Leon Henkin on Completeness Henkins renowned proofs of PhDs in - - PowerPoint PPT Presentation

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Leon Henkin on Completeness PhDs in Logic 2017 Bochum

Manzano, M

USAL

May 2017

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

1 Henkin’s renowned proofs of completeness 2 The completeness of FOL in Henkin’s course 3 References

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Leon Henkin (1921-2006)

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Leon Henkin (1921-2006) 1947 The Completeness of Formal Systems PhD thesis, under direction of Alonzo Church

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Leon Henkin (1921-2006) 1947 The Completeness of Formal Systems PhD thesis, under direction of Alonzo Church from 1952 Henkin was part of The Group in Logic and the Methodology of Science created by Alfred Tarski in Berkeley

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Leon Henkin (1921-2006) 1947 The Completeness of Formal Systems PhD thesis, under direction of Alonzo Church from 1952 Henkin was part of The Group in Logic and the Methodology of Science created by Alfred Tarski in Berkeley 2014, The Life and Work of Leon Henkin

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

The Life and Work of Leon Henkin (2014)

in Henkin on Completeness, [1], Manzano, M. A detailed account of the three proofs Henkin published last century. In [4].

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

The Life and Work of Leon Henkin (2014)

in Henkin on Completeness, [1], Manzano, M. A detailed account of the three proofs Henkin published last century. In [4]. the paper Changing a Semantics: Opportunism or Courage?, [2] by Andreka, van Benthem, and others gives a systematic view of general models in mathematical and philosophical terms. In [4].

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

The Life and Work of Leon Henkin (2014)

in Henkin on Completeness, [1], Manzano, M. A detailed account of the three proofs Henkin published last century. In [4]. the paper Changing a Semantics: Opportunism or Courage?, [2] by Andreka, van Benthem, and others gives a systematic view of general models in mathematical and philosophical terms. In [4]. the paper Henkin and Hybrid Logic [3] explains the relationship between this logic and Henkin previous results, by Blackburn and others. In [4]

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin on completeness, 2014 (M. Manzano)

Apart from his well known published results on completeness

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin on completeness, 2014 (M. Manzano)

Apart from his well known published results on completeness

For first-order logic, 1949, The completeness of the first-order functional calculus

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin on completeness, 2014 (M. Manzano)

Apart from his well known published results on completeness

For first-order logic, 1949, The completeness of the first-order functional calculus Type theory, 1950, Completeness in the theory of types

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin on completeness, 2014 (M. Manzano)

Apart from his well known published results on completeness

For first-order logic, 1949, The completeness of the first-order functional calculus Type theory, 1950, Completeness in the theory of types Propositional type theory, 1963, A theory of propositional types

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin on completeness, 2014 (M. Manzano)

Apart from his well known published results on completeness

For first-order logic, 1949, The completeness of the first-order functional calculus Type theory, 1950, Completeness in the theory of types Propositional type theory, 1963, A theory of propositional types A new completeness proof for FOL, 1963, An Extension of the Craig-Lyndon Interpolation Theorem

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin on completeness, 2014 (M. Manzano)

Apart from his well known published results on completeness

For first-order logic, 1949, The completeness of the first-order functional calculus Type theory, 1950, Completeness in the theory of types Propositional type theory, 1963, A theory of propositional types A new completeness proof for FOL, 1963, An Extension of the Craig-Lyndon Interpolation Theorem

Henkin wrote an extremely interesting autobiographical paper, 1996, The discovery of my completeness proofs

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin on completeness, 2014 (M. Manzano)

Apart from his well known published results on completeness

For first-order logic, 1949, The completeness of the first-order functional calculus Type theory, 1950, Completeness in the theory of types Propositional type theory, 1963, A theory of propositional types A new completeness proof for FOL, 1963, An Extension of the Craig-Lyndon Interpolation Theorem

Henkin wrote an extremely interesting autobiographical paper, 1996, The discovery of my completeness proofs As well as two expository papers on that subject,

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin on completeness, 2014 (M. Manzano)

Apart from his well known published results on completeness

For first-order logic, 1949, The completeness of the first-order functional calculus Type theory, 1950, Completeness in the theory of types Propositional type theory, 1963, A theory of propositional types A new completeness proof for FOL, 1963, An Extension of the Craig-Lyndon Interpolation Theorem

Henkin wrote an extremely interesting autobiographical paper, 1996, The discovery of my completeness proofs As well as two expository papers on that subject,

