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

Henkin’s renowned proofs of completeness Henkin on completeness, 2014 (M. Manzano) Henkin on Completeness Following Henkin’s autobiographical paper, The discovery Manzano, M of my completeness proofs, 1996, I point out some of Henkin’s renowned Henkin’s stated influences, especially three of them: proofs of completeness The completeness of FOL in Henkin’s course References

Henkin’s renowned proofs of completeness Henkin on completeness, 2014 (M. Manzano) Henkin on Completeness Following Henkin’s autobiographical paper, The discovery Manzano, M of my completeness proofs, 1996, I point out some of Henkin’s renowned Henkin’s stated influences, especially three of them: proofs of completeness 1 Gödel’s completeness theorem, as well as his article on the The consistency of the axiom of choice (where he builds a completeness of FOL in constructible universe), Henkin’s course References

Henkin’s renowned proofs of completeness Henkin on completeness, 2014 (M. Manzano) Henkin on Completeness Following Henkin’s autobiographical paper, The discovery Manzano, M of my completeness proofs, 1996, I point out some of Henkin’s renowned Henkin’s stated influences, especially three of them: proofs of completeness 1 Gödel’s completeness theorem, as well as his article on the The consistency of the axiom of choice (where he builds a completeness of FOL in constructible universe), Henkin’s 2 Russell’s theory of types and his expository explanation of course the axiom of choice, and References

Henkin’s renowned proofs of completeness Henkin on completeness, 2014 (M. Manzano) Henkin on Completeness Following Henkin’s autobiographical paper, The discovery Manzano, M of my completeness proofs, 1996, I point out some of Henkin’s renowned Henkin’s stated influences, especially three of them: proofs of completeness 1 Gödel’s completeness theorem, as well as his article on the The consistency of the axiom of choice (where he builds a completeness of FOL in constructible universe), Henkin’s 2 Russell’s theory of types and his expository explanation of course the axiom of choice, and References 3 Church’s formulation of the theory of types and the important role played by both

Henkin’s renowned proofs of completeness Henkin on completeness, 2014 (M. Manzano) Henkin on Completeness Following Henkin’s autobiographical paper, The discovery Manzano, M of my completeness proofs, 1996, I point out some of Henkin’s renowned Henkin’s stated influences, especially three of them: proofs of completeness 1 Gödel’s completeness theorem, as well as his article on the The consistency of the axiom of choice (where he builds a completeness of FOL in constructible universe), Henkin’s 2 Russell’s theory of types and his expository explanation of course the axiom of choice, and References 3 Church’s formulation of the theory of types and the important role played by both the lambda operator and 1

Henkin’s renowned proofs of completeness Henkin on completeness, 2014 (M. Manzano) Henkin on Completeness Following Henkin’s autobiographical paper, The discovery Manzano, M of my completeness proofs, 1996, I point out some of Henkin’s renowned Henkin’s stated influences, especially three of them: proofs of completeness 1 Gödel’s completeness theorem, as well as his article on the The consistency of the axiom of choice (where he builds a completeness of FOL in constructible universe), Henkin’s 2 Russell’s theory of types and his expository explanation of course the axiom of choice, and References 3 Church’s formulation of the theory of types and the important role played by both the lambda operator and 1 the description operators in foundational issues. 2

Henkin’s renowned proofs of completeness Henkin on completeness, 2014 (M. Manzano) Henkin on Completeness Following Henkin’s autobiographical paper, The discovery Manzano, M of my completeness proofs, 1996, I point out some of Henkin’s renowned Henkin’s stated influences, especially three of them: proofs of completeness 1 Gödel’s completeness theorem, as well as his article on the The consistency of the axiom of choice (where he builds a completeness of FOL in constructible universe), Henkin’s 2 Russell’s theory of types and his expository explanation of course the axiom of choice, and References 3 Church’s formulation of the theory of types and the important role played by both the lambda operator and 1 the description operators in foundational issues. 2 4 His declared interest in the nameable types

Henkin’s renowned proofs of completeness Henkin’s Ph.D. Thesis, 1947. Alonzo Church Henkin on Completeness in the theory of types, 1950 Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness of FOL in Henkin’s course References

Henkin’s renowned proofs of completeness Henkin’s Ph.D. Thesis, 1947. Alonzo Church Henkin on Completeness in the theory of types, 1950 Completeness Manzano, M Henkin’s proofs marked the beginning of the new method. Henkin’s renowned proofs of completeness The completeness of FOL in Henkin’s course References

