A Formally Verified Abstract Account of Gödel’s Incompleteness Theorems Andrei Popescu Dmitriy Traytel e HOL l l e b a ∀ s I = α λ β →
Gödel’s Incompleteness Theorems 1931
Gödel’s Incompleteness Theorems 1931 Fix a consistent logical theory that - contains enough arithmetic, - can itself be arithmetized.
Gödel’s Incompleteness Theorems 1931 Fix a consistent logical theory that - contains enough arithmetic, - can itself be arithmetized. There are sentences that the theory cannot decide (i.e., neither prove nor disprove).
Gödel’s Incompleteness Theorems 1931 Fix a consistent logical theory that - contains enough arithmetic, - can itself be arithmetized. There are sentences that the theory cannot decide (i.e., neither prove nor disprove). The theory cannot prove (an internal formulation of) its own consistency.
Pen and Paper Proofs of � and �
Pen and Paper Proofs of � and � … … … …
Pen and Paper Proofs of � and � … … … … The reader who does not like incomplete and (apparently) irremediably messy proofs of syntactic facts may wish to skim over the rest of this chapter and take it for granted that …
Formal Verifications of � and �
Formal Verifications of � and � TEM NQTHM HOL Light Coq Isabelle Sieg Shankar Harrison O’Connor Paulson 1986 2004 2005 2015 1978
End of story
End of story?
End of story?
Formal Verifications of � and � Shared structure
Formal Verifications of � and � Shared structure - Fix a particular logic: Classical FOL
Formal Verifications of � and � Shared structure - Fix a particular logic: Classical FOL - Fix a particular theory (+ finite extensions of it) - Arithmetic (Harrison, O’Connor) - Hereditarily finite set theory (Sieg, Shankar, Paulson)
Formal Verifications of � and � Shared structure - Fix a particular logic: Classical FOL - Fix a particular theory (+ finite extensions of it) - Arithmetic (Harrison, O’Connor) - Hereditarily finite set theory (Sieg, Shankar, Paulson)
Formal Verifications of � and � Shared structure - Fix a particular logic: Classical FOL - Fix a particular theory (+ finite extensions of it) - Arithmetic (Harrison, O’Connor) - Hereditarily finite set theory (Sieg, Shankar, Paulson) - Tour de force for the particular combination
Formal Verifications of � and � Shared structure - Fix a particular logic: Classical FOL - Fix a particular theory (+ finite extensions of it) - Arithmetic (Harrison, O’Connor) - Hereditarily finite set theory (Sieg, Shankar, Paulson) - Tour de force for the particular combination Scope of and remains largely unexploded
Formal Verifications of � and � Shared structure - Fix a particular logic: Classical FOL - Fix a particular theory (+ finite extensions of it) - Arithmetic (Harrison, O’Connor) - Hereditarily finite set theory (Sieg, Shankar, Paulson) - Tour de force for the particular combination Scope of and remains largely unexploded E.g. do they hold for Intuitionistic FOL , HOL , CIC ?
Our Motto:
Our Motto: Don’t Fix, Gather!
Our Contributions
Our Contributions e HOL l l e b a ∀ s - Abstract I formalization of and = α λ β → - Answer “What must/may a logic/theory offer?” - Understand variants and distill trade-offs from the literature - Correct a mistake in a pen and paper proof
Our Contributions e HOL l l e b a ∀ s - Abstract I formalization of and = α λ β → - Answer “What must/may a logic/theory offer?” - Understand variants and distill trade-offs from the literature - Correct a mistake in a pen and paper proof - Concrete instantiation to hereditarily finite set theory - Reproduce (for ) and improve (for ) Paulson’s formalization
What must a logic/theory o ff er? Generic Provability Connectives Numerals Syntax Relation
What must a logic/theory o ff er? Generic Provability Connectives Numerals Syntax Relation What may a logic/theory o ff er? Classical Order-like Proofs Encodings Logic Relation Represent- Derivability Standard Soundness ability Conditions Model Omega- Completeness Proofs vs. Consistency Consistency of Provability Provability
Generic Syntax
• sets: Var, Term, Fmla with Var ⊆ Term Generic Syntax
• sets: Var, Term, Fmla with Var ⊆ Term Generic • operators: Syntax FV_Term : Term → 2 Var FV : Fmla → 2 Var subst_Term : Term → Var → Term → Term subst : Fmla → Var → Term → Fmla
• sets: Var, Term, Fmla with Var ⊆ Term Generic • operators: Syntax FV_Term : Term → 2 Var FV : Fmla → 2 Var subst_Term : Term → Var → Term → Term subst : Fmla → Var → Term → Fmla • properties, e.g.: x ∈ FV( φ ) implies FV(subst φ x s) = FV( φ ) - {x} ∪ FV_Term(s)
• sets: Var, Term, Fmla with Var ⊆ Term Generic • operators: Syntax FV_Term : Term → 2 Var FV : Fmla → 2 Var subst_Term : Term → Var → Term → Term subst : Fmla → Var → Term → Fmla • properties, e.g.: x ∈ FV( φ ) implies FV(subst φ x s) = FV( φ ) - {x} ∪ FV_Term(s) We require unary substitution only. We derive parallel substitution from it.
