction from right to left is the Correctness Lemma.

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Completeness for L–consequence

Main theorem

Theorem (Correctness and Completeness) Let L be a countable logical language and T a countable set of L–formulas. Then T | =L F iff there is a countable ordinal α and a countable subset T0 ⊆ T such that T0

α L F.

Proof The direction from right to left is the Correctness Lemma. For the opposite direction assume T0

α L F for all countable

subsets T0 ⊆ T and all countable ordinals α. Then by the Syntactial Main Lemma SL

T,F cannot be well–founded. By the

Semantical Main Lemma there is a term model TL and a TL–assignment Φ that verifies all the formulas in T but falsifies F[Φ]. Hence T | =L F.

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L–Validity

We call an L–formula F valid in an L–structure M iff M | = F[Φ] holds true for all M–assignments Φ.

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L–Validity

We call an L–formula F valid in an L–structure M iff M | = F[Φ] holds true for all M–assignments Φ. F is valid if F is valid in all L–structures, i.e. if ∅ | =L F.

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L–Validity

We call an L–formula F valid in an L–structure M iff M | = F[Φ] holds true for all M–assignments Φ. F is valid if F is valid in all L–structures, i.e. if ∅ | =L F. As a corollary to the Correctness and Completeness Theorem we get that the term–models for L are distinguished in the following sense.

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L–Validity

We call an L–formula F valid in an L–structure M iff M | = F[Φ] holds true for all M–assignments Φ. F is valid if F is valid in all L–structures, i.e. if ∅ | =L F. As a corollary to the Correctness and Completeness Theorem we get that the term–models for L are distinguished in the following sense. Corollary Let L be a countable logical language. An L–formula is valid iff it is valid in all term–models of L.

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L–Validity

We call an L–formula F valid in an L–structure M iff M | = F[Φ] holds true for all M–assignments Φ. F is valid if F is valid in all L–structures, i.e. if ∅ | =L F. As a corollary to the Correctness and Completeness Theorem we get that the term–models for L are distinguished in the following sense. Corollary Let L be a countable logical language. An L–formula is valid iff it is valid in all term–models of L. Proof If F is valid it is valid in all term–models.

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L–Validity

We call an L–formula F valid in an L–structure M iff M | = F[Φ] holds true for all M–assignments Φ. F is valid if F is valid in all L–structures, i.e. if ∅ | =L F. As a corollary to the Correctness and Completeness Theorem we get that the term–models for L are distinguished in the following sense. Corollary Let L be a countable logical language. An L–formula is valid iff it is valid in all term–models of L. Proof If F is valid it is valid in all term–models. If F is not valid we have ∅ | =L F. Then there is no countable

• rdinal α such that

α L F. By the Syntactial Main Lemma SL F

is not well–founded and by the Semantical Main Lemma we

• btain a term–model T and a T–assignment that falsifies

F.

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The Cut Rule

As an aside we want to remark that the correctness and completeness theorem imply that the cut-rule T

α L ∆, F and T α L ∆, F c imply β L ∆ for some ordinal β

is a permissible rule of the verfication calculus.

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The langugage LM Search Trees Truth Complexity Term–models Admissible fragments

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Term–models

Substitution

Lemma (Substitution Lemma) Let T

α L ∆, F(t) for a term t that does neither occur in T nor

in ∆. Then T

α L ∆, F(s) holds true for all L–terms t.

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Term–models

Substitution

Lemma (Substitution Lemma) Let T

α L ∆, F(t) for a term t that does neither occur in T nor

in ∆. Then T

α L ∆, F(s) holds true for all L–terms t.

This lemma has some relevance for the complexity of the search tree. If the redex of a node s in SL

T,∆ is (∀x)F(x) then

we have nodes s⌢t above s for all L-terms t. According to the Substitution Lemma it suffices to choose the branch s⌢t0 for the first L–term t0 that does not occur in T ∪ ∆s. Therefore infinite branching could be avoided in this case.

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Term–models

Admissible fragments

If LA is an admissible fragment the language (LA)T of its term–model is, striktly speaking, not an admissible fragment. The reason is that the characteristic subformula sequence of CS((Qx)F(x)) is not necessarily an element of A.

