Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Feature 3: Stratified Nonstandard Analysis The usual picture of ∗ N : ∗ N , the hypernatural numbers � �� � 0 1 . . . ω 1 . . . ω 2 . . . ω k . . . ✲ � �� � � �� � N , the natural/finite numbers Ω = ∗ N \ N , the infinite numbers
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Feature 3: Stratified Nonstandard Analysis The usual picture of ∗ N : In NSA , the infinite numbers are split into ‘small’ and ‘large’. ∗ N , the hypernatural numbers � �� � 0 1 . . . ω 1 . . . ω 2 . . . ω k . . . ✲ � �� � � �� � N , the natural/finite numbers Ω = ∗ N \ N , the infinite numbers
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Feature 3: Stratified Nonstandard Analysis The usual picture of ∗ N : In NSA , the infinite numbers are split into ‘small’ and ‘large’. ∗ N , the hypernatural numbers � �� � the large infinite numbers the small infinite numbers � �� � � �� � 0 1 . . . ω 1 . . . ω 2 . . . ω k . . . ✲ � �� � � �� � N , the natural/finite numbers Ω = ∗ N \ N , the infinite numbers
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Feature 3: Stratified Nonstandard Analysis The usual picture of ∗ N : In NSA , the infinite numbers are split into ‘small’ and ‘large’. ∗ N , the hypernatural numbers � �� � Ω 1 = ∗ N \ N 1 , the large infinite numbers N 1 = N ∪ the small infinite numbers � �� � � �� � 0 1 . . . ω 1 . . . ω 2 . . . ω k . . . ✲ � �� � � �� � N , the natural/finite numbers Ω = ∗ N \ N , the infinite numbers
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Feature 2: Ω-invariance
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Feature 2: Ω-invariance Ω-invariance ≈ algorithm ≈ finite procedure Definition (Ω-invariance) For ψ ( n , m ) ∈ ∆ 0 and ω ∈ Ω , the formula ψ ( n , ω ) is Ω -invariant if
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Feature 2: Ω-invariance Ω-invariance ≈ algorithm ≈ finite procedure Definition (Ω-invariance) For ψ ( n , m ) ∈ ∆ 0 and ω ∈ Ω , the formula ψ ( n , ω ) is Ω -invariant if ( ∀ n ∈ N )( ∀ ω ′ ∈ Ω)[ ψ ( n , ω ) ↔ ψ ( n , ω ′ )] .
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Feature 2: Ω-invariance Ω-invariance ≈ algorithm ≈ finite procedure Definition (Ω-invariance) For ψ ( n , m ) ∈ ∆ 0 and ω ∈ Ω , the formula ψ ( n , ω ) is Ω -invariant if ( ∀ n ∈ N )( ∀ ω ′ ∈ Ω)[ ψ ( n , ω ) ↔ ψ ( n , ω ′ )] . Note that ψ ( n , ω ) depends on ω ∈ Ω, but not on the choice of ω ∈ Ω.
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Feature 2: Ω-invariance Ω-invariance ≈ algorithm ≈ finite procedure Definition (Ω-invariance) For ψ ( n , m ) ∈ ∆ 0 and ω ∈ Ω , the formula ψ ( n , ω ) is Ω -invariant if ( ∀ n ∈ N )( ∀ ω ′ ∈ Ω)[ ψ ( n , ω ) ↔ ψ ( n , ω ′ )] . Note that ψ ( n , ω ) depends on ω ∈ Ω, but not on the choice of ω ∈ Ω. NSA has Ω-CA instead of ∆ 1 -CA. Principle (Ω-CA) For all Ω -invariant ψ ( n , ω ) , we have ( ∃ X ⊂ N )( ∀ n ∈ N )( n ∈ X ↔ ψ ( n , ω )) .