Henkin’s renowned proofs of completeness Henkin’s Ph.D. Thesis, 1947. Alonzo Church Henkin on Completeness in the theory of types, 1950 Completeness Manzano, M Henkin’s proofs marked the beginning of the new method. Henkin’s renowned Right at the beginning of his paper Henkin recalls Gödel’s proofs of completeness results: The completeness of FOL in Henkin’s course References

Henkin’s renowned proofs of completeness Henkin’s Ph.D. Thesis, 1947. Alonzo Church Henkin on Completeness in the theory of types, 1950 Completeness Manzano, M Henkin’s proofs marked the beginning of the new method. Henkin’s renowned Right at the beginning of his paper Henkin recalls Gödel’s proofs of completeness results: The 1 First order calculus is complete , 1930 completeness of FOL in Henkin’s course References

Henkin’s renowned proofs of completeness Henkin’s Ph.D. Thesis, 1947. Alonzo Church Henkin on Completeness in the theory of types, 1950 Completeness Manzano, M Henkin’s proofs marked the beginning of the new method. Henkin’s renowned Right at the beginning of his paper Henkin recalls Gödel’s proofs of completeness results: The 1 First order calculus is complete , 1930 completeness of FOL in 2 Second order calculus is incomplete , 1931 Henkin’s course “ ... no matter what (recursive) set of axioms are chosen, References the system will contain a formula which is valid but not a formal theorem.”

Henkin’s renowned proofs of completeness Henkin’s Ph.D. Thesis, 1947. Alonzo Church Henkin on Completeness in the theory of types, 1950 Completeness Manzano, M Henkin’s proofs marked the beginning of the new method. Henkin’s renowned Right at the beginning of his paper Henkin recalls Gödel’s proofs of completeness results: The 1 First order calculus is complete , 1930 completeness of FOL in 2 Second order calculus is incomplete , 1931 Henkin’s course “ ... no matter what (recursive) set of axioms are chosen, References the system will contain a formula which is valid but not a formal theorem.” The standard semantics is being determined by structures D = �� D α � α ∈ TS , ... � where D 0 = { T , F } , D 1 � = ∅ , D ( 0 , 1 ) = ℘ ( D 1 ) , etc.

Henkin’s renowned proofs of completeness Henkin’s Ph.D. Thesis, 1947. Alonzo Church Henkin on Completeness in the theory of types, 1950 Completeness Manzano, M Henkin’s proofs marked the beginning of the new method. Henkin’s renowned Right at the beginning of his paper Henkin recalls Gödel’s proofs of completeness results: The 1 First order calculus is complete , 1930 completeness of FOL in 2 Second order calculus is incomplete , 1931 Henkin’s course “ ... no matter what (recursive) set of axioms are chosen, References the system will contain a formula which is valid but not a formal theorem.” The standard semantics is being determined by structures D = �� D α � α ∈ TS , ... � where D 0 = { T , F } , D 1 � = ∅ , D ( 0 , 1 ) = ℘ ( D 1 ) , etc. Picture

Henkin’s renowned proofs of completeness Henkin’s Ph.D. Thesis, 1947. Alonzo Church Henkin on Completeness in the theory of types, 1950 Completeness Manzano, M Henkin’s proofs marked the beginning of the new method. Henkin’s renowned Right at the beginning of his paper Henkin recalls Gödel’s proofs of completeness results: The 1 First order calculus is complete , 1930 completeness of FOL in 2 Second order calculus is incomplete , 1931 Henkin’s course “ ... no matter what (recursive) set of axioms are chosen, References the system will contain a formula which is valid but not a formal theorem.” The standard semantics is being determined by structures D = �� D α � α ∈ TS , ... � where D 0 = { T , F } , D 1 � = ∅ , D ( 0 , 1 ) = ℘ ( D 1 ) , etc. Picture Example

Henkin’s renowned proofs of completeness Completeness in the theory of types (1950) Henkin’s Theorem has the well known form: Henkin on Completeness Theorem 1 . If Λ is any consistent set of cwffs there is a Manzano, M general model (in which each domain D α is denumerable) Henkin’s renowned with respect to which Λ is satisfiable. proofs of completeness The completeness of 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: Henkin on Completeness Theorem 1 . If Λ is any consistent set of cwffs there is a Manzano, M general model (in which each domain D α is denumerable) Henkin’s renowned with respect to which Λ is satisfiable. proofs of completeness The proof follows the following steps: The completeness of 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: Henkin on Completeness Theorem 1 . If Λ is any consistent set of cwffs there is a Manzano, M general model (in which each domain D α is denumerable) Henkin’s renowned with respect to which Λ is satisfiable. proofs of completeness The proof follows the following steps: The completeness 1 “...to construct a maximal consistent set Γ such that Γ of FOL in contains Λ ...” Henkin’s course Maximal consistent sets describe with enormous precision a References possible model for themselves