Truth and Provability

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin on completeness, 2014 (M. Manzano)

Apart from his well known published results on completeness

For first-order logic, 1949, The completeness of the first-order functional calculus Type theory, 1950, Completeness in the theory of types Propositional type theory, 1963, A theory of propositional types A new completeness proof for FOL, 1963, An Extension of the Craig-Lyndon Interpolation Theorem

Henkin wrote an extremely interesting autobiographical paper, 1996, The discovery of my completeness proofs As well as two expository papers on that subject,

Truth and Provability Completeness

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin on completeness, 2014 (M. Manzano)

Apart from his well known published results on completeness

For first-order logic, 1949, The completeness of the first-order functional calculus Type theory, 1950, Completeness in the theory of types Propositional type theory, 1963, A theory of propositional types A new completeness proof for FOL, 1963, An Extension of the Craig-Lyndon Interpolation Theorem

Henkin wrote an extremely interesting autobiographical paper, 1996, The discovery of my completeness proofs As well as two expository papers on that subject,

Truth and Provability Completeness in: Morgenbesser, S. (ed.). Philosophy of Science today (1967)

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin on completeness, 2014 (M. Manzano)

Following Henkin’s autobiographical paper, The discovery

• f my completeness proofs, 1996, I point out some of

Henkin’s stated influences, especially three of them:

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin on completeness, 2014 (M. Manzano)

Following Henkin’s autobiographical paper, The discovery

• f my completeness proofs, 1996, I point out some of

Henkin’s stated influences, especially three of them:

1 Gödel’s completeness theorem, as well as his article on the

consistency of the axiom of choice (where he builds a constructible universe),

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin on completeness, 2014 (M. Manzano)

Following Henkin’s autobiographical paper, The discovery

• f my completeness proofs, 1996, I point out some of

Henkin’s stated influences, especially three of them:

1 Gödel’s completeness theorem, as well as his article on the

consistency of the axiom of choice (where he builds a constructible universe),

2 Russell’s theory of types and his expository explanation of

the axiom of choice, and

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin on completeness, 2014 (M. Manzano)

Following Henkin’s autobiographical paper, The discovery

• f my completeness proofs, 1996, I point out some of

Henkin’s stated influences, especially three of them:

1 Gödel’s completeness theorem, as well as his article on the

consistency of the axiom of choice (where he builds a constructible universe),

2 Russell’s theory of types and his expository explanation of

the axiom of choice, and

3 Church’s formulation of the theory of types and the

important role played by both

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin on completeness, 2014 (M. Manzano)

Following Henkin’s autobiographical paper, The discovery

• f my completeness proofs, 1996, I point out some of

Henkin’s stated influences, especially three of them:

1 Gödel’s completeness theorem, as well as his article on the

consistency of the axiom of choice (where he builds a constructible universe),

2 Russell’s theory of types and his expository explanation of

the axiom of choice, and

3 Church’s formulation of the theory of types and the

important role played by both

1

the lambda operator and

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin on completeness, 2014 (M. Manzano)

Following Henkin’s autobiographical paper, The discovery

• f my completeness proofs, 1996, I point out some of

Henkin’s stated influences, especially three of them:

1 Gödel’s completeness theorem, as well as his article on the

consistency of the axiom of choice (where he builds a constructible universe),

2 Russell’s theory of types and his expository explanation of

the axiom of choice, and

3 Church’s formulation of the theory of types and the

important role played by both

1

the lambda operator and

2

the description operators in foundational issues.

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin on completeness, 2014 (M. Manzano)

Following Henkin’s autobiographical paper, The discovery

• f my completeness proofs, 1996, I point out some of

Henkin’s stated influences, especially three of them:

1 Gödel’s completeness theorem, as well as his article on the

consistency of the axiom of choice (where he builds a constructible universe),

2 Russell’s theory of types and his expository explanation of

the axiom of choice, and

3 Church’s formulation of the theory of types and the

important role played by both

1

the lambda operator and

2

the description operators in foundational issues.

4 His declared interest in the nameable types

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin’s Ph.D. Thesis, 1947. Alonzo Church

Completeness in the theory of types, 1950

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin’s Ph.D. Thesis, 1947. Alonzo Church

Completeness in the theory of types, 1950 Henkin’s proofs marked the beginning of the new method.