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA BISH (based on BHK)
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA BISH (based on BHK) NSA (based on CL)
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA BISH (based on BHK) NSA (based on CL) Central: algorithm and proof
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA BISH (based on BHK) NSA (based on CL) Central: algorithm and proof A ∨ B : an algo yields a proof of A or of B
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA BISH (based on BHK) NSA (based on CL) Central: Ω-invariance and Transfer ( T ) Central: algorithm and proof A ∨ B : an algo yields a proof of A or of B
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA BISH (based on BHK) NSA (based on CL) Central: Ω-invariance and Transfer ( T ) Central: algorithm and proof A ∨ B : A V B : an algo yields a proof of A or of B
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA BISH (based on BHK) NSA (based on CL) Central: Ω-invariance and Transfer ( T ) Central: algorithm and proof A ∨ B : A V B : There is Ω-invariant ψ ( � x , ω ) s.t. an algo yields a proof of A or of B ψ ( � x , ω ) → [ A ( � x ) ∧ [ A ( � x ) ∈ T ]] ∧ ¬ ψ ( � x , ω ) → [ B ( � x ) ∧ [ B ( � x ) ∈ T ]]
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA BISH (based on BHK) NSA (based on CL) Central: Ω-invariance and Transfer ( T ) Central: algorithm and proof A ∨ B : A V B : There is Ω-invariant ψ ( � x , ω ) s.t. an algo yields a proof of A or of B ψ ( � x , ω ) → [ A ( � x ) ∧ [ A ( � x ) ∈ T ]] ∧ ¬ ψ ( � x , ω ) → [ B ( � x ) ∧ [ B ( � x ) ∈ T ]] A → B : an algo converts a proof of A to a proof of B
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA BISH (based on BHK) NSA (based on CL) Central: Ω-invariance and Transfer ( T ) Central: algorithm and proof A ∨ B : A V B : There is Ω-invariant ψ ( � x , ω ) s.t. an algo yields a proof of A or of B ψ ( � x , ω ) → [ A ( � x ) ∧ [ A ( � x ) ∈ T ]] ∧ ¬ ψ ( � x , ω ) → [ B ( � x ) ∧ [ B ( � x ) ∈ T ]] � � � � A → B : an algo converts a proof of A A ⇛ B : A ∧ [ A ∈ T ] → B ∧ [ B ∈ T ] to a proof of B
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA BISH (based on BHK) NSA (based on CL) Central: Ω-invariance and Transfer ( T ) Central: algorithm and proof A ∨ B : A V B : There is Ω-invariant ψ ( � x , ω ) s.t. an algo yields a proof of A or of B ψ ( � x , ω ) → [ A ( � x ) ∧ [ A ( � x ) ∈ T ]] ∧ ¬ ψ ( � x , ω ) → [ B ( � x ) ∧ [ B ( � x ) ∈ T ]] � � � � A → B : an algo converts a proof of A A ⇛ B : A ∧ [ A ∈ T ] → B ∧ [ B ∈ T ] to a proof of B ‘ A ∈ T ’ means ‘ A satisfies Transfer’.