Henkin’s renowned proofs of completeness Completeness in the theory of types (1950) Henkin’s Theorem has the well known form: Henkin on Completeness Theorem 1 . If Λ is any consistent set of cwffs there is a Manzano, M general model (in which each domain D α is denumerable) Henkin’s renowned with respect to which Λ is satisfiable. proofs of completeness The proof follows the following steps: The completeness 1 “...to construct a maximal consistent set Γ such that Γ of FOL in contains Λ ...” Henkin’s course Maximal consistent sets describe with enormous precision a References possible model for themselves 2 “Two cwffs A α and B α of type α will be called equivalent iff Γ � A α = B α ” This is a genuine congruence relation

Henkin’s renowned proofs of completeness Completeness in the theory of types (1950) Henkin’s Theorem has the well known form: Henkin on Completeness Theorem 1 . If Λ is any consistent set of cwffs there is a Manzano, M general model (in which each domain D α is denumerable) Henkin’s renowned with respect to which Λ is satisfiable. proofs of completeness The proof follows the following steps: The completeness 1 “...to construct a maximal consistent set Γ such that Γ of FOL in contains Λ ...” Henkin’s course Maximal consistent sets describe with enormous precision a References possible model for themselves 2 “Two cwffs A α and B α of type α will be called equivalent iff Γ � A α = B α ” This is a genuine congruence relation 3 “We now define by induction on α a frame of domains { D α } and simultaneously a one-to-one mapping Φ of equivalence classes onto the domains D α such that Φ ([ A α ]) is in D α ”

Henkin’s renowned proofs of completeness Completeness in the theory of types 1950 Henkin on Completeness Manzano, M Henkin’s renowned Concerning completeness of TT , Henkin’s idea was: proofs of completeness The completeness of FOL in Henkin’s course References

Henkin’s renowned proofs of completeness Completeness in the theory of types 1950 Henkin on Completeness Manzano, M Henkin’s renowned Concerning completeness of TT , Henkin’s idea was: proofs of completeness 1 To change the semantics The completeness of FOL in Henkin’s course References

Henkin’s renowned proofs of completeness Completeness in the theory of types 1950 Henkin on Completeness Manzano, M Henkin’s renowned Concerning completeness of TT , Henkin’s idea was: proofs of completeness 1 To change the semantics The 2 we accept a wider class of models (including standard and completeness of FOL in non-standard models) Henkin’s course References

Henkin’s renowned proofs of completeness Completeness in the theory of types 1950 Henkin on Completeness Manzano, M Henkin’s renowned Concerning completeness of TT , Henkin’s idea was: proofs of completeness 1 To change the semantics The 2 we accept a wider class of models (including standard and completeness of FOL in non-standard models) Henkin’s course 3 caution: not so wide as to question comprehension axiom References

Henkin’s renowned proofs of completeness Completeness in the theory of types 1950 Henkin on Completeness Manzano, M Henkin’s renowned Concerning completeness of TT , Henkin’s idea was: proofs of completeness 1 To change the semantics The 2 we accept a wider class of models (including standard and completeness of FOL in non-standard models) Henkin’s course 3 caution: not so wide as to question comprehension axiom References 4 redefine the concept of validity

Henkin’s renowned proofs of completeness Completeness in the theory of types 1950 Henkin on Completeness Manzano, M Henkin’s renowned Concerning completeness of TT , Henkin’s idea was: proofs of completeness 1 To change the semantics The 2 we accept a wider class of models (including standard and completeness of FOL in non-standard models) Henkin’s course 3 caution: not so wide as to question comprehension axiom References 4 redefine the concept of validity Picture

Henkin’s renowned proofs of completeness The Little Mermaid Henkin on Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness of FOL in Henkin’s course References

Henkin’s renowned proofs of completeness Completeness in the theory of types 1950 Henkin on Shall be associated with a Nominalist position Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness of FOL in Henkin’s course References

Henkin’s renowned proofs of completeness Completeness in the theory of types 1950 Henkin on Shall be associated with a Nominalist position Completeness came after focussing on the definable elements of the full Manzano, M hierarchy of types Henkin’s renowned proofs of completeness The completeness of FOL in Henkin’s course References

Henkin’s renowned proofs of completeness Completeness in the theory of types 1950 Henkin on Shall be associated with a Nominalist position Completeness came after focussing on the definable elements of the full Manzano, M hierarchy of types Henkin’s In case we start with a countable infinite set as the renowned proofs of universe of individuals D, completeness The completeness of FOL in Henkin’s course References