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin’s Ph.D. Thesis, 1947. Alonzo Church

Completeness in the theory of types, 1950 Henkin’s proofs marked the beginning of the new method. Right at the beginning of his paper Henkin recalls Gödel’s results:

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin’s Ph.D. Thesis, 1947. Alonzo Church

Completeness in the theory of types, 1950 Henkin’s proofs marked the beginning of the new method. Right at the beginning of his paper Henkin recalls Gödel’s results:

1 First order calculus is complete, 1930

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin’s Ph.D. Thesis, 1947. Alonzo Church

Completeness in the theory of types, 1950 Henkin’s proofs marked the beginning of the new method. Right at the beginning of his paper Henkin recalls Gödel’s results:

1 First order calculus is complete, 1930 2 Second order calculus is incomplete, 1931

“... no matter what (recursive) set of axioms are chosen, the system will contain a formula which is valid but not a formal theorem.”

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin’s Ph.D. Thesis, 1947. Alonzo Church

Completeness in the theory of types, 1950 Henkin’s proofs marked the beginning of the new method. Right at the beginning of his paper Henkin recalls Gödel’s results:

1 First order calculus is complete, 1930 2 Second order calculus is incomplete, 1931

“... no matter what (recursive) set of axioms are chosen, the system will contain a formula which is valid but not a formal theorem.”

The standard semantics is being determined by structures D = Dαα∈TS, ... where D0 = {T, F} , D1 = ∅, D(0,1) = ℘(D1),etc.

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin’s Ph.D. Thesis, 1947. Alonzo Church

Completeness in the theory of types, 1950 Henkin’s proofs marked the beginning of the new method. Right at the beginning of his paper Henkin recalls Gödel’s results:

1 First order calculus is complete, 1930 2 Second order calculus is incomplete, 1931

“... no matter what (recursive) set of axioms are chosen, the system will contain a formula which is valid but not a formal theorem.”

The standard semantics is being determined by structures D = Dαα∈TS, ... where D0 = {T, F} , D1 = ∅, D(0,1) = ℘(D1),etc. Picture

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Henkin’s Ph.D. Thesis, 1947. Alonzo Church

Completeness in the theory of types, 1950 Henkin’s proofs marked the beginning of the new method. Right at the beginning of his paper Henkin recalls Gödel’s results:

1 First order calculus is complete, 1930 2 Second order calculus is incomplete, 1931

“... no matter what (recursive) set of axioms are chosen, the system will contain a formula which is valid but not a formal theorem.”

The standard semantics is being determined by structures D = Dαα∈TS, ... where D0 = {T, F} , D1 = ∅, D(0,1) = ℘(D1),etc. Picture

Example

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Completeness in the theory of types (1950)

Henkin’s Theorem has the well known form: Theorem 1. If Λ is any consistent set of cwffs there is a general model (in which each domain Dα is denumerable) with respect to which Λ is satisfiable.

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Completeness in the theory of types (1950)

Henkin’s Theorem has the well known form: Theorem 1. If Λ is any consistent set of cwffs there is a general model (in which each domain Dα is denumerable) with respect to which Λ is satisfiable. The proof follows the following steps:

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Completeness in the theory of types (1950)

Henkin’s Theorem has the well known form: Theorem 1. If Λ is any consistent set of cwffs there is a general model (in which each domain Dα is denumerable) with respect to which Λ is satisfiable. The proof follows the following steps:

1 “...to construct a maximal consistent set Γ such that Γ

contains Λ...” Maximal consistent sets describe with enormous precision a possible model for themselves

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Completeness in the theory of types (1950)

Henkin’s Theorem has the well known form: Theorem 1. If Λ is any consistent set of cwffs there is a general model (in which each domain Dα is denumerable) with respect to which Λ is satisfiable. The proof follows the following steps:

1 “...to construct a maximal consistent set Γ such that Γ

contains Λ...” Maximal consistent sets describe with enormous precision a possible model for themselves

2 “Two cwffs Aα and Bα of type α will be called equivalent

iff Γ Aα = Bα” This is a genuine congruence relation

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Completeness in the theory of types (1950)

Henkin’s Theorem has the well known form: Theorem 1. If Λ is any consistent set of cwffs there is a general model (in which each domain Dα is denumerable) with respect to which Λ is satisfiable. The proof follows the following steps:

1 “...to construct a maximal consistent set Γ such that Γ

contains Λ...” Maximal consistent sets describe with enormous precision a possible model for themselves