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Term–models

Admissible fragments

If LA is an admissible fragment the language (LA)T of its term–model is, striktly speaking, not an admissible fragment. The reason is that the characteristic subformula sequence of CS((Qx)F(x)) is not necessarily an element of A. However, with the help of the Substitution Lemma we can modify the proof of the addendum to the Syntactical Main Lemma that it also holds true for (LA)T.

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Term–models

Admissible fragments

If LA is an admissible fragment the language (LA)T of its term–model is, striktly speaking, not an admissible fragment. The reason is that the characteristic subformula sequence of CS((Qx)F(x)) is not necessarily an element of A. However, with the help of the Substitution Lemma we can modify the proof of the addendum to the Syntactical Main Lemma that it also holds true for (LA)T. This has a series of consequences.

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Admissible fragments

Barwise compactness and completeness

Theorem (Barwise compactness (first version)) Let LA be a countable admissible fragment. If T | =LA F for a Σ1–definable set T of LA–formulas then there is an A–finite subset T0 ⊆ T such that T0 | =LA F.

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Admissible fragments

Barwise compactness and completeness (continued)

We may reformulate the first version of Barwise compactness in a more familiar form Theorem (Barwise completeness) Let LA be a countable admissible fragment and T a Σ1–definable set of LA–formulas. Then the following statements are equivalent T | =LA F (∃α)[T

α LA F]

A | = (∃α)(∃T)[T

α LA F].

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Admissible fragments

Barwise compactness and completeness (continued)

Theorem (Barwise compactness) Let LA be a countable admissible language. A countable set T

• f LA–formula which is Σ–definable in A is LA–consistent iff

every A–finite subset of T is LA–consistent.

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Admissible fragments

Barwise compactness and completeness (continued)

Theorem (Barwise compactness) Let LA be a countable admissible language. A countable set T

• f LA–formula which is Σ–definable in A is LA–consistent iff

every A–finite subset of T is LA–consistent. Proof T is inconsistent iff there is a formula F ∈ T such that T | =LA F c. By the first version of Barwise Compactness there is an A–finite subset of T0 ⊆ T such that T0 | =LA F c. So T0 is inconsistent.

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M–Consequences

The language L+

M

Definition Let M be a countable L–structure. We obtain the language L+

M by adding a unary predicate constant M to the language

LM.

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The langugage LM Search Trees Truth Complexity Term–models Admissible fragments

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M–Consequences

The language L+

M

Definition Let M be a countable L–structure. We obtain the language L+

M by adding a unary predicate constant M to the language

• LM. We define

M(t) ≃ t ≡ m m ∈ |M|

• .

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M–Consequences

The language L+

M

Definition Let M be a countable L–structure. We obtain the language L+

M by adding a unary predicate constant M to the language

• LM. We define

M(t) ≃ t ≡ m m ∈ |M|

• .

Mc(t) ≃ t ≡ m m ∈ |M|

• .

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M–Consequences

The language L+

M

Definition Let M be a countable L–structure. We obtain the language L+

M by adding a unary predicate constant M to the language

• LM. We define

M(t) ≃ t ≡ m m ∈ |M|

• .

Mc(t) ≃ t ≡ m m ∈ |M|

• .

An LM–structure is a structure N such that M is a substructure of N↾LM.

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M–Consequences

The language L+

M

Definition Let M be a countable L–structure. We obtain the language L+

M by adding a unary predicate constant M to the language

• LM. We define

M(t) ≃ t ≡ m m ∈ |M|

• .

Mc(t) ≃ t ≡ m m ∈ |M|

• .

An LM–structure is a structure N such that M is a substructure of N↾LM. MN = |M|

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The langugage LM Search Trees Truth Complexity Term–models Admissible fragments

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M–Consequences

The language L+

M

Definition Let M be a countable L–structure. We obtain the language L+

M by adding a unary predicate constant M to the language

• LM. We define

M(t) ≃ t ≡ m m ∈ |M|

• .

Mc(t) ≃ t ≡ m m ∈ |M|

• .