Henkin’s renowned proofs of completeness Completeness in the theory of types 1950 Henkin on Shall be associated with a Nominalist position Completeness came after focussing on the definable elements of the full Manzano, M hierarchy of types Henkin’s In case we start with a countable infinite set as the renowned proofs of universe of individuals D, completeness the universe of subsets of the universe of individuals will The have both: objects with a name and without one. completeness of FOL in Henkin’s course References

Henkin’s renowned proofs of completeness Completeness in the theory of types 1950 Henkin on Shall be associated with a Nominalist position Completeness came after focussing on the definable elements of the full Manzano, M hierarchy of types Henkin’s In case we start with a countable infinite set as the renowned proofs of universe of individuals D, completeness the universe of subsets of the universe of individuals will The have both: objects with a name and without one. completeness of FOL in Example of undefinable subset of N Henkin’s course definable subsets X 1 , ..., X n , ... (1) References undefinable set Y = { n ∈ N | n �∈ X n }

Henkin’s renowned proofs of completeness Completeness in the theory of types 1950 Henkin on Shall be associated with a Nominalist position Completeness came after focussing on the definable elements of the full Manzano, M hierarchy of types Henkin’s In case we start with a countable infinite set as the renowned proofs of universe of individuals D, completeness the universe of subsets of the universe of individuals will The have both: objects with a name and without one. completeness of FOL in Example of undefinable subset of N Henkin’s course definable subsets X 1 , ..., X n , ... (1) References undefinable set Y = { n ∈ N | n �∈ X n } the definable types form a proper subset of the standard hierarchy.

Henkin’s renowned proofs of completeness Completeness in the theory of types 1950 Henkin on Shall be associated with a Nominalist position Completeness came after focussing on the definable elements of the full Manzano, M hierarchy of types Henkin’s In case we start with a countable infinite set as the renowned proofs of universe of individuals D, completeness the universe of subsets of the universe of individuals will The have both: objects with a name and without one. completeness of FOL in Example of undefinable subset of N Henkin’s course definable subsets X 1 , ..., X n , ... (1) References undefinable set Y = { n ∈ N | n �∈ X n } the definable types form a proper subset of the standard hierarchy. this restricted class itself form a hierarchy

Henkin’s renowned proofs of completeness Completeness in the theory of types 1950 Henkin on Shall be associated with a Nominalist position Completeness came after focussing on the definable elements of the full Manzano, M hierarchy of types Henkin’s In case we start with a countable infinite set as the renowned proofs of universe of individuals D, completeness the universe of subsets of the universe of individuals will The have both: objects with a name and without one. completeness of FOL in Example of undefinable subset of N Henkin’s course definable subsets X 1 , ..., X n , ... (1) References undefinable set Y = { n ∈ N | n �∈ X n } 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’s renowned proofs of completeness Completeness in the theory of types 1950 Henkin on Shall be associated with a Nominalist position Completeness came after focussing on the definable elements of the full Manzano, M hierarchy of types Henkin’s In case we start with a countable infinite set as the renowned proofs of universe of individuals D, completeness the universe of subsets of the universe of individuals will The have both: objects with a name and without one. completeness of FOL in Example of undefinable subset of N Henkin’s course definable subsets X 1 , ..., X n , ... (1) References undefinable set Y = { n ∈ N | n �∈ X n } 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’s renowned proofs of completeness The Completeness of the First Order Functional Calculus. 1949 Henkin on The new method of proof can be used for FOL Completeness Manzano, M Henkin’s renowned proofs of completeness The completeness of FOL in Henkin’s course References

Henkin’s renowned proofs of completeness The Completeness of the First Order Functional Calculus. 1949 Henkin on The new method of proof can be used for FOL Completeness Completeness for first order logic was obtained readapting Manzano, M the method (not the reverse) Henkin’s renowned proofs of completeness The completeness of FOL in Henkin’s course References

Henkin’s renowned proofs of completeness The Completeness of the First Order Functional Calculus. 1949 Henkin on The new method of proof can be used for FOL Completeness Completeness for first order logic was obtained readapting Manzano, M the method (not the reverse) Henkin’s renowned proofs of The proof follows the following steps: completeness The completeness of FOL in Henkin’s course References

Henkin’s renowned proofs of completeness The Completeness of the First Order Functional Calculus. 1949 Henkin on The new method of proof can be used for FOL Completeness Completeness for first order logic was obtained readapting Manzano, M the method (not the reverse) Henkin’s renowned proofs of The proof follows the following steps: completeness As in the previous proof for type theory a maximal The completeness consistent set Γ ω is build. of FOL in Henkin’s “ It is easy to see that Γ ω possesses the following course properties: References