2 “Two cwffs Aα and Bα of type α will be called equivalent

iff Γ Aα = Bα” This is a genuine congruence relation

3 “We now define by induction on α a frame of domains

{Dα} and simultaneously a one-to-one mapping Φ of equivalence classes onto the domains Dα such that Φ([Aα]) is in Dα”

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Completeness in the theory of types 1950

Concerning completeness of TT, Henkin’s idea was:

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Completeness in the theory of types 1950

Concerning completeness of TT, Henkin’s idea was:

1 To change the semantics

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Completeness in the theory of types 1950

Concerning completeness of TT, Henkin’s idea was:

1 To change the semantics 2 we accept a wider class of models (including standard and

non-standard models)

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Completeness in the theory of types 1950

Concerning completeness of TT, Henkin’s idea was:

1 To change the semantics 2 we accept a wider class of models (including standard and

non-standard models)

3 caution: not so wide as to question comprehension axiom

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Completeness in the theory of types 1950

Concerning completeness of TT, Henkin’s idea was:

1 To change the semantics 2 we accept a wider class of models (including standard and

non-standard models)

3 caution: not so wide as to question comprehension axiom 4 redefine the concept of validity

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Completeness in the theory of types 1950

Concerning completeness of TT, Henkin’s idea was:

1 To change the semantics 2 we accept a wider class of models (including standard and

non-standard models)

3 caution: not so wide as to question comprehension axiom 4 redefine the concept of validity

Picture

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

The Little Mermaid

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Completeness in the theory of types 1950

Shall be associated with a Nominalist position

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Completeness in the theory of types 1950

Shall be associated with a Nominalist position

came after focussing on the definable elements of the full hierarchy of types

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Completeness in the theory of types 1950

Shall be associated with a Nominalist position

came after focussing on the definable elements of the full hierarchy of types In case we start with a countable infinite set as the universe of individuals D,

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Completeness in the theory of types 1950

Shall be associated with a Nominalist position

came after focussing on the definable elements of the full hierarchy of types In case we start with a countable infinite set as the universe of individuals D, the universe of subsets of the universe of individuals will have both: objects with a name and without one.

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Completeness in the theory of types 1950

Shall be associated with a Nominalist position

came after focussing on the definable elements of the full hierarchy of types In case we start with a countable infinite set as the universe of individuals D, the universe of subsets of the universe of individuals will have both: objects with a name and without one. Example of undefinable subset of N definable subsets X1, ..., Xn, ... (1) undefinable set Y = {n ∈ N | n ∈ Xn}

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Completeness in the theory of types 1950

Shall be associated with a Nominalist position

came after focussing on the definable elements of the full hierarchy of types In case we start with a countable infinite set as the universe of individuals D, the universe of subsets of the universe of individuals will have both: objects with a name and without one. Example of undefinable subset of N definable subsets X1, ..., Xn, ... (1) undefinable set Y = {n ∈ N | n ∈ Xn}

the definable types form a proper subset of the standard hierarchy.

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Completeness in the theory of types 1950

Shall be associated with a Nominalist position

came after focussing on the definable elements of the full hierarchy of types In case we start with a countable infinite set as the universe of individuals D, the universe of subsets of the universe of individuals will have both: objects with a name and without one. Example of undefinable subset of N definable subsets X1, ..., Xn, ... (1) undefinable set Y = {n ∈ N | n ∈ Xn}

the definable types form a proper subset of the standard hierarchy. this restricted class itself form a hierarchy

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Completeness in the theory of types 1950

Shall be associated with a Nominalist position

came after focussing on the definable elements of the full hierarchy of types In case we start with a countable infinite set as the universe of individuals D, the universe of subsets of the universe of individuals will have both: objects with a name and without one. Example of undefinable subset of N definable subsets X1, ..., Xn, ... (1) undefinable set Y = {n ∈ N | n ∈ Xn}

the definable types form a proper subset of the standard hierarchy. this restricted class itself form a hierarchy

the proof involves the axiom of choice

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

Completeness in the theory of types 1950

Shall be associated with a Nominalist position

came after focussing on the definable elements of the full hierarchy of types In case we start with a countable infinite set as the universe of individuals D, the universe of subsets of the universe of individuals will have both: objects with a name and without one. Example of undefinable subset of N definable subsets X1, ..., Xn, ... (1) undefinable set Y = {n ∈ N | n ∈ Xn}

the definable types form a proper subset of the standard hierarchy. this restricted class itself form a hierarchy

the proof involves the axiom of choice the rules of the calculus are also involved.