An LM–structure is a structure N such that M is a substructure of N↾LM. MN = |M| mN = m for all m ∈ |M|

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M–Consequences

Definition Let T be a set of L+

M–formulas. We say that an L+ M–formula F

is a M–consequence of T, denoted by T | =M F, iff for every L+

M–structure N and every N–assignment Φ that satisfies all

the formulas in T we also get N | = F[Φ].

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The langugage LM Search Trees Truth Complexity Term–models Admissible fragments

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M–Consequences

Definition Let T be a set of L+

M–formulas. We say that an L+ M–formula F

is a M–consequence of T, denoted by T | =M F, iff for every L+

M–structure N and every N–assignment Φ that satisfies all

the formulas in T we also get N | = F[Φ]. Having fixed all the types L+

M–formulas and their characteristic

subformula sequences we obtain a verification calculus T

α L+

M ∆.

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M–Consequences

Definition Let T be a set of L+

M–formulas. We say that an L+ M–formula F

is a M–consequence of T, denoted by T | =M F, iff for every L+

M–structure N and every N–assignment Φ that satisfies all

the formulas in T we also get N | = F[Φ]. Having fixed all the types L+

M–formulas and their characteristic

subformula sequences we obtain a verification calculus T

α L+

M ∆.

By induction on α we get T

α L+

M F implies T |

=M F for all L+

M–formulas F.

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The langugage LM Search Trees Truth Complexity Term–models Admissible fragments

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M–Consequences

The L+

M–term–model

To obtain also the converse implication we have to extend the notion of a term–model to the language L+

• M. Again we define

The domain of TL+

M is the set {t

t is an L+

M–term }

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M–Consequences

The L+

M–term–model

To obtain also the converse implication we have to extend the notion of a term–model to the language L+

• M. Again we define

The domain of TL+

M is the set {t

t is an L+

M–term }

The diagram of TL+

M comprises all sentences t ≡ t and

(Rm1, . . . , mn) for wich RM(m1, . . . , mn) belongs to the diagram of M.

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The langugage LM Search Trees Truth Complexity Term–models Admissible fragments

M–

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M–Consequences

The L+

M–term–model

To obtain also the converse implication we have to extend the notion of a term–model to the language L+

• M. Again we define

The domain of TL+

M is the set {t

t is an L+

M–term }

The diagram of TL+

M comprises all sentences t ≡ t and

(Rm1, . . . , mn) for wich RM(m1, . . . , mn) belongs to the diagram of M. f L+

M(t1, . . . , tn) = (ft1, . . . , tn)

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The langugage LM Search Trees Truth Complexity Term–models Admissible fragments

M–

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Applications to Logic Applications to Structure Theory Bibliography

M–Consequences

The L+

M–term–model

To obtain also the converse implication we have to extend the notion of a term–model to the language L+

• M. Again we define

The domain of TL+

M is the set {t

t is an L+

M–term }

The diagram of TL+

M comprises all sentences t ≡ t and

(Rm1, . . . , mn) for wich RM(m1, . . . , mn) belongs to the diagram of M. f L+

M(t1, . . . , tn) = (ft1, . . . , tn)

Relation constants are treated as relation variables

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M–Consequences

Correctness and Completeness

The LM+TL+

M

–search tree S

L+

M

T,∆ only depends on the basis

structure M and not on TL+

• M. The Sematical Main Lemma

carries over and we get

Some applications of semi–formal systems Wolfram Pohlers Abstract semi–formal systems M–Logic

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M–

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M–Consequences

Correctness and Completeness

The LM+TL+

M

–search tree S

L+

M

T,∆ only depends on the basis

structure M and not on TL+

• M. The Sematical Main Lemma

carries over and we get Theorem Let M be a countable structure, L+

M a countable language and

T a countable set of L+

M–formulas. Then F is a consequence of

T in M–logic iff there is countable ordinal α and a subset T0 ⊆ Tsuch that T0

α M F.

In case that L+

M is an A–admissible fragment and T is

Σ–definable on A the set T0 can be chosen A–finite.

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Applications to Logic

First Order Logic

Looking at the term model of first order logic we observe that

• nly formulas whose outmost logical symbol is a quantifier have

infinite characteristic subformula sequences.

