An Almost Constructive Proof of Classical First-Order Completeness First Bachelor Seminar Talk Dominik Wehr Advisors: Dominik Kirst, Yannick Forster Saarland University 14th December 2018
Syntax, Deduction, and Semantics Model Existence Completeness Outro Partial History of First-Order Completeness First formal statement by Hilbert and Ackermann 1 1928 odel 2 1929 First proven by G¨ Greatly simplified by Henkin 3 1947 . . . Constructive analysis by Herbelin and Ilik 4 2016 1Ackermann and Hilbert. “Grundz¨ uge der theoretischen Logik” 2G¨ odel. “¨ Uber die Vollst¨ andigkeit des Logikkalk¨ uls” 3Henkin. “The Completeness of the First-Order Functional Calculus” 4Herbelin and Ilik. An analysis of the constructive content of Henkin’s proof of G¨ odel’s completeness theorem 2 / 19
Syntax, Deduction, and Semantics Model Existence Completeness Outro Definition (Syntax) s, t : T ::= e | f t | x | p x, p : N ˙ ˙ ψ | ˙ ϕ, ψ : F ::= ⊥ | P s t | ϕ → ∀ x.ϕ x : N ˙ ˙ ¬ ˙ ϕ ˙ ¬ ϕ := ϕ → ˙ ˙ ⊥ ∃ x.ϕ := ˙ ∀ x. ˙ ¬ ϕ ∨ ψ := ˙ ¬ ϕ → ˙ ψ 3 / 19
Syntax, Deduction, and Semantics Model Existence Completeness Outro Definition (Deduction system) ϕ ∈ A ϕ :: A ⊢ ψ Ctx II A ⊢ ϕ A ⊢ ϕ → ˙ ψ A ⊢ ϕ → ˙ ψ A ⊢ ϕ A ⊢ ˙ ¬ ˙ ¬ ϕ IE DN A ⊢ ψ A ⊢ ϕ A ⊢ ϕ x p fresh for ϕ and A p AllI A ⊢ ˙ ∀ x.ϕ A ⊢ ˙ ∀ x.ϕ t closed AllE A ⊢ ϕ x t 4 / 19
Syntax, Deduction, and Semantics Model Existence Completeness Outro Definition (Interpretation) An interpretation I on a domain D consists of: e I : D f I : D → D · I : N → D P I : D → D → P Definition (Evaluation) Given ρ : N → D , we extend I to t I ,ρ : D and ρ � I ϕ : P : ρ � I ˙ ⊥ = ⊥ ρ � I P s t = P I s I ,ρ t I ,ρ ρ � I ϕ → ˙ ψ = ρ � I ϕ → ρ � I ψ ρ � I ˙ ∀ x.ϕ = ∀ d : D . ρ [ x �→ d ] � I ϕ A � ϕ := ∀ I ρ. ρ � I A → ρ � I ϕ 5 / 19
Syntax, Deduction, and Semantics Model Existence Completeness Outro Definition (Theories) We extend the previous notions to theories T : F → P : T � ϕ := ∀ I ρ. ρ � I T → ρ � I ϕ T ⊢ ϕ := A ⊢ ϕ ∃ A. A ⊆ T ∧ A ⊢ ϕ Definition (Consistency) We call T : F → P consistent if T � ˙ ⊥ maximally consistent if T � ˙ ⊥ and ϕ ∈ T if T ∪ { ϕ } � ˙ ⊥ 6 / 19
Syntax, Deduction, and Semantics Model Existence Completeness Outro Proof Outline Model Existence ? � ? T T consistent A � ϕ → A ⊢ ϕ 7 / 19
Syntax, Deduction, and Semantics Model Existence Completeness Outro Quantifier-free Model Existence Lindenbaum Herbrandt T Ω � Ω consistent maximally model for T consistent closed 8 / 19
Syntax, Deduction, and Semantics Model Existence Completeness Outro Definition Given a consistent T , we fix an enumeration E F and define � Ω n ∪ {E F n } Ω n ∪ {E F n } consistent Ω 0 = T Ω n +1 = Ω n otherwise � Ω := Ω n Lemma (Lindenbaum) Ω is a maximally consistent extension of T . 9 / 19
Syntax, Deduction, and Semantics Model Existence Completeness Outro Quantifier-free Model Existence Lindenbaum Herbrandt T Ω � Ω consistent maximally model for T consistent closed 10 / 19