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

The Completeness of the First Order Functional Calculus. 1949

The new method of proof can be used for FOL

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

The Completeness of the First Order Functional Calculus. 1949

The new method of proof can be used for FOL Completeness for first order logic was obtained readapting the method (not the reverse)

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

The Completeness of the First Order Functional Calculus. 1949

The new method of proof can be used for FOL Completeness for first order logic was obtained readapting the method (not the reverse) The proof follows the following steps:

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

The Completeness of the First Order Functional Calculus. 1949

The new method of proof can be used for FOL Completeness for first order logic was obtained readapting the method (not the reverse) The proof follows the following steps: As in the previous proof for type theory a maximal consistent set Γω is build. “It is easy to see that Γω possesses the following properties:

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

The Completeness of the First Order Functional Calculus. 1949

The new method of proof can be used for FOL Completeness for first order logic was obtained readapting the method (not the reverse) The proof follows the following steps: As in the previous proof for type theory a maximal consistent set Γω is build. “It is easy to see that Γω possesses the following properties:

1 Γω is a maximal consistent set of cwffs of Sω

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

The Completeness of the First Order Functional Calculus. 1949

The new method of proof can be used for FOL Completeness for first order logic was obtained readapting the method (not the reverse) The proof follows the following steps: As in the previous proof for type theory a maximal consistent set Γω is build. “It is easy to see that Γω possesses the following properties:

1 Γω is a maximal consistent set of cwffs of Sω 2 If a formula (∃x)A is in Γω then Γω also contains a

formula A‘ ... by substituting some constant for each free occurrence of the variable x”

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

The Completeness of the First Order Functional Calculus. 1949

The new method of proof can be used for FOL Completeness for first order logic was obtained readapting the method (not the reverse) The proof follows the following steps: As in the previous proof for type theory a maximal consistent set Γω is build. “It is easy to see that Γω possesses the following properties:

1 Γω is a maximal consistent set of cwffs of Sω 2 If a formula (∃x)A is in Γω then Γω also contains a

formula A‘ ... by substituting some constant for each free occurrence of the variable x”

An interpretation is build on top of this set using the set

• f individual constants.

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

A Theory of Propositional Types 1963

The theory of propositional types only uses λ and ≡

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

A Theory of Propositional Types 1963

The theory of propositional types only uses λ and ≡ Names and denotations do match: ‘In particular, we shall associate, which each element x of an arbitrary type Dα, a closed formula xn of type α such that (xn)d = x.’

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

A Theory of Propositional Types 1963

The theory of propositional types only uses λ and ≡ Names and denotations do match: ‘In particular, we shall associate, which each element x of an arbitrary type Dα, a closed formula xn of type α such that (xn)d = x.’ Henkin uses a different method to prove completeness

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

A Theory of Propositional Types 1963

The theory of propositional types only uses λ and ≡ Names and denotations do match: ‘In particular, we shall associate, which each element x of an arbitrary type Dα, a closed formula xn of type α such that (xn)d = x.’ Henkin uses a different method to prove completeness The important result from where the completeness theorem easily follows has the amazing form:

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Henkin’s renowned proofs of completeness

A Theory of Propositional Types 1963

The theory of propositional types only uses λ and ≡ Names and denotations do match: ‘In particular, we shall associate, which each element x of an arbitrary type Dα, a closed formula xn of type α such that (xn)d = x.’ Henkin uses a different method to prove completeness The important result from where the completeness theorem easily follows has the amazing form: Lemma ‘Let Aα be any formula and ϕ an assignment. Let A(ϕ)

α

be the formula obtained from Aα by substituting, for each free

• ccurrence of any variable Xβ in Aα, the formula (ϕXβ)n.

Then A(ϕ)

α

≡ (V (Aα, ϕ))n.’

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Completeness

Proof

Theorem If | = A0 then A0 Proof. If A0 is closed then | = A0 implies V (A0, ϕ) = T for any assignment ϕ.

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Completeness

Proof

Theorem If | = A0 then A0 Proof. If A0 is closed then | = A0 implies V (A0, ϕ) = T for any assignment ϕ. Thus the lemma gives A0 ≡ (V (A0, ϕ))n which turns to be A0 ≡ T n, where T n is the name of the truth value true.