Connectives
• operators: Connectives ≡ : Term → Term → Fmla → , ∧ , ∨ : Fmla → Fmla → Fmla ¬ : Fmla → Fmla ⊥ , ⊤ : Fmla ∃ , ∀ : Var → Fmla → Fmla
• operators: Connectives ≡ : Term → Term → Fmla → , ∧ , ∨ : Fmla → Fmla → Fmla ¬ : Fmla → Fmla ⊥ , ⊤ : Fmla ∃ , ∀ : Var → Fmla → Fmla We require a minimal list w.r.t. intuitionistic deduction and define the rest. Note: operators, not constructors
• unary relation: Provability Relation ⊢ ⊆ Fmla we write ⊢ φ if φ ∈ ⊢ • properties: ⊢ contains the standard (Hilbert-style) intuitionistic FOL axioms about the connectives
• unary relation: Provability Relation ⊢ ⊆ Fmla we write ⊢ φ if φ ∈ ⊢ • properties: ⊢ contains the standard (Hilbert-style) intuitionistic FOL axioms about the connectives • nonempty set: Numerals Num ⊆ Fmla 0
• property: ⊢ ¬ ¬ φ → φ Classical Logic
• property: ⊢ ¬ ¬ φ → φ Classical Logic • formula: ≺ ∈ Fmla 2 Order-like • properties, e.g.: Relation for all φ ∈ Fmla 1 and n ∈ Num, if ⊢ φ (m) for all m ∈ Num, then ⊢∀ x. x ≺ n → φ (x)
• property: ⊢ ¬ ¬ φ → φ Classical Logic • formula: ≺ ∈ Fmla 2 Order-like • properties, e.g.: Relation for all φ ∈ Fmla 1 and n ∈ Num, if ⊢ φ (m) for all m ∈ Num, then ⊢∀ x. x ≺ n → φ (x) • set: Proof Proofs • binary relation: ⊩ ∈ Proof × Fmla we write p ⊩ φ if (p, φ ) ∈⊩
• operators: Encodings ⟨ _ ⟩ : Fmla → Num and ⟨ _ ⟩ : Proof → Num • formulas subst, ⊩ , ¬ Represent- • property: ability behave like operators/relations (subst, ⊩ , ¬) on encodings
• operators: Encodings ⟨ _ ⟩ : Fmla → Num and ⟨ _ ⟩ : Proof → Num • formulas subst, ⊩ , ¬ Represent- • property: ability behave like operators/relations (subst, ⊩ , ¬) on encodings • property: ⊬⊥ Consistency
• operators: Encodings ⟨ _ ⟩ : Fmla → Num and ⟨ _ ⟩ : Proof → Num • formulas subst, ⊩ , ¬ Represent- • property: ability behave like operators/relations (subst, ⊩ , ¬) on encodings • property: ⊬⊥ Consistency • property: For all φ ∈ Fmla 1 , Omega- Consistency if ⊢ ¬ φ ( n ) for all n ∈ Num then ⊬ ¬¬( ∃ x . φ ( x ))
What must a logic/theory o ff er? Generic Provability Connectives Numerals Syntax Relation What may a logic/theory o ff er? Classical Order-like Proofs Encodings Logic Relation Represent- Derivability Standard Soundness ability Conditions Model Omega- Completeness Proofs vs. Consistency Consistency of Provability Provability
Omega- Proofs Consistency subst, ⊩ ⊢ φ implies ⊢⊢⟨ φ ⟩ Represent- Derivability Encodings ability Conditions There exists φ ∈ Fmla 0 such that ⊬ φ and ⊬ ¬ φ
Rosser’s Consistency Proofs Trick subst, ⊩ ⊢ φ implies ⊢⊢⟨ φ ⟩ Represent- Derivability Encodings ability Conditions There exists φ ∈ Fmla 0 such that ⊬ φ and ⊬ ¬ φ a la Rosser
Rosser’s Consistency Proofs Trick subst, ⊩ ⊢ φ implies ⊢⊢⟨ φ ⟩ Represent- Derivability Encodings ability Conditions ¬ There exists φ ∈ Fmla 0 such that ⊬ φ and ⊬ ¬ φ a la Rosser
Rosser’s Order-like Consistency Proofs Trick Relation subst, ⊩ ⊢ φ implies ⊢⊢⟨ φ ⟩ Represent- Derivability Encodings ability Conditions ¬ There exists φ ∈ Fmla 0 such that ⊬ φ and ⊬ ¬ φ a la Rosser
Standard Completeness Proofs vs. Soundness Model of Provability Provability ⊢ φ implies ⊢⊢⟨ φ ⟩ subst Represent- Derivability Encodings ability Conditions ⊢⊢⟨ φ ⟩ implies ⊢ φ There exists φ ∈ Fmla 0 such that ⊬ φ and ⊬ ¬ φ semantic and φ is true in the standard model
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