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA BISH (based on BHK) NSA (based on CL) Central: Ω-invariance and Transfer ( T ) Central: algorithm and proof A ∨ B : A V B : There is Ω-invariant ψ ( � x , ω ) s.t. an algo yields a proof of A or of B ψ ( � x , ω ) → [ A ( � x ) ∧ [ A ( � x ) ∈ T ]] ∧ ¬ ψ ( � x , ω ) → [ B ( � x ) ∧ [ B ( � x ) ∈ T ]] � � � � A → B : an algo converts a proof of A A ⇛ B : A ∧ [ A ∈ T ] → B ∧ [ B ∈ T ] to a proof of B ‘ A ∈ T ’ means ‘ A satisfies Transfer’. E.g. ‘( ∀ n ∈ N ) ϕ ( n ) ∈ T ’ is [( ∀ n ∈ N ) ϕ ( n ) → ( ∀ n ∈ ∗ N ) ϕ ( n )]
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA BISH (based on BHK) NSA (based on CL) Central: Ω-invariance and Transfer ( T ) Central: algorithm and proof A ∨ B : A V B : There is Ω-invariant ψ ( � x , ω ) s.t. an algo yields a proof of A or of B ψ ( � x , ω ) → [ A ( � x ) ∧ [ A ( � x ) ∈ T ]] ∧ ¬ ψ ( � x , ω ) → [ B ( � x ) ∧ [ B ( � x ) ∈ T ]] � � � � A → B : an algo converts a proof of A A ⇛ B : A ∧ [ A ∈ T ] → B ∧ [ B ∈ T ] to a proof of B ‘ A ∈ T ’ means ‘ A satisfies Transfer’. E.g. ‘( ∀ n ∈ N ) ϕ ( n ) ∈ T ’ is [( ∀ n ∈ N ) ϕ ( n ) → ( ∀ n ∈ ∗ N ) ϕ ( n )] E.g. ‘( ∃ n ∈ ∗ N ) ϕ ( n ) ∈ T ’ is [( ∃ n ∈ ∗ N ) ϕ ( n ) → ( ∃ n ∈ N 1 ) ϕ ( n )]
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA BISH (based on BHK) NSA (based on CL) Central: Ω-invariance and Transfer ( T ) Central: algorithm and proof A ∨ B : A V B : There is Ω-invariant ψ ( � x , ω ) s.t. an algo yields a proof of A or of B ψ ( � x , ω ) → [ A ( � x ) ∧ [ A ( � x ) ∈ T ]] ∧ ¬ ψ ( � x , ω ) → [ B ( � x ) ∧ [ B ( � x ) ∈ T ]] � � � � A → B : an algo converts a proof of A A ⇛ B : A ∧ [ A ∈ T ] → B ∧ [ B ∈ T ] to a proof of B ¬ A : A → (0 = 1)
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA BISH (based on BHK) NSA (based on CL) Central: Ω-invariance and Transfer ( T ) Central: algorithm and proof A ∨ B : A V B : There is Ω-invariant ψ ( � x , ω ) s.t. an algo yields a proof of A or of B ψ ( � x , ω ) → [ A ( � x ) ∧ [ A ( � x ) ∈ T ]] ∧ ¬ ψ ( � x , ω ) → [ B ( � x ) ∧ [ B ( � x ) ∈ T ]] � � � � A → B : an algo converts a proof of A A ⇛ B : A ∧ [ A ∈ T ] → B ∧ [ B ∈ T ] to a proof of B ¬ A : A → (0 = 1) ∼ A : A ⇛ (0 = 1)
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA BISH (based on BHK) NSA (based on CL) Central: Ω-invariance and Transfer ( T ) Central: algorithm and proof A ∨ B : A V B : There is Ω-invariant ψ ( � x , ω ) s.t. an algo yields a proof of A or of B ψ ( � x , ω ) → [ A ( � x ) ∧ [ A ( � x ) ∈ T ]] ∧ ¬ ψ ( � x , ω ) → [ B ( � x ) ∧ [ B ( � x ) ∈ T ]] � � � � A → B : an algo converts a proof of A A ⇛ B : A ∧ [ A ∈ T ] → B ∧ [ B ∈ T ] to a proof of B ¬ A : A → (0 = 1) ∼ A : A ⇛ (0 = 1) ( ∃ x ) A ( x ): an algo computes x 0 such that A ( x 0 )