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Applications to Logic

First Order Logic

Looking at the term model of first order logic we observe that

• nly formulas whose outmost logical symbol is a quantifier have

infinite characteristic subformula sequences. The only rule with infinitely many premises is thus the –rule for an quantifier ∀.

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Applications to Logic

First Order Logic

Looking at the term model of first order logic we observe that

• nly formulas whose outmost logical symbol is a quantifier have

infinite characteristic subformula sequences. The only rule with infinitely many premises is thus the –rule for an quantifier ∀. In view of the Substitution Lemma we can therefore define a new calculus T

α L ∆ by

T

α L ∆ entails T α L ∆ if the main formula of the last

inference is not universally quantified If (∀x)F(x) ∈ ∆ and T

α0 L

∆, F(s) for some term s that does neither occur in T nor in ∆, then T

α L ∆ for all

α > α0.

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Applications to Logic

First Order Logic

Lemma Let T be an arbitrary and ∆ a finite set of first order formulas. Then T

α L ∆ is equivalent to T α L ∆.

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Applications to Logic

First Order Logic

Lemma Let T be an arbitrary and ∆ a finite set of first order formulas. Then T

α L ∆ is equivalent to T α L ∆.

Theorem Let T be a countable set of first order formulas. Then T | =L F iff there is a finite ordinal n and a finite subset T0 ⊆ T such that T0

n L F where F is a first order formula.

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Applications to Logic

First Order Logic

Corollary A countable set of first order formulas is consistent iff every finite subset is consistent.

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Applications to Logic

First Order Logic

Corollary A countable set of first order formulas is consistent iff every finite subset is consistent. Corollary A formula of a countable first order language is valid iff it is valid in the term model.

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Applications to Logic

First Order Logic

Corollary A countable set of first order formulas is consistent iff every finite subset is consistent. Corollary A formula of a countable first order language is valid iff it is valid in the term model. Corollary A Π1

1–sentence that is valid in pure (first order) logic has finite

truth complexity.

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Applications

The logic L∞,ω

Definition (The language Lκ,ω) The logical operators of the language are ∀, ∃, and .

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Applications

The logic L∞,ω

Definition (The language Lκ,ω) The logical operators of the language are ∀, ∃, and . Besides atomic formulas and the first order operations we have the formation rules If

• Fι

ι < λ

• for λ < κ is a sequence of Lκ,ω–formula

which contains at most finitely many free variables, then

• Fι

ι < λ

• and

Fι ι < λ

• are Lκ,ω–formulas.

and are dual operations.

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First order logic Infinitary Logic Lκ,ω Further Possible Applications

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To turn Lκ,ω into an abstract logical language we reagard its term model T and identify Lκ,ω and (Lκ,ω)T. We define all formulas whose outmost logical symbols is or ∀ are in –type.

Some applications of semi–formal systems Wolfram Pohlers Abstract semi–formal systems M–Logic Applications to Logic

First order logic Infinitary Logic Lκ,ω Further Possible Applications

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To turn Lκ,ω into an abstract logical language we reagard its term model T and identify Lκ,ω and (Lκ,ω)T. We define all formulas whose outmost logical symbols is or ∀ are in –type. all formulas whose outmost logical symbols is or ∃ are in –type.

Some applications of semi–formal systems Wolfram Pohlers Abstract semi–formal systems M–Logic Applications to Logic

First order logic Infinitary Logic Lκ,ω Further Possible Applications

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To turn Lκ,ω into an abstract logical language we reagard its term model T and identify Lκ,ω and (Lκ,ω)T. We define all formulas whose outmost logical symbols is or ∀ are in –type. all formulas whose outmost logical symbols is or ∃ are in –type. CS((Qx)F(x)) =

• F(t)

t is a Lκ,ω–term

• .

Some applications of semi–formal systems Wolfram Pohlers Abstract semi–formal systems M–Logic Applications to Logic

First order logic Infinitary Logic Lκ,ω Further Possible Applications

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To turn Lκ,ω into an abstract logical language we reagard its term model T and identify Lκ,ω and (Lκ,ω)T. We define all formulas whose outmost logical symbols is or ∀ are in –type. all formulas whose outmost logical symbols is or ∃ are in –type. CS((Qx)F(x)) =

• F(t)

t is a Lκ,ω–term

• .

