Negative Church’s Thesis and Russian Constructivism SATO Kentaro sato@inf.unibe.ch Institution for Computer Science, University of Bern * partially supported by John Templeton Foundation Negative Church’s Thesis and Russian Constructivism – p. 1
Plan of This Talk • Russian Recursive Constructive Mathematics (vs. Classical Mathematics and vs. Brouwer’s Intuitionism); • New Principle: Negative Church’s Thesis NCT — representing RRCM-spirit better than CT ? • Realizability model of NCT + BCP + MP and “any f : R → R is continuous”; • Consequences of NCT + MP . Negative Church’s Thesis and Russian Constructivism – p. 2
Plan of This Talk • Russian Recursive Constructive Mathematics (vs. Classical Mathematics and vs. Brouwer’s Intuitionism); • New Principle: Negative Church’s Thesis NCT — representing RRCM-spirit better than CT ? • Realizability model of NCT + BCP + MP and “any f : R → R is continuous”; • Consequences of NCT + MP . All the technical results are proved in: T. Nemoto and K. Sato “A marriage of Brouwer’s Intuitionism and Hilbert’s Finitism I: Arithmetic”, to appear in The Journal of Symbolic Logic Negative Church’s Thesis and Russian Constructivism – p. 2
Varieties of Mathematics Classical Mathematics Brouwer’s Intuitionistic Mathematics Negative Church’s Thesis and Russian Constructivism – p. 3
Varieties of Mathematics Classical Mathematics Brouwer’s Intuitionistic Mathematics • the precursor of (serious) constructivism; Negative Church’s Thesis and Russian Constructivism – p. 3
Varieties of Mathematics Classical Mathematics Brouwer’s Intuitionistic Mathematics • the precursor of (serious) constructivism; • rejection of LEM (intuitionistic logic); Negative Church’s Thesis and Russian Constructivism – p. 3
Varieties of Mathematics Classical Mathematics Brouwer’s Intuitionistic Mathematics • the precursor of (serious) constructivism; • rejection of LEM (intuitionistic logic); • infinite sequences can be captured only via finite fragments → Brouwer’s continuous principle; • Bar Induction. Negative Church’s Thesis and Russian Constructivism – p. 3
Varieties of Mathematics Classical Mathematics Brouwer’s Intuitionistic Mathematics • the precursor of (serious) constructivism; • rejection of LEM (intuitionistic logic); • infinite sequences can be captured only via finite fragments → Brouwer’s continuous principle; • Bar Induction. Russian Recursive Constructive Mathematics • rejection of LEM (intuitionistic logic); Negative Church’s Thesis and Russian Constructivism – p. 3
Varieties of Mathematics Classical Mathematics Brouwer’s Intuitionistic Mathematics • the precursor of (serious) constructivism; • rejection of LEM (intuitionistic logic); • infinite sequences can be captured only via finite fragments → Brouwer’s continuous principle; • Bar Induction. Russian Recursive Constructive Mathematics • rejection of LEM (intuitionistic logic); • everything is computable/recursive; Negative Church’s Thesis and Russian Constructivism – p. 3