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA BISH (based on BHK) NSA (based on CL) Central: Ω-invariance and Transfer ( T ) Central: algorithm and proof A ∨ B : A V B : There is Ω-invariant ψ ( � x , ω ) s.t. an algo yields a proof of A or of B ψ ( � x , ω ) → [ A ( � x ) ∧ [ A ( � x ) ∈ T ]] ∧ ¬ ψ ( � x , ω ) → [ B ( � x ) ∧ [ B ( � x ) ∈ T ]] � � � � A → B : an algo converts a proof of A A ⇛ B : A ∧ [ A ∈ T ] → B ∧ [ B ∈ T ] to a proof of B ¬ A : A → (0 = 1) ∼ A : A ⇛ (0 = 1) ( ∃ x ) A ( x ): an algo computes x 0 ( ∃ x ) A ( x ): “an Ω-inv. proc. computes x 0 such that A ( x 0 ) such that A ( x 0 )”
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA BISH (based on BHK) NSA (based on CL) Central: Ω-invariance and Transfer ( T ) Central: algorithm and proof A ∨ B : A V B : There is Ω-invariant ψ ( � x , ω ) s.t. an algo yields a proof of A or of B ψ ( � x , ω ) → [ A ( � x ) ∧ [ A ( � x ) ∈ T ]] ∧ ¬ ψ ( � x , ω ) → [ B ( � x ) ∧ [ B ( � x ) ∈ T ]] � � � � A → B : an algo converts a proof of A A ⇛ B : A ∧ [ A ∈ T ] → B ∧ [ B ∈ T ] to a proof of B ¬ A : A → (0 = 1) ∼ A : A ⇛ (0 = 1) ( ∃ x ) A ( x ): an algo computes x 0 ( ∃ x ) A ( x ): “an Ω-inv. proc. computes x 0 such that A ( x 0 ) such that A ( x 0 )” ∼ [( ∀ n ∈ N ) A ( n )]
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA BISH (based on BHK) NSA (based on CL) Central: Ω-invariance and Transfer ( T ) Central: algorithm and proof A ∨ B : A V B : There is Ω-invariant ψ ( � x , ω ) s.t. an algo yields a proof of A or of B ψ ( � x , ω ) → [ A ( � x ) ∧ [ A ( � x ) ∈ T ]] ∧ ¬ ψ ( � x , ω ) → [ B ( � x ) ∧ [ B ( � x ) ∈ T ]] � � � � A → B : an algo converts a proof of A A ⇛ B : A ∧ [ A ∈ T ] → B ∧ [ B ∈ T ] to a proof of B ¬ A : A → (0 = 1) ∼ A : A ⇛ (0 = 1) ( ∃ x ) A ( x ): an algo computes x 0 ( ∃ x ) A ( x ): “an Ω-inv. proc. computes x 0 such that A ( x 0 ) such that A ( x 0 )” ∼ [( ∀ n ∈ N ) A ( n )] ≡ ( ∃ n ∈ N 1 ) ∼ A ( n ) WEAKER than ( ∃ n ∈ N ) ∼ A ( n ).
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA BISH (based on BHK) NSA (based on CL) Central: Ω-invariance and Transfer ( T ) Central: algorithm and proof A ∨ B : A V B : There is Ω-invariant ψ ( � x , ω ) s.t. an algo yields a proof of A or of B ψ ( � x , ω ) → [ A ( � x ) ∧ [ A ( � x ) ∈ T ]] ∧ ¬ ψ ( � x , ω ) → [ B ( � x ) ∧ [ B ( � x ) ∈ T ]] � � � � A → B : an algo converts a proof of A A ⇛ B : A ∧ [ A ∈ T ] → B ∧ [ B ∈ T ] to a proof of B ¬ A : A → (0 = 1) ∼ A : A ⇛ (0 = 1) ( ∃ x ) A ( x ): an algo computes x 0 ( ∃ x ) A ( x ): “an Ω-inv. proc. computes x 0 such that A ( x 0 ) such that A ( x 0 )” ¬ [( ∀ n ∈ N ) A ( n )] is WEAKER ∼ [( ∀ n ∈ N ) A ( n )] ≡ ( ∃ n ∈ N 1 ) ∼ A ( n ) than ( ∃ n ∈ N ) ¬ A ( n ). WEAKER than ( ∃ n ∈ N ) ∼ A ( n ).