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Completeness

Proof

Theorem If | = A0 then A0 Proof. If A0 is closed then | = A0 implies V (A0, ϕ) = T for any assignment ϕ. Thus the lemma gives A0 ≡ (V (A0, ϕ))n which turns to be A0 ≡ T n, where T n is the name of the truth value true. Using the calculus (axiom 2: (A0 ≡ T n) ≡ A0) and the rule of replacement R we obtain the desired result, A0.

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Completeness

Proof

Theorem If | = A0 then A0 Proof. If A0 is closed then | = A0 implies V (A0, ϕ) = T for any assignment ϕ. Thus the lemma gives A0 ≡ (V (A0, ϕ))n which turns to be A0 ≡ T n, where T n is the name of the truth value true. Using the calculus (axiom 2: (A0 ≡ T n) ≡ A0) and the rule of replacement R we obtain the desired result, A0. When A0 is a valid formula but not a sentence, we pass from A0 to the sentence ∀Xγ1...Xγr A0

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Completeness

Proof

Theorem If | = A0 then A0 Proof. If A0 is closed then | = A0 implies V (A0, ϕ) = T for any assignment ϕ. Thus the lemma gives A0 ≡ (V (A0, ϕ))n which turns to be A0 ≡ T n, where T n is the name of the truth value true. Using the calculus (axiom 2: (A0 ≡ T n) ≡ A0) and the rule of replacement R we obtain the desired result, A0. When A0 is a valid formula but not a sentence, we pass from A0 to the sentence ∀Xγ1...Xγr A0 Applying the rules of the calculus, we obtain, A0.

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

The completeness of FOL in Henkin’s course

Herbrand’s Theorem

Berkeley 1977, Henkin’s Metamathematics course for doctorate students,

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

The completeness of FOL in Henkin’s course

Herbrand’s Theorem

Before each class Henkin would give us a text of some 4-5 pages that summarized what was to be addressed in the class

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

The completeness of FOL in Henkin’s course

Herbrand’s Theorem

Before each class Henkin would give us a text of some 4-5 pages that summarized what was to be addressed in the class Henkin was proving completeness of FOL by a different method

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

The completeness of FOL in Henkin’s course

Herbrand’s Theorem

Before each class Henkin would give us a text of some 4-5 pages that summarized what was to be addressed in the class Henkin was proving completeness of FOL by a different method Henkin was using a reduction to sentential logic PL.

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

The completeness of FOL in Henkin’s course

Herbrand’s Theorem

Before each class Henkin would give us a text of some 4-5 pages that summarized what was to be addressed in the class Henkin was proving completeness of FOL by a different method Henkin was using a reduction to sentential logic PL. Theorem (Extended Herbrand’s) For any set of sentences Γ ∪ {A} ⊆ Sent(L) we have: Γ A iff Γ ∪ ∆ PL A where ∆ ⊆ Sent(L) effectively given. L = L∪ C (new individual constants)

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

The completeness of FOL in Henkin’s course

Proof. We take ∆ = ∆1 ∪ ∆2 ∪ ∆3

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

The completeness of FOL in Henkin’s course

Proof. We take ∆ = ∆1 ∪ ∆2 ∪ ∆3

1 ∆1 sentences of form ∃xiB → B(ci,B ) each

∃xiB ∈ Sent(L)

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

The completeness of FOL in Henkin’s course

Proof. We take ∆ = ∆1 ∪ ∆2 ∪ ∆3

1 ∆1 sentences of form ∃xiB → B(ci,B ) each

∃xiB ∈ Sent(L)

2 ∆2 axioms for quantifier

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

The completeness of FOL in Henkin’s course

Proof. We take ∆ = ∆1 ∪ ∆2 ∪ ∆3

1 ∆1 sentences of form ∃xiB → B(ci,B ) each

∃xiB ∈ Sent(L)

2 ∆2 axioms for quantifier 3 ∆3 axioms for the equality symbol

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

The completeness of FOL in Henkin’s course

Proof. We take ∆ = ∆1 ∪ ∆2 ∪ ∆3

1 ∆1 sentences of form ∃xiB → B(ci,B ) each

∃xiB ∈ Sent(L)

2 ∆2 axioms for quantifier 3 ∆3 axioms for the equality symbol

Γ ∪ ∆ PL A implies Γ A is Herbrand’s theorem.