Varieties of Mathematics Classical Mathematics Brouwer’s Intuitionistic Mathematics • the precursor of (serious) constructivism; • rejection of LEM (intuitionistic logic); • infinite sequences can be captured only via finite fragments → Brouwer’s continuous principle; • Bar Induction. Russian Recursive Constructive Mathematics • rejection of LEM (intuitionistic logic); • everything is computable/recursive; • Markov’s Principle: ¬¬∃ xϕ [ x ] → ∃ xϕ [ x ] , for decidable ϕ . Negative Church’s Thesis and Russian Constructivism – p. 3
Standard Formalization of RRCM 0. Heyting Arithmetic HA plus Negative Church’s Thesis and Russian Constructivism – p. 4
Standard Formalization of RRCM 0. Heyting Arithmetic HA plus 1. Markov’s Principle MP : • ¬¬∃ xϕ [ x ] → ∃ xϕ [ x ] for a ∆ 0 formula ϕ [ x ] or • ∀ x ( ϕ ( x ) ∨ ¬ ϕ [ x ]) → ( ¬¬∃ xϕ [ x ] → ∃ xϕ [ x ]) for any formula ϕ [ x ] ; Negative Church’s Thesis and Russian Constructivism – p. 4
Standard Formalization of RRCM 0. Heyting Arithmetic HA plus 1. Markov’s Principle MP : • ¬¬∃ xϕ [ x ] → ∃ xϕ [ x ] for a ∆ 0 formula ϕ [ x ] or • ∀ x ( ϕ ( x ) ∨ ¬ ϕ [ x ]) → ( ¬¬∃ xϕ [ x ] → ∃ xϕ [ x ]) for any formula ϕ [ x ] ; 2. and Church’s Thesis: • ∀ x ∃ yϕ [ x, y ] → ∃ e ∀ x ( { e } ( x ) ↓ ∧ ϕ [ x, { e } ( x )]) or • ∀ α ∃ e ∀ x ( { e } ( x ) ↓ ∧ { e } ( x ) = α ( x )) + choice ∀ x ∃ yϕ [ x, y ] → ∃ α ∀ xϕ [ x, α ( x )] (in the function-based 2nd order setting). Negative Church’s Thesis and Russian Constructivism – p. 4
Kleene’s Number Realizability For a formula ϕ , define another formula e nr ϕ by e nr ϕ ≡ ϕ for atomic ϕ ; e nr ϕ ∧ ψ ≡ (( e ) 0 nr ϕ ) ∧ (( e ) 1 nr ψ ); e nr ϕ ∨ ψ ≡ (( e ) 0 = 0 ∧ ( e ) 1 nr ϕ ) ∨ (( e ) 0 � = 0 ∧ ( e ) 1 nr ψ ); e nr ϕ → ψ ≡ ∀ x (( x nr ϕ ) → ( { e } ( x ) ↓ ∧ { e } ( x ) nr ψ )); e nr ∃ xϕ [ x ] ≡ ( e ) 1 nr ϕ [( e ) 0 ]; e nr ∀ xϕ [ x ] ≡ ∀ x ( { e } ( x ) ↓ ∧ { e } ( x ) nr ϕ [ x ]) . Negative Church’s Thesis and Russian Constructivism – p. 5
Kleene’s Number Realizability For a formula ϕ , define another formula e nr ϕ by e nr ϕ ≡ ϕ for atomic ϕ ; e nr ϕ ∧ ψ ≡ (( e ) 0 nr ϕ ) ∧ (( e ) 1 nr ψ ); e nr ϕ ∨ ψ ≡ (( e ) 0 = 0 ∧ ( e ) 1 nr ϕ ) ∨ (( e ) 0 � = 0 ∧ ( e ) 1 nr ψ ); e nr ϕ → ψ ≡ ∀ x (( x nr ϕ ) → ( { e } ( x ) ↓ ∧ { e } ( x ) nr ψ )); e nr ∃ xϕ [ x ] ≡ ( e ) 1 nr ϕ [( e ) 0 ]; e nr ∀ xϕ [ x ] ≡ ∀ x ( { e } ( x ) ↓ ∧ { e } ( x ) nr ϕ [ x ]) . A straightforward formalization of BHK interpretation by “algorithms” = “partial recursive functions”. Negative Church’s Thesis and Russian Constructivism – p. 5