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA BISH (based on BHK) NSA (based on CL) Central: Ω-invariance and Transfer ( T ) Central: algorithm and proof A ∨ B : A V B : There is Ω-invariant ψ ( � x , ω ) s.t. an algo yields a proof of A or of B ψ ( � x , ω ) → [ A ( � x ) ∧ [ A ( � x ) ∈ T ]] ∧ ¬ ψ ( � x , ω ) → [ B ( � x ) ∧ [ B ( � x ) ∈ T ]] � � � � A → B : an algo converts a proof of A A ⇛ B : A ∧ [ A ∈ T ] → B ∧ [ B ∈ T ] to a proof of B ¬ A : A → (0 = 1) ∼ A : A ⇛ (0 = 1) ( ∃ x ) A ( x ): an algo computes x 0 ( ∃ x ) A ( x ): “an Ω-inv. proc. computes x 0 such that A ( x 0 ) such that A ( x 0 )” We know: If BISH ⊢ X then X �→ LPO, LLPO, MP, . . . (princ. rejected in BISH)
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA BISH (based on BHK) NSA (based on CL) Central: Ω-invariance and Transfer ( T ) Central: algorithm and proof A ∨ B : A V B : There is Ω-invariant ψ ( � x , ω ) s.t. an algo yields a proof of A or of B ψ ( � x , ω ) → [ A ( � x ) ∧ [ A ( � x ) ∈ T ]] ∧ ¬ ψ ( � x , ω ) → [ B ( � x ) ∧ [ B ( � x ) ∈ T ]] � � � � A → B : an algo converts a proof of A A ⇛ B : A ∧ [ A ∈ T ] → B ∧ [ B ∈ T ] to a proof of B ¬ A : A → (0 = 1) ∼ A : A ⇛ (0 = 1) ( ∃ x ) A ( x ): an algo computes x 0 ( ∃ x ) A ( x ): “an Ω-inv. proc. computes x 0 such that A ( x 0 ) such that A ( x 0 )” We know: If BISH ⊢ X then X �→ LPO, LLPO, MP, . . . (princ. rejected in BISH) We show: If NSA ⊢ Y then Y � ⇛ LPO , LLPO , MP , . . .
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion The translation B from BISH to NSA BISH (based on BHK) NSA (based on CL) Central: Ω-invariance and Transfer ( T ) Central: algorithm and proof A ∨ B : A V B : There is Ω-invariant ψ ( � x , ω ) s.t. an algo yields a proof of A or of B ψ ( � x , ω ) → [ A ( � x ) ∧ [ A ( � x ) ∈ T ]] ∧ ¬ ψ ( � x , ω ) → [ B ( � x ) ∧ [ B ( � x ) ∈ T ]] � � � � A → B : an algo converts a proof of A A ⇛ B : A ∧ [ A ∈ T ] → B ∧ [ B ∈ T ] to a proof of B ¬ A : A → (0 = 1) ∼ A : A ⇛ (0 = 1) ( ∃ x ) A ( x ): an algo computes x 0 ( ∃ x ) A ( x ): “an Ω-inv. proc. computes x 0 such that A ( x 0 ) such that A ( x 0 )” We know: If BISH ⊢ X then X �→ LPO, LLPO, MP, . . . (princ. rejected in BISH) We show: If NSA ⊢ Y then Y � ⇛ LPO , LLPO , MP , . . . (e.g. LPO is B (LPO), unprovable in NSA
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ 1 , P ∨ ¬ P �
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ 1 , P ∨ ¬ P � LPR: ( ∀ x ∈ R )( x > 0 ∨ ¬ ( x > 0)) �
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ 1 , P ∨ ¬ P � LPR: ( ∀ x ∈ R )( x > 0 ∨ ¬ ( x > 0)) � MCT: monotone convergence thm �
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ 1 , P ∨ ¬ P � LPR: ( ∀ x ∈ R )( x > 0 ∨ ¬ ( x > 0)) � MCT: monotone convergence thm � CIT: Cantor intersection thm