Henkin’s renowned proofs of completeness The Completeness of the First Order Functional Calculus. 1949 Henkin on The new method of proof can be used for FOL Completeness Completeness for first order logic was obtained readapting Manzano, M the method (not the reverse) Henkin’s renowned proofs of The proof follows the following steps: completeness As in the previous proof for type theory a maximal The completeness consistent set Γ ω is build. of FOL in Henkin’s “ It is easy to see that Γ ω possesses the following course properties: References 1 Γ ω is a maximal consistent set of cwffs of S ω

Henkin’s renowned proofs of completeness The Completeness of the First Order Functional Calculus. 1949 Henkin on The new method of proof can be used for FOL Completeness Completeness for first order logic was obtained readapting Manzano, M the method (not the reverse) Henkin’s renowned proofs of The proof follows the following steps: completeness As in the previous proof for type theory a maximal The completeness consistent set Γ ω is build. of FOL in Henkin’s “ It is easy to see that Γ ω possesses the following course properties: References 1 Γ ω is a maximal consistent set of cwffs of S ω 2 If a formula ( ∃ x ) A is in Γ ω then Γ ω also contains a formula A ‘ � ... by substituting some constant for each free occurrence of the variable x”

Henkin’s renowned proofs of completeness The Completeness of the First Order Functional Calculus. 1949 Henkin on The new method of proof can be used for FOL Completeness Completeness for first order logic was obtained readapting Manzano, M the method (not the reverse) Henkin’s renowned proofs of The proof follows the following steps: completeness As in the previous proof for type theory a maximal The completeness consistent set Γ ω is build. of FOL in Henkin’s “ It is easy to see that Γ ω possesses the following course properties: References 1 Γ ω is a maximal consistent set of cwffs of S ω 2 If a formula ( ∃ x ) A is in Γ ω then Γ ω 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 of individual constants.

Henkin’s renowned proofs of completeness A Theory of Propositional Types 1963 Henkin on Completeness The theory of propositional types only uses λ and ≡ Manzano, M Henkin’s renowned proofs of completeness The completeness of FOL in Henkin’s course References

Henkin’s renowned proofs of completeness A Theory of Propositional Types 1963 Henkin on Completeness The theory of propositional types only uses λ and ≡ Manzano, M Names and denotations do match: ‘In particular, we shall Henkin’s associate, which each element x of an arbitrary type D α , a renowned closed formula x n of type α such that ( x n ) d = x .’ proofs of completeness The completeness of FOL in Henkin’s course References

Henkin’s renowned proofs of completeness A Theory of Propositional Types 1963 Henkin on Completeness The theory of propositional types only uses λ and ≡ Manzano, M Names and denotations do match: ‘In particular, we shall Henkin’s associate, which each element x of an arbitrary type D α , a renowned closed formula x n of type α such that ( x n ) d = x .’ proofs of completeness The Henkin uses a different method to prove completeness completeness of FOL in Henkin’s course References

Henkin’s renowned proofs of completeness A Theory of Propositional Types 1963 Henkin on Completeness The theory of propositional types only uses λ and ≡ Manzano, M Names and denotations do match: ‘In particular, we shall Henkin’s associate, which each element x of an arbitrary type D α , a renowned closed formula x n of type α such that ( x n ) d = x .’ proofs of completeness The Henkin uses a different method to prove completeness completeness of FOL in The important result from where the completeness Henkin’s course theorem easily follows has the amazing form: References

Henkin’s renowned proofs of completeness A Theory of Propositional Types 1963 Henkin on Completeness The theory of propositional types only uses λ and ≡ Manzano, M Names and denotations do match: ‘In particular, we shall Henkin’s associate, which each element x of an arbitrary type D α , a renowned closed formula x n of type α such that ( x n ) d = x .’ proofs of completeness The Henkin uses a different method to prove completeness completeness of FOL in The important result from where the completeness Henkin’s course theorem easily follows has the amazing form: References Lemma ‘Let A α be any formula and ϕ an assignment. Let A ( ϕ ) be the α formula obtained from A α by substituting, for each free occurrence of any variable X β in A α , the formula ( ϕ X β ) n . Then � A ( ϕ ) ≡ ( V ( A α , ϕ )) n .’ α

Completeness Proof Henkin on Theorem Completeness If | = A 0 then � A 0 Manzano, M Henkin’s Proof. renowned proofs of completeness If A 0 is closed then | = A 0 implies V ( A 0 , ϕ ) = T for any The completeness assignment ϕ . of FOL in Henkin’s course References

Completeness Proof Henkin on Theorem Completeness If | = A 0 then � A 0 Manzano, M Henkin’s Proof. renowned proofs of completeness If A 0 is closed then | = A 0 implies V ( A 0 , ϕ ) = T for any The completeness assignment ϕ . of FOL in Thus the lemma gives � A 0 ≡ ( V ( A 0 , ϕ )) n which turns to Henkin’s course be � A 0 ≡ T n , where T n is the name of the truth value References true.