CS(O

• Fι

ι ∈ I

• ) =
• Fι

ι ∈ I

• where O is one of the
• perators , .

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Lκ,ω–Logic

Correctness and Completeness for sentences

Applying the abstract correctness and completeness theorems we obtain Theorem (Correctness for Π1

1–sentences)

Let M be an Lκ,ω–structure and F be an Lκ,ω–formula whose free predicate variables are among X1, . . . , Xn then

α Lκ,ω F

implies M | = (∀X1) . . . (∀Xn)F.

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Lκ,ω–Logic

Correctness and Completeness for sentences

Applying the abstract correctness and completeness theorems we obtain Theorem (Correctness for Π1

1–sentences)

Let M be an Lκ,ω–structure and F be an Lκ,ω–formula whose free predicate variables are among X1, . . . , Xn then

α Lκ,ω F

implies M | = (∀X1) . . . (∀Xn)F. Theorem Let M be an Lκ,ω–structure of cardinality σ and F be an (Lκ,ω)M–sentence. Then M | = F iff there is an ordinal α < max {σ+, κ} such that

α Lκ,ω F.

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Lκ,ω–Logic

Completeness for Π1

1–sentences

Theorem Let M be a countable Lω1,ω–structure and (∀X1) . . . (∀Xn)F a Π1

1–sentence. Then M |

= (∀X1) . . . (∀Xn)F iff there is a countable ordinal α such that

α Lω1,ω F.

This theorem becomes false if we replace ω1 by larger cardinals.

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Lκ,ω–Logic

Completeness for Π1

1–sentences

Theorem Let M be a countable Lω1,ω–structure and (∀X1) . . . (∀Xn)F a Π1

1–sentence. Then M |

= (∀X1) . . . (∀Xn)F iff there is a countable ordinal α such that

α Lω1,ω F.

This theorem becomes false if we replace ω1 by larger cardinals. Clearly also all the result for Lω1,ω–conclusions and countable admissible fragments carry over to Lω1,ω.

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Lκ,ω–Logic

Completeness for Π1

1–sentences

Theorem Let M be a countable Lω1,ω–structure and (∀X1) . . . (∀Xn)F a Π1

1–sentence. Then M |

= (∀X1) . . . (∀Xn)F iff there is a countable ordinal α such that

α Lω1,ω F.

This theorem becomes false if we replace ω1 by larger cardinals. Clearly also all the result for Lω1,ω–conclusions and countable admissible fragments carry over to Lω1,ω. E.g. Theorem Let LA is an admissible fragment of Lω1,ω, T a countable set of LA–formulas which is Σ–definable on A and F an LA–formula such that T | =Lω1,ω F then there is ´ a A–finite subset T0 of T such that T0 | =Lω1,ω F.

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First order logic Infinitary Logic Lκ,ω Further Possible Applications

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Further possible applications to logics

Without going into details we just want to mention that the methods inotroduced here also apply to Weak second order logic (over a given logic L)

Some applications of semi–formal systems Wolfram Pohlers Abstract semi–formal systems M–Logic Applications to Logic

First order logic Infinitary Logic Lκ,ω Further Possible Applications

Applications to Structure Theory Bibliography

Further possible applications to logics

Without going into details we just want to mention that the methods inotroduced here also apply to Weak second order logic (over a given logic L) Logical languages with the Game–quantifier

Some applications of semi–formal systems Wolfram Pohlers Abstract semi–formal systems M–Logic Applications to Logic

First order logic Infinitary Logic Lκ,ω Further Possible Applications

Applications to Structure Theory Bibliography

Further possible applications to logics

Without going into details we just want to mention that the methods inotroduced here also apply to Weak second order logic (over a given logic L) Logical languages with the Game–quantifier Hopeless are full second order logics. The techniques used here hinge on countability.

Some applications of semi–formal systems Wolfram Pohlers Abstract semi–formal systems M–Logic Applications to Logic

First order logic Infinitary Logic Lκ,ω Further Possible Applications

Applications to Structure Theory Bibliography

Further possible applications to logics

Without going into details we just want to mention that the methods inotroduced here also apply to Weak second order logic (over a given logic L) Logical languages with the Game–quantifier Hopeless are full second order logics. The techniques used here hinge on countability. What might be possible is a restricted second order logic in which the characteristic subformula sequence of a second order quantifier is defined by CS((QX)F(X)) :=

• F(S)

S is an L–definable set term

• .