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant LPO: For P ∈ Σ 1 , P ∨ ¬ P � LPR: ( ∀ x ∈ R )( x > 0 ∨ ¬ ( x > 0)) � MCT: monotone convergence thm � CIT: Cantor intersection thm
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant LPO: For P ∈ Σ 1 , P ∨ ¬ P LPO : For P ∈ Σ 1 , P V ∼ P � ⇚ ⇛ LPR: ( ∀ x ∈ R )( x > 0 ∨ ¬ ( x > 0)) � MCT: monotone convergence thm � CIT: Cantor intersection thm
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant LPO: For P ∈ Σ 1 , P ∨ ¬ P LPO : For P ∈ Σ 1 , P V ∼ P � ⇚ ⇛ LPR: ( ∀ x ∈ R )( x > 0 ∨ ¬ ( x > 0)) LPR : ( ∀ x ∈ R )( x > 0 V ∼ ( x > 0)) � ⇚ ⇛ MCT: monotone convergence thm � CIT: Cantor intersection thm
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant LPO: For P ∈ Σ 1 , P ∨ ¬ P LPO : For P ∈ Σ 1 , P V ∼ P � ⇚ ⇛ LPR: ( ∀ x ∈ R )( x > 0 ∨ ¬ ( x > 0)) LPR : ( ∀ x ∈ R )( x > 0 V ∼ ( x > 0)) � ⇚ ⇛ MCT: monotone convergence thm MCT : monotone convergence thm � ⇚ ⇛ CIT: Cantor intersection thm
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant LPO: For P ∈ Σ 1 , P ∨ ¬ P LPO : For P ∈ Σ 1 , P V ∼ P � ⇚ ⇛ LPR: ( ∀ x ∈ R )( x > 0 ∨ ¬ ( x > 0)) LPR : ( ∀ x ∈ R )( x > 0 V ∼ ( x > 0)) � ⇚ ⇛ MCT: monotone convergence thm MCT : monotone convergence thm � ⇚ ⇛ CIT: Cantor intersection thm CIT : Cantor intersection thm
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant LPO: For P ∈ Σ 1 , P ∨ ¬ P LPO : For P ∈ Σ 1 , P V ∼ P � ⇚ ⇛ LPR: ( ∀ x ∈ R )( x > 0 ∨ ¬ ( x > 0)) LPR : ( ∀ x ∈ R )( x > 0 V ∼ ( x > 0)) � ⇚ ⇛ MCT: monotone convergence thm MCT : monotone convergence thm � (limit computed by algo) ⇚ ⇛ CIT: Cantor intersection thm CIT : Cantor intersection thm
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant LPO: For P ∈ Σ 1 , P ∨ ¬ P LPO : For P ∈ Σ 1 , P V ∼ P � ⇚ ⇛ LPR: ( ∀ x ∈ R )( x > 0 ∨ ¬ ( x > 0)) LPR : ( ∀ x ∈ R )( x > 0 V ∼ ( x > 0)) � ⇚ ⇛ MCT: monotone convergence thm MCT : monotone convergence thm � (limit computed by algo) (limit computed by Ω-inv. proc.) ⇚ ⇛ CIT: Cantor intersection thm CIT : Cantor intersection thm
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant LPO: For P ∈ Σ 1 , P ∨ ¬ P LPO : For P ∈ Σ 1 , P V ∼ P � ⇚ ⇛ LPR: ( ∀ x ∈ R )( x > 0 ∨ ¬ ( x > 0)) LPR : ( ∀ x ∈ R )( x > 0 V ∼ ( x > 0)) � ⇚ ⇛ MCT: monotone convergence thm MCT : monotone convergence thm � (limit computed by algo) (limit computed by Ω-inv. proc.) ⇚ ⇛ CIT: Cantor intersection thm CIT : Cantor intersection thm (point in intersection computed by algo)
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant LPO: For P ∈ Σ 1 , P ∨ ¬ P LPO : For P ∈ Σ 1 , P V ∼ P � ⇚ ⇛ LPR: ( ∀ x ∈ R )( x > 0 ∨ ¬ ( x > 0)) LPR : ( ∀ x ∈ R )( x > 0 V ∼ ( x > 0)) � ⇚ ⇛ MCT: monotone convergence thm MCT : monotone convergence thm � (limit computed by algo) (limit computed by Ω-inv. proc.) ⇚ ⇛ CIT: Cantor intersection thm CIT : Cantor intersection thm (point in intersection computed by algo) (point in intersection computed by Ω-inv. proc.)