Kleene’s Number Realizability For a formula ϕ , define another formula e nr ϕ by e nr ϕ ≡ ϕ for atomic ϕ ; e nr ϕ ∧ ψ ≡ (( e ) 0 nr ϕ ) ∧ (( e ) 1 nr ψ ); e nr ϕ ∨ ψ ≡ (( e ) 0 = 0 ∧ ( e ) 1 nr ϕ ) ∨ (( e ) 0 � = 0 ∧ ( e ) 1 nr ψ ); e nr ϕ → ψ ≡ ∀ x (( x nr ϕ ) → ( { e } ( x ) ↓ ∧ { e } ( x ) nr ψ )); e nr ∃ xϕ [ x ] ≡ ( e ) 1 nr ϕ [( e ) 0 ]; e nr ∀ xϕ [ x ] ≡ ∀ x ( { e } ( x ) ↓ ∧ { e } ( x ) nr ϕ [ x ]) . A straightforward formalization of BHK interpretation by “algorithms” = “partial recursive functions”. But “BHK interpretation”=“Brouwer’s world”? Negative Church’s Thesis and Russian Constructivism – p. 5
Standard Formalization of BIM Brouwer’s Intuitionistic Mathematics is formalized, within the function-based 2nd order language, by 0. Heyting Arithmetic HA plus 1. Induction: for any formula ϕ ϕ [0] ∧ ∀ x ( ϕ [ x ] → ϕ [ x +1) → ∀ xϕ [ x ] ; Negative Church’s Thesis and Russian Constructivism – p. 6
Standard Formalization of BIM Brouwer’s Intuitionistic Mathematics is formalized, within the function-based 2nd order language, by 0. Heyting Arithmetic HA plus 1. Induction: for any formula ϕ ϕ [0] ∧ ∀ x ( ϕ [ x ] → ϕ [ x +1) → ∀ xϕ [ x ] ; 2. Axiom of Choice AC : for any formula ϕ ∀ x ∃ yϕ [ x, y ] → ∃ α ∀ xϕ [ x, α ( x )] ; Negative Church’s Thesis and Russian Constructivism – p. 6
Standard Formalization of BIM Brouwer’s Intuitionistic Mathematics is formalized, within the function-based 2nd order language, by 0. Heyting Arithmetic HA plus 1. Induction: for any formula ϕ ϕ [0] ∧ ∀ x ( ϕ [ x ] → ϕ [ x +1) → ∀ xϕ [ x ] ; 2. Axiom of Choice AC : for any formula ϕ ∀ x ∃ yϕ [ x, y ] → ∃ α ∀ xϕ [ x, α ( x )] ; 3. Brouwer Continuous Principle: for any formula ϕ ∀ α ∃ xϕ [ α, x ] → ∀ α ∃ n, x ∀ β ( β ↾ n = α ↾ n → ϕ [ β, x ]) ; Negative Church’s Thesis and Russian Constructivism – p. 6
Standard Formalization of BIM Brouwer’s Intuitionistic Mathematics is formalized, within the function-based 2nd order language, by 0. Heyting Arithmetic HA plus 1. Induction: for any formula ϕ ϕ [0] ∧ ∀ x ( ϕ [ x ] → ϕ [ x +1) → ∀ xϕ [ x ] ; 2. Axiom of Choice AC : for any formula ϕ ∀ x ∃ yϕ [ x, y ] → ∃ α ∀ xϕ [ x, α ( x )] ; 3. Brouwer Continuous Principle: for any formula ϕ ∀ α ∃ xϕ [ α, x ] → ∀ α ∃ n, x ∀ β ( β ↾ n = α ↾ n → ϕ [ β, x ]) ; 4. Brouwer’s Bar Induction: for a ∆ 0 0 formula ϕ ∀ α ∃ nϕ [ α ↾ n ] ∧ ∀ u ( ∀ xϕ [ u ∗� x � ] → ϕ [ u ]) → ϕ [ � � ] Negative Church’s Thesis and Russian Constructivism – p. 6
Church’s Thesis Contradicts BCP • Church’s Thesis: ∀ α ∃ e ( α = { e } ) . • Brouwer’s Continuity Principle implies: ∀ α ∃ n ∀ β ( β ↾ n = α ↾ n → β = { e } ) . Negative Church’s Thesis and Russian Constructivism – p. 7
Church’s Thesis Contradicts BCP • Church’s Thesis: ∀ α ∃ e ( α = { e } ) . • Brouwer’s Continuity Principle implies: ∀ α ∃ n ∀ β ( β ↾ n = α ↾ n → β = { e } ) . • Particularly, ∃ n ∀ β ( β ↾ n = 0 ↾ n → β = { e } ) . Negative Church’s Thesis and Russian Constructivism – p. 7
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