Completeness Proof Henkin on Theorem Completeness If | = A 0 then � A 0 Manzano, M Henkin’s Proof. renowned proofs of completeness If A 0 is closed then | = A 0 implies V ( A 0 , ϕ ) = T for any The completeness assignment ϕ . of FOL in Thus the lemma gives � A 0 ≡ ( V ( A 0 , ϕ )) n which turns to Henkin’s course be � A 0 ≡ T n , where T n is the name of the truth value References true. Using the calculus (axiom 2: ( A 0 ≡ T n ) ≡ A 0 ) and the rule of replacement R we obtain the desired result, � A 0 .

Completeness Proof Henkin on Theorem Completeness If | = A 0 then � A 0 Manzano, M Henkin’s Proof. renowned proofs of completeness If A 0 is closed then | = A 0 implies V ( A 0 , ϕ ) = T for any The completeness assignment ϕ . of FOL in Thus the lemma gives � A 0 ≡ ( V ( A 0 , ϕ )) n which turns to Henkin’s course be � A 0 ≡ T n , where T n is the name of the truth value References true. Using the calculus (axiom 2: ( A 0 ≡ T n ) ≡ A 0 ) and the rule of replacement R we obtain the desired result, � A 0 . When A 0 is a valid formula but not a sentence, we pass from A 0 to the sentence ∀ X γ 1 ... X γ r A 0

Completeness Proof Henkin on Theorem Completeness If | = A 0 then � A 0 Manzano, M Henkin’s Proof. renowned proofs of completeness If A 0 is closed then | = A 0 implies V ( A 0 , ϕ ) = T for any The completeness assignment ϕ . of FOL in Thus the lemma gives � A 0 ≡ ( V ( A 0 , ϕ )) n which turns to Henkin’s course be � A 0 ≡ T n , where T n is the name of the truth value References true. Using the calculus (axiom 2: ( A 0 ≡ T n ) ≡ A 0 ) and the rule of replacement R we obtain the desired result, � A 0 . When A 0 is a valid formula but not a sentence, we pass from A 0 to the sentence ∀ X γ 1 ... X γ r A 0 Applying the rules of the calculus, we obtain, � A 0 .

The completeness of FOL in Henkin’s course Herbrand’s Theorem Henkin on Completeness Manzano, M Berkeley 1977, Henkin’s Metamathematics course for Henkin’s doctorate students, renowned proofs of completeness The completeness of FOL in Henkin’s course References

The completeness of FOL in Henkin’s course Herbrand’s Theorem Henkin on Completeness Before each class Henkin would give us a text of some 4-5 Manzano, M pages that summarized what was to be addressed in the class Henkin’s renowned proofs of completeness The completeness of FOL in Henkin’s course References

The completeness of FOL in Henkin’s course Herbrand’s Theorem Henkin on Completeness Before each class Henkin would give us a text of some 4-5 Manzano, M pages that summarized what was to be addressed in the class Henkin’s renowned proofs of Henkin was proving completeness of FOL by a different completeness method The completeness of FOL in Henkin’s course References

The completeness of FOL in Henkin’s course Herbrand’s Theorem Henkin on Completeness Before each class Henkin would give us a text of some 4-5 Manzano, M pages that summarized what was to be addressed in the class Henkin’s renowned proofs of Henkin was proving completeness of FOL by a different completeness method The completeness Henkin was using a reduction to sentential logic PL . of FOL in Henkin’s course References

The completeness of FOL in Henkin’s course Herbrand’s Theorem Henkin on Completeness Before each class Henkin would give us a text of some 4-5 Manzano, M pages that summarized what was to be addressed in the class Henkin’s renowned proofs of Henkin was proving completeness of FOL by a different completeness method The completeness Henkin was using a reduction to sentential logic PL . of FOL in Henkin’s course Theorem (Extended Herbrand’s) References 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)

The completeness of FOL in Henkin’s course Henkin on Completeness Proof. Manzano, M We take ∆ = ∆ 1 ∪ ∆ 2 ∪ ∆ 3 Henkin’s renowned proofs of completeness The completeness of FOL in Henkin’s course References

The completeness of FOL in Henkin’s course Henkin on Completeness Proof. Manzano, M We take ∆ = ∆ 1 ∪ ∆ 2 ∪ ∆ 3 Henkin’s renowned 1 ∆ 1 sentences of form ∃ x i B → B ( c i , B ) each proofs of completeness ∃ x i B ∈ Sent ( L � ) The completeness of FOL in Henkin’s course References