Search trees for a similar logic have been used by Sch¨ utte in [6].

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Characteristic Ordinals of Structures

Definition (Characteristic ordinals) Given a structure M there is variety of characteristic ordinals. For structures in the language of arithmetic we define the following characteristic ordinals

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Structure Theory

Characteristic Ordinals of Structures

Definition (Characteristic ordinals) Given a structure M there is variety of characteristic ordinals. For structures in the language of arithmetic we define the following characteristic ordinals δ1

0(M) is the supremum of the ordertypes of well-ordings

that are elementarily definable in M.

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Structure Theory

Characteristic Ordinals of Structures

Definition (Characteristic ordinals) Given a structure M there is variety of characteristic ordinals. For structures in the language of arithmetic we define the following characteristic ordinals δ1

0(M) is the supremum of the ordertypes of well-ordings

that are elementarily definable in M. δ1

1(M) (σ1 1(M)) is the sup of ordertypes of well–orderings

that are ∆1

1– (Σ1 1–)definable on M

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Structure Theory

Characteristic Ordinals of Structures

Definition (Characteristic ordinals) Given a structure M there is variety of characteristic ordinals. For structures in the language of arithmetic we define the following characteristic ordinals δ1

0(M) is the supremum of the ordertypes of well-ordings

that are elementarily definable in M. δ1

1(M) (σ1 1(M)) is the sup of ordertypes of well–orderings

that are ∆1

1– (Σ1 1–)definable on M

κM is the sup of the closure ordinals of inductive definitions that are elementarily definable on M.

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Structure Theory

Characteristic Ordinals of Structures

Definition (Characteristic ordinals) Given a structure M there is variety of characteristic ordinals. For structures in the language of arithmetic we define the following characteristic ordinals δ1

0(M) is the supremum of the ordertypes of well-ordings

that are elementarily definable in M. δ1

1(M) (σ1 1(M)) is the sup of ordertypes of well–orderings

that are ∆1

1– (Σ1 1–)definable on M

κM is the sup of the closure ordinals of inductive definitions that are elementarily definable on M. πM is the sup of truth complexities of Π–sentences that are valid in M

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Characteristic Ordinals of Structures

¸ Definition (Characteristic ordinals continued) Let A be a structure in the language of set theory

• (A) is the supremum of the ordertypes of the ordinals in

M, i.e. the least ordinal that does not belong to A.

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Characteristic Ordinals of Structures

¸ Definition (Characteristic ordinals continued) Let A be a structure in the language of set theory

• (A) is the supremum of the ordertypes of the ordinals in

M, i.e. the least ordinal that does not belong to A. O(M) := o(HYP(M)), where HYP(M) is the least admissible structure above M.

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Structure Theory

The Boundedness Theorem

The connection between verification calculi and characteristic

• rdinals is given by the following lemma.

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Structure Theory

The Boundedness Theorem

The connection between verification calculi and characteristic

• rdinals is given by the following lemma.

To fix notations let Prog(≺, X) be the formula (∀x)[(∀y)[y ≺ x → (Xy)] → (Xx)]. Then (∀X)[Prog(≺, X) → (∀x ∈ field(≺))(Xx)] expresses the well-foundedness of the relation ≺.

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Structure Theory

The Boundedness Theorem

The connection between verification calculi and characteristic

• rdinals is given by the following lemma.

To fix notations let Prog(≺, X) be the formula (∀x)[(∀y)[y ≺ x → (Xy)] → (Xx)]. Then (∀X)[Prog(≺, X) → (∀x ∈ field(≺))(Xx)] expresses the well-foundedness of the relation ≺. Lemma (Boundedness Lemma) If ≺ is a transitive ordering that is Σ1

1–definable in a structure

M then

α LM (Prog(≺, X))c, (∀x ∈ field(≺))(Xx) implies

• typ(≺) ≤ α.