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

The completeness of FOL in Henkin’s course

Proof. We take ∆ = ∆1 ∪ ∆2 ∪ ∆3

1 ∆1 sentences of form ∃xiB → B(ci,B ) each

∃xiB ∈ Sent(L)

2 ∆2 axioms for quantifier 3 ∆3 axioms for the equality symbol

Γ ∪ ∆ PL A implies Γ A is Herbrand’s theorem. Γ A implies Γ ∪ ∆ PL A proven by contraposition.

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

The completeness of FOL in Henkin’s course

Proof. We take ∆ = ∆1 ∪ ∆2 ∪ ∆3

1 ∆1 sentences of form ∃xiB → B(ci,B ) each

∃xiB ∈ Sent(L)

2 ∆2 axioms for quantifier 3 ∆3 axioms for the equality symbol

Γ ∪ ∆ PL A implies Γ A is Herbrand’s theorem. Γ A implies Γ ∪ ∆ PL A proven by contraposition.

1 Γ ∪ ∆ PL A implies Γ ∪ ∆ |

=PL A (completeness PL)

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

The completeness of FOL in Henkin’s course

Proof. We take ∆ = ∆1 ∪ ∆2 ∪ ∆3

1 ∆1 sentences of form ∃xiB → B(ci,B ) each

∃xiB ∈ Sent(L)

2 ∆2 axioms for quantifier 3 ∆3 axioms for the equality symbol

Γ ∪ ∆ PL A implies Γ A is Herbrand’s theorem. Γ A implies Γ ∪ ∆ PL A proven by contraposition.

1 Γ ∪ ∆ PL A implies Γ ∪ ∆ |

=PL A (completeness PL)

2 From propositional interpretation we obtain a first order

structure B such that | =B Γ but | =B A and so, Γ | = A

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

The completeness of FOL in Henkin’s course

Proof. We take ∆ = ∆1 ∪ ∆2 ∪ ∆3

1 ∆1 sentences of form ∃xiB → B(ci,B ) each

∃xiB ∈ Sent(L)

2 ∆2 axioms for quantifier 3 ∆3 axioms for the equality symbol

Γ ∪ ∆ PL A implies Γ A is Herbrand’s theorem. Γ A implies Γ ∪ ∆ PL A proven by contraposition.

1 Γ ∪ ∆ PL A implies Γ ∪ ∆ |

=PL A (completeness PL)

2 From propositional interpretation we obtain a first order

structure B such that | =B Γ but | =B A and so, Γ | = A

3 Thus, Γ A (soundness FOL)

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

The completeness of FOL in Henkin’s course

Predicate logic: Reduction to sentential logic

Note that a proof of the kind described above, includes a completeness proof for first order logic. Theorem (Completeness of FOL) Previous theorem (using completeness of PL) includes completeness of FOL Proof.

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

The completeness of FOL in Henkin’s course

Predicate logic: Reduction to sentential logic

Note that a proof of the kind described above, includes a completeness proof for first order logic. Theorem (Completeness of FOL) Previous theorem (using completeness of PL) includes completeness of FOL Proof. For the theorem shows Γ A implies Γ ∪ ∆ PL A

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

The completeness of FOL in Henkin’s course

Predicate logic: Reduction to sentential logic

Note that a proof of the kind described above, includes a completeness proof for first order logic. Theorem (Completeness of FOL) Previous theorem (using completeness of PL) includes completeness of FOL Proof. For the theorem shows Γ A implies Γ ∪ ∆ PL A On the other hand, using the structure B we show that Γ ∪ ∆ PL A implies Γ | = A

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

The completeness of FOL in Henkin’s course

Predicate logic: Reduction to sentential logic

Note that a proof of the kind described above, includes a completeness proof for first order logic. Theorem (Completeness of FOL) Previous theorem (using completeness of PL) includes completeness of FOL Proof. For the theorem shows Γ A implies Γ ∪ ∆ PL A On the other hand, using the structure B we show that Γ ∪ ∆ PL A implies Γ | = A Therefore, Γ | = A implies Γ A, which is completeness for first order logic.