The completeness of FOL in Henkin’s course Henkin on Completeness Proof. Manzano, M We take ∆ = ∆ 1 ∪ ∆ 2 ∪ ∆ 3 Henkin’s renowned 1 ∆ 1 sentences of form ∃ x i B → B ( c i , B ) each proofs of completeness ∃ x i B ∈ Sent ( L � ) The 2 ∆ 2 axioms for quantifier completeness of FOL in Henkin’s course References

The completeness of FOL in Henkin’s course Henkin on Completeness Proof. Manzano, M We take ∆ = ∆ 1 ∪ ∆ 2 ∪ ∆ 3 Henkin’s renowned 1 ∆ 1 sentences of form ∃ x i B → B ( c i , B ) each proofs of completeness ∃ x i B ∈ Sent ( L � ) The 2 ∆ 2 axioms for quantifier completeness 3 ∆ 3 axioms for the equality symbol of FOL in Henkin’s course References

The completeness of FOL in Henkin’s course Henkin on Completeness Proof. Manzano, M We take ∆ = ∆ 1 ∪ ∆ 2 ∪ ∆ 3 Henkin’s renowned 1 ∆ 1 sentences of form ∃ x i B → B ( c i , B ) each proofs of completeness ∃ x i B ∈ Sent ( L � ) The 2 ∆ 2 axioms for quantifier completeness 3 ∆ 3 axioms for the equality symbol of FOL in Henkin’s course Γ ∪ ∆ � PL A implies Γ � A is Herbrand’s theorem. References

The completeness of FOL in Henkin’s course Henkin on Completeness Proof. Manzano, M We take ∆ = ∆ 1 ∪ ∆ 2 ∪ ∆ 3 Henkin’s renowned 1 ∆ 1 sentences of form ∃ x i B → B ( c i , B ) each proofs of completeness ∃ x i B ∈ Sent ( L � ) The 2 ∆ 2 axioms for quantifier completeness 3 ∆ 3 axioms for the equality symbol of FOL in Henkin’s course Γ ∪ ∆ � PL A implies Γ � A is Herbrand’s theorem. References Γ � A implies Γ ∪ ∆ � PL A proven by contraposition.

The completeness of FOL in Henkin’s course Henkin on Completeness Proof. Manzano, M We take ∆ = ∆ 1 ∪ ∆ 2 ∪ ∆ 3 Henkin’s renowned 1 ∆ 1 sentences of form ∃ x i B → B ( c i , B ) each proofs of completeness ∃ x i B ∈ Sent ( L � ) The 2 ∆ 2 axioms for quantifier completeness 3 ∆ 3 axioms for the equality symbol of FOL in Henkin’s course Γ ∪ ∆ � PL A implies Γ � A is Herbrand’s theorem. References Γ � A implies Γ ∪ ∆ � PL A proven by contraposition. 1 Γ ∪ ∆ �� PL A implies Γ ∪ ∆ �| = PL A (completeness PL )

The completeness of FOL in Henkin’s course Henkin on Completeness Proof. Manzano, M We take ∆ = ∆ 1 ∪ ∆ 2 ∪ ∆ 3 Henkin’s renowned 1 ∆ 1 sentences of form ∃ x i B → B ( c i , B ) each proofs of completeness ∃ x i B ∈ Sent ( L � ) The 2 ∆ 2 axioms for quantifier completeness 3 ∆ 3 axioms for the equality symbol of FOL in Henkin’s course Γ ∪ ∆ � PL A implies Γ � A is Herbrand’s theorem. References 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

The completeness of FOL in Henkin’s course Henkin on Completeness Proof. Manzano, M We take ∆ = ∆ 1 ∪ ∆ 2 ∪ ∆ 3 Henkin’s renowned 1 ∆ 1 sentences of form ∃ x i B → B ( c i , B ) each proofs of completeness ∃ x i B ∈ Sent ( L � ) The 2 ∆ 2 axioms for quantifier completeness 3 ∆ 3 axioms for the equality symbol of FOL in Henkin’s course Γ ∪ ∆ � PL A implies Γ � A is Herbrand’s theorem. References 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 )

The completeness of FOL in Henkin’s course Predicate logic: Reduction to sentential logic Henkin on Note that a proof of the kind described above, includes a Completeness completeness proof for first order logic. Manzano, M Theorem (Completeness of FOL) Henkin’s renowned proofs of Previous theorem (using completeness of PL) includes completeness completeness of FOL The completeness of FOL in Henkin’s Proof. course References