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Characteristic Ordinals The Boundedness Theorems Applications of Boundedness Inductive Definitions

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Structure Theory

The Boundedness Theorem

The connection between verification calculi and characteristic

• rdinals is given by the following lemma.

To fix notations let Prog(≺, X) be the formula (∀x)[(∀y)[y ≺ x → (Xy)] → (Xx)]. Then (∀X)[Prog(≺, X) → (∀x ∈ field(≺))(Xx)] expresses the well-foundedness of the relation ≺. Lemma (Boundedness Lemma) If ≺ is a transitive ordering that is Σ1

1–definable in a structure

M then

α LM (Prog(≺, X))c, (∀x ∈ field(≺))(Xx) implies

• typ(≺) ≤ α.

Theorem (Boundedness) Let ≺ be a well–ordering of limit order–type that is Σ1

1–definable in a countable structure M. Then

• typ(≺) = tcLM(Wf(≺)).

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Theorem Let M be a countable structure. Then δ1

0(M) ≤ δ1 1(M) ≤ σ1 1(M) ≤ πM.

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Theorem Let M be a countable structure. Then δ1

0(M) ≤ δ1 1(M) ≤ σ1 1(M) ≤ πM.

Theorem Let M be a countable structure. Then πM ≤ O(M).

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Structure Theory

Theorem Let M be a countable structure. Then δ1

0(M) ≤ δ1 1(M) ≤ σ1 1(M) ≤ πM.

Theorem Let M be a countable structure. Then πM ≤ O(M). Proof Since |M| ∈ HYP(M) we obtain LM as a countable HYP(M)–admissible set. By the theorem on admissible fragments we obtain tcLM(F) < o(HYP(M)) = O(M) for all Π1

1–sentences that are valid in M.

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Structure Theory

A reinspection of the definition of the search tree shows that it is definable by course of values recursion. Therefore there are structures in which the search trees are definable. Theorem Let M be a countable structure which allows for a definition of search trees. Then δ1

0(M) = δ1 1(M) = σ1 1(M) = πM ≤ O(M).

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Structure Theory

A reinspection of the definition of the search tree shows that it is definable by course of values recursion. Therefore there are structures in which the search trees are definable. Theorem Let M be a countable structure which allows for a definition of search trees. Then δ1

0(M) = δ1 1(M) = σ1 1(M) = πM ≤ O(M).

Proof For a Π1

1–sentence (∀X)F we have M |

= (∀X)F iff

α LM F iff the search tree SLM F

is well–founded. For any α < πM there is a Π1

1–sentence (∀X)F such that M |

= (∀X)F and α ≤ tcLM((∀X)F). Since tcLM((∀X)F) is less or equal than the order–type of the search tree SLM

F

in the Kleene–Brouwer ordering and SLM

F

is elementarily definable in M we get α < δ1

0(M).

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Theorem Let M be a countable structure that allows for a HYP(M)–recursive pairing function. Then σ1

1(M) = πM = O(M).

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Structure Theory

Theorem Let M be a countable structure that allows for a HYP(M)–recursive pairing function. Then σ1

1(M) = πM = O(M).

Proof The proof needs the fact that HYP(M) is projectible into M which entails that O(M) is the sup of the order–types

• f well–orderings of |M| that are elements of HYP(M). Since

every ordering on |M| in HYP(M) is Σ1

1–definable in M we get

α ≤ otyp(≺) ≤ σ1

1(M) ≤ πM for all α < O(M).

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Structure Theory

Theorem Let M be a countable structure that admits the definition of the search tree (which needs an elementarily definable pairing machinery). Then δ1

0(M) = δ1 1(M) = σ1 1(M) = O(M).

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Structure Theory

Theorem Let M be a countable structure that admits the definition of the search tree (which needs an elementarily definable pairing machinery). Then δ1

0(M) = δ1 1(M) = σ1 1(M) = O(M).

Theorem Let M be a countable structure which allows for a definition of search trees. Then tcLM(F) < πM entails that F is ∆1

1.

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Structure Theory

Theorem Let M be a countable structure that admits the definition of the search tree (which needs an elementarily definable pairing machinery). Then δ1

0(M) = δ1 1(M) = σ1 1(M) = O(M).

Theorem Let M be a countable structure which allows for a definition of search trees. Then tcLM(F) < πM entails that F is ∆1

1.