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Offspring of Henkin’s papers

Extensions of First Order Logic (1996) Manzano, M. CUP

General method to translate logics into MSL logic

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Offspring of Henkin’s papers

Extensions of First Order Logic (1996) Manzano, M. CUP

General method to translate logics into MSL logic I like to credit most of my ideas on translation between logics to two papers of Henkin

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Offspring of Henkin’s papers

Extensions of First Order Logic (1996) Manzano, M. CUP

General method to translate logics into MSL logic I like to credit most of my ideas on translation between logics to two papers of Henkin

1 Completeness in the theory of types of 1950

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Offspring of Henkin’s papers

Extensions of First Order Logic (1996) Manzano, M. CUP

General method to translate logics into MSL logic I like to credit most of my ideas on translation between logics to two papers of Henkin

1 Completeness in the theory of types of 1950 2 and to his paper of 1953, Banishing the rule of

substitution for functional variables

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Offspring of Henkin’s papers

Extensions of First Order Logic (1996) Manzano, M. CUP

General method to translate logics into MSL logic I like to credit most of my ideas on translation between logics to two papers of Henkin

1 Completeness in the theory of types of 1950 2 and to his paper of 1953, Banishing the rule of

substitution for functional variables

From 1: we learn that a modification of the semantics can adapt validities (in the new semantics) to logical theorems

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Offspring of Henkin’s papers

Extensions of First Order Logic (1996) Manzano, M. CUP

General method to translate logics into MSL logic I like to credit most of my ideas on translation between logics to two papers of Henkin

1 Completeness in the theory of types of 1950 2 and to his paper of 1953, Banishing the rule of

substitution for functional variables

From 1: we learn that a modification of the semantics can adapt validities (in the new semantics) to logical theorems However, you do not find in his paper of 1950 translations

• f formulas nor the open appearance of a many-sorted

calculus

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Offspring of Henkin’s papers

Extensions of First Order Logic: Manzano, M. CUP

In connection with higher order logic, the many-sorted calculus was introduced in the paper of 1953.

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Offspring of Henkin’s papers

Extensions of First Order Logic: Manzano, M. CUP

In connection with higher order logic, the many-sorted calculus was introduced in the paper of 1953. Henkin proposes the comprehension axiom as a way to avoid the rule of substitution. The new calculus allows me:

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Offspring of Henkin’s papers

Extensions of First Order Logic: Manzano, M. CUP

In connection with higher order logic, the many-sorted calculus was introduced in the paper of 1953. Henkin proposes the comprehension axiom as a way to avoid the rule of substitution. The new calculus allows me: To prove completeness for HOL with the general semantics, using completeness of MSL.

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Offspring of Henkin’s papers

Extensions of First Order Logic: Manzano, M. CUP

In connection with higher order logic, the many-sorted calculus was introduced in the paper of 1953. Henkin proposes the comprehension axiom as a way to avoid the rule of substitution. The new calculus allows me: To prove completeness for HOL with the general semantics, using completeness of MSL. To isolate calculi between the MSL calculus and HOL, by weakening comprehension.

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

Offspring of Henkin’s papers

Extensions of First Order Logic: Manzano, M. CUP

In connection with higher order logic, the many-sorted calculus was introduced in the paper of 1953. Henkin proposes the comprehension axiom as a way to avoid the rule of substitution. The new calculus allows me: To prove completeness for HOL with the general semantics, using completeness of MSL. To isolate calculi between the MSL calculus and HOL, by weakening comprehension. Picture

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

References

Henkin, L. [1949]. The completeness of the first order functional calculus. The Journal of Symbolic Logic. vol. 14,

• pp. 159-166.

Henkin, L. [1950]. Completeness in the theory of types.The Journal of Symbolic Logic. vol. 15. pp. 81-91. Henkin, L. [1953]. Some Notes on Nominalism. The Journal of Symbolic Logic. vol. 18. pp. 19-29. Henkin, L. [1953]. Some Notes on Nominalism. The Journal of Symbolic Logic. vol. 18. pp. 19-29. Henkin, L., A theory of propositional types, Fundam.

• Math. 52 (1963), pp. 323—344.

Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness

• f FOL in

Henkin’s course References

References

Manzano, M. [2014]. Henkin on Completeness. In The Life and Work of Leon Henkin. Andreka, H. van Benthem, J. Bezhanishvili, N. and Németi. I. [2014] Changing a Semantics: Opportunism or Courage? In The Life and Work of Leon Henkin. Blackburn, P., Huertas, A. Manzano, M. and Jørgensen, K.

• F. [2014]. Henkin and Hybrid Logic. In The Life and Work
• f Leon Henkin.

Manzano, M. Sain, I. and Alonso, E. (eds) [2014]. The Life and Work of Leon Henkin. Essays on His Contributions. Springer Basil.