The completeness of FOL in Henkin’s course Predicate logic: Reduction to sentential logic Henkin on Note that a proof of the kind described above, includes a Completeness completeness proof for first order logic. Manzano, M Theorem (Completeness of FOL) Henkin’s renowned proofs of Previous theorem (using completeness of PL) includes completeness completeness of FOL The completeness of FOL in Henkin’s Proof. course References For the theorem shows Γ �� A implies Γ ∪ ∆ �� PL A

The completeness of FOL in Henkin’s course Predicate logic: Reduction to sentential logic Henkin on Note that a proof of the kind described above, includes a Completeness completeness proof for first order logic. Manzano, M Theorem (Completeness of FOL) Henkin’s renowned proofs of Previous theorem (using completeness of PL) includes completeness completeness of FOL The completeness of FOL in Henkin’s Proof. course References 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

The completeness of FOL in Henkin’s course Predicate logic: Reduction to sentential logic Henkin on Note that a proof of the kind described above, includes a Completeness completeness proof for first order logic. Manzano, M Theorem (Completeness of FOL) Henkin’s renowned proofs of Previous theorem (using completeness of PL) includes completeness completeness of FOL The completeness of FOL in Henkin’s Proof. course References 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.

Offspring of Henkin’s papers Extensions of First Order Logic (1996) Manzano, M. CUP Henkin on Completeness Manzano, M General method to translate logics into MSL logic Henkin’s renowned proofs of completeness The completeness of FOL in Henkin’s course References

Offspring of Henkin’s papers Extensions of First Order Logic (1996) Manzano, M. CUP Henkin on Completeness Manzano, M General method to translate logics into MSL logic Henkin’s I like to credit most of my ideas on translation between renowned proofs of logics to two papers of Henkin completeness The completeness of FOL in Henkin’s course References

Offspring of Henkin’s papers Extensions of First Order Logic (1996) Manzano, M. CUP Henkin on Completeness Manzano, M General method to translate logics into MSL logic Henkin’s I like to credit most of my ideas on translation between renowned proofs of logics to two papers of Henkin completeness The 1 Completeness in the theory of types of 1950 completeness of FOL in Henkin’s course References

Offspring of Henkin’s papers Extensions of First Order Logic (1996) Manzano, M. CUP Henkin on Completeness Manzano, M General method to translate logics into MSL logic Henkin’s I like to credit most of my ideas on translation between renowned proofs of logics to two papers of Henkin completeness The 1 Completeness in the theory of types of 1950 completeness 2 and to his paper of 1953, Banishing the rule of of FOL in Henkin’s substitution for functional variables course References

Offspring of Henkin’s papers Extensions of First Order Logic (1996) Manzano, M. CUP Henkin on Completeness Manzano, M General method to translate logics into MSL logic Henkin’s I like to credit most of my ideas on translation between renowned proofs of logics to two papers of Henkin completeness The 1 Completeness in the theory of types of 1950 completeness 2 and to his paper of 1953, Banishing the rule of of FOL in Henkin’s substitution for functional variables course References From 1: we learn that a modification of the semantics can adapt validities (in the new semantics) to logical theorems

Offspring of Henkin’s papers Extensions of First Order Logic (1996) Manzano, M. CUP Henkin on Completeness Manzano, M General method to translate logics into MSL logic Henkin’s I like to credit most of my ideas on translation between renowned proofs of logics to two papers of Henkin completeness The 1 Completeness in the theory of types of 1950 completeness 2 and to his paper of 1953, Banishing the rule of of FOL in Henkin’s substitution for functional variables course References 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 of formulas nor the open appearance of a many-sorted calculus

Offspring of Henkin’s papers Extensions of First Order Logic: Manzano, M. CUP Henkin on Completeness Manzano, M In connection with higher order logic, the many-sorted calculus was introduced in the paper of 1953. Henkin’s renowned proofs of completeness The completeness of FOL in Henkin’s course References

Offspring of Henkin’s papers Extensions of First Order Logic: Manzano, M. CUP Henkin on Completeness Manzano, M In connection with higher order logic, the many-sorted calculus was introduced in the paper of 1953. Henkin’s renowned proofs of completeness Henkin proposes the comprehension axiom as a way to The avoid the rule of substitution. The new calculus allows me: completeness of FOL in Henkin’s course References

Offspring of Henkin’s papers Extensions of First Order Logic: Manzano, M. CUP Henkin on Completeness Manzano, M In connection with higher order logic, the many-sorted calculus was introduced in the paper of 1953. Henkin’s renowned proofs of completeness Henkin proposes the comprehension axiom as a way to The avoid the rule of substitution. The new calculus allows me: completeness of FOL in Henkin’s course To prove completeness for HOL with the general References semantics , using completeness of MSL .