Proof If tcLM((∀X)G(X)) < πM there is a Π1

1–sentence

(∀X)H(X) such that M | = (∀X)H(X). Then we get M | = (∀X)G(X) iff SLM

G

is well–founded iff there is a function from SLM

G

into SLM

H

which preserves the tree structure.

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Inductive Definitions

The equivalent of the Boundedness Lemma in terms of inductive definitions is given by Lemma Let F(X, x) and ∆ be finite set of X–positive formulas. Then

α LM ((∀y)[F(X, y) → (Xy)])c, ∆(X) for a set ∆ of X–positive

formulas implies M | = ∆[I 2α

F ].

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Structure Theory

Inductive Definitions

The equivalent of the Boundedness Lemma in terms of inductive definitions is given by Lemma Let F(X, x) and ∆ be finite set of X–positive formulas. Then

α LM ((∀y)[F(X, y) → (Xy)])c, ∆(X) for a set ∆ of X–positive

formulas implies M | = ∆[I 2α

F ].

Let ClF(X) be the formula (∀X)[F(X, x) → (Xx)] and IndF(z) the formula (∀X)[ClF(X) → (Xz)].

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Structure Theory

Inductive Definitions

The equivalent of the Boundedness Lemma in terms of inductive definitions is given by Lemma Let F(X, x) and ∆ be finite set of X–positive formulas. Then

α LM ((∀y)[F(X, y) → (Xy)])c, ∆(X) for a set ∆ of X–positive

formulas implies M | = ∆[I 2α

F ].

Let ClF(X) be the formula (∀X)[F(X, x) → (Xx)] and IndF(z) the formula (∀X)[ClF(X) → (Xz)]. Then z ∈ IF iff (∀X)[ClF(X) → x ∈ X] iff IndF(z). For all elements n of the fixed–point IF we have |n|F ≤ 2tcLM(IndF (n))

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Inductive Definitions

Lemma Let M be an infinite countable structure. Then κM ≤ πM.

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Structure Theory

Inductive Definitions

Lemma Let M be an infinite countable structure. Then κM ≤ πM. Theorem Let M be an infinite countable structure that admits definitions

• f search trees. Then all the characteristic ordinals coinc´

ıde.

Some applications of semi–formal systems Wolfram Pohlers Abstract semi–formal systems M–Logic Applications to Logic Applications to Structure Theory Bibliography

• J. Barwise, Admissible sets and structures,

Perspectives in Mathematical Logic, Springer-Verlag, Berlin/Heidelberg/New York, 1975.

• A. Beckmann and W. Pohlers, Application of

cut–free infinitary derivations to generalized recursion theory, Annals of Pure and Applied Logic, vol. 94 (1998), pp. 1–19.

• L. Henkin, A generalization of the notion of

ω-consistency, Journal of Symbolic Logic, vol. 19 (1954), pp. 183–196. , A generalization of the concept of ω–completeness, Journal of Symbolic Logic, vol. 22 (1957), pp. 1–14.

Some applications of semi–formal systems Wolfram Pohlers Abstract semi–formal systems M–Logic Applications to Logic Applications to Structure Theory Bibliography

• D. Hilbert, Die Grundlegung der elementaren

Zahlenlehre, Mathematische Annalen, vol. 104 (1931),

• pp. 485–494.
• K. Sch¨

utte, Syntactical and semantical properties of simple type theory, Journal of Symbolic Logic, vol. 25 (1960), pp. 305–326.

• G. Takeuti, On a generalized logic calculus, Japanese

Journal of Mathematics, vol. 24 (1953), pp. 149–156.

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Closure ordinals of inductive definitions

Definition (Inductive definitions) An X–positive formula F(X, x) be an defines an operator ∆F : Pow(|M|) − → Pow(|M|) sending a set S ⊆ |M| to the set {x ∈ |M| M | = F(S, x)}. The least fixed–point IF of ∆F is the set inductively defined by F. We therefore call F an inductive definition. We recursively define the stage I α

F by I α F := ∆F( ξ<α I ξ F)

The closure ordinal of the inductive definition F is the least ordinal such that I α

F = I α+1 F

.