Syntax, Deduction, and Semantics Model Existence Completeness Outro Definition (Herbrandt model) Given a theory Ω we define its Herbrandt model on closed terms T c : t Ω ,ρ := t P Ω s t := P s t ∈ Ω Lemma (Model correctness) Let Ω be maximally consistent and ϕ be closed and quantifier-free, then � Ω ϕ ↔ ϕ ∈ Ω Corollary (Model existence) Let T be consistent and closed, then � Ω T . 11 / 19
Syntax, Deduction, and Semantics Model Existence Completeness Outro Lemma (Maximally consistent membership) Let Ω be maximally consistent. Then ϕ ∈ Ω ↔ Ω ⊢ ϕ . Lemma (Model correctness) Let Ω be maximally consistent and ϕ be closed and quantifier-free, then � Ω ϕ ↔ ϕ ∈ Ω Proof. Proof per induction on the size of ϕ . There are three cases: P s t ∈ Ω ↔ P s t ∈ Ω ⊥ ↔ Ω ⊢ ˙ ⊥ (Ω ⊢ ϕ → Ω ⊢ ψ ) ↔ Ω ⊢ ϕ → ˙ ψ 12 / 19
Syntax, Deduction, and Semantics Model Existence Completeness Outro First-Order Model Existence Lindenbaum Henkin Herbrandt T H � Ω Ω consistent consistent model for T parameter-free not closed closed 13 / 19
Syntax, Deduction, and Semantics Model Existence Completeness Outro Definition (Henkin axioms) Let T be consistent and parameter-free. Then define H as follows: ˙ ˙ if E F n = ˙ H n ∪ { ϕ x p → ∀ x.ϕ } ∀ x.ϕ H 0 = T H n +1 = with p fresh in H n H n otherwise � H := H n Lemma (Henkin correctness) H is consistent t ) ↔ H ⊢ ˙ ( ∀ t : T c . H ⊢ ϕ x ∀ x. ϕ 14 / 19
Syntax, Deduction, and Semantics Model Existence Completeness Outro Proof Outline Model Existence � Ω T T consistent parameter-free closed A � ϕ → A ⊢ ϕ 15 / 19
Syntax, Deduction, and Semantics Model Existence Completeness Outro Theorem (Strong quasi-completeness) Let both T and ϕ be closed and parameter-free. T � ϕ → ¬¬T ⊢ ϕ Theorem (Refutation completeness) ¬ ϕ } ⊢ ˙ T ⊢ ϕ ↔ T ∪ { ˙ ⊥ 16 / 19
Syntax, Deduction, and Semantics Model Existence Completeness Outro Theorem (Strong quasi-completeness) Let both T and ϕ be closed and parameter-free. T � ϕ → ¬¬T ⊢ ϕ Definition (Stability of ⊢ ) ¬¬ A ⊢ ϕ → A ⊢ ϕ Theorem (Completeness) Assume the stability of ⊢ . Let A and ϕ be closed and parameter-free. A � ϕ → A ⊢ ϕ 17 / 19
Syntax, Deduction, and Semantics Model Existence Completeness Outro Future Work Establish Soundness and use AutoSubst Completeness of an intuitionistic Gentzen system Cut free completeness of intuitionistic ND Multiple possibilities: Cut elimination for classical ND Game semantics ... 18 / 19
Syntax, Deduction, and Semantics Model Existence Completeness Outro References Hugo Herbelin and Danko Ilik. An analysis of the constructive content of Henkin’s proof of G¨ odel’s completeness theorem . Draft. 2016. George F. Schumm. A Henkin-style completeness proof for the pure implicational calculus. Vol. 16. 3. Duke University Press, July 1975, pp. 402–404. Melvin Fitting. First-Order Logic and Automated Theorem Proving . Springer, 1996. Yannik Forster, Dominik Kirst, and Gert Smolka. On Synthetic Undecidability in Coq, with an Application to the Entscheidungsproblem . CPP’19. 2018. 19 / 19
Recommend
More recommend