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant LPO: For P ∈ Σ 1 , P ∨ ¬ P LPO : For P ∈ Σ 1 , P V ∼ P � ⇚ ⇛ LPR: ( ∀ x ∈ R )( x > 0 ∨ ¬ ( x > 0)) LPR : ( ∀ x ∈ R )( x > 0 V ∼ ( x > 0)) � ⇚ ⇛ MCT: monotone convergence thm MCT : monotone convergence thm � (limit computed by algo) (limit computed by Ω-inv. proc.) ⇚ ⇛ CIT: Cantor intersection thm CIT : Cantor intersection thm ⇚ ⇛ Universal Transfer: For all ϕ ∈ ∆ 0 ( ∀ n ∈ N ) ϕ ( n ) → ( ∀ n ∈ ∗ N ) ϕ ( n )
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant LPO: For P ∈ Σ 1 , P ∨ ¬ P LPO : For P ∈ Σ 1 , P V ∼ P � ⇚ ⇛ LPR: ( ∀ x ∈ R )( x > 0 ∨ ¬ ( x > 0)) LPR : ( ∀ x ∈ R )( x > 0 V ∼ ( x > 0)) � ⇚ ⇛ MCT: monotone convergence thm MCT : monotone convergence thm � (limit computed by algo) (limit computed by Ω-inv. proc.) ⇚ ⇛ CIT: Cantor intersection thm CIT : Cantor intersection thm ⇚ ⇛ Universal Transfer: For all ϕ ∈ ∆ 0 ( ∀ n ∈ N ) ϕ ( n ) → ( ∀ n ∈ ∗ N ) ϕ ( n ) � � NSA does prove ( ∀ δ ∈ R ) δ > 0 ⇛ ( x > 0) V ( x < δ ) . � � BISH does prove ( ∀ δ ∈ R ) δ > 0 → ( x > 0) ∨ ( x < δ ) .
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B II
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B II BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B II BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LLPO For P , Q ∈ Σ 1 , ¬ ( P ∧ Q ) → ¬ P ∨ ¬ Q �
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B II BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LLPO For P , Q ∈ Σ 1 , ¬ ( P ∧ Q ) → ¬ P ∨ ¬ Q � LLPR: ( ∀ x ∈ R )( x ≥ 0 ∨ x ≤ 0) �
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B II BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LLPO For P , Q ∈ Σ 1 , ¬ ( P ∧ Q ) → ¬ P ∨ ¬ Q � LLPR: ( ∀ x ∈ R )( x ≥ 0 ∨ x ≤ 0) � NIL ( ∀ x , y ∈ R )( xy = 0 → x = 0 ∨ y = 0) �
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B II BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LLPO For P , Q ∈ Σ 1 , ¬ ( P ∧ Q ) → ¬ P ∨ ¬ Q � LLPR: ( ∀ x ∈ R )( x ≥ 0 ∨ x ≤ 0) � NIL ( ∀ x , y ∈ R )( xy = 0 → x = 0 ∨ y = 0) � IVT: Intermediate value theorem
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B II BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant LLPO For P , Q ∈ Σ 1 , ¬ ( P ∧ Q ) → ¬ P ∨ ¬ Q � LLPR: ( ∀ x ∈ R )( x ≥ 0 ∨ x ≤ 0) � NIL ( ∀ x , y ∈ R )( xy = 0 → x = 0 ∨ y = 0) � IVT: Intermediate value theorem
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B II BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant LLPO LLPO For P , Q ∈ Σ 1 , ¬ ( P ∧ Q ) → ¬ P ∨ ¬ Q For P , Q ∈ Σ 1 , ∼ ( P ∧ Q ) ⇛ ∼ P V ∼ Q � ⇚ ⇛ LLPR: ( ∀ x ∈ R )( x ≥ 0 ∨ x ≤ 0) � NIL ( ∀ x , y ∈ R )( xy = 0 → x = 0 ∨ y = 0) � IVT: Intermediate value theorem
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B II BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant LLPO LLPO For P , Q ∈ Σ 1 , ¬ ( P ∧ Q ) → ¬ P ∨ ¬ Q For P , Q ∈ Σ 1 , ∼ ( P ∧ Q ) ⇛ ∼ P V ∼ Q � ⇚ ⇛ LLPR: ( ∀ x ∈ R )( x ≥ 0 ∨ x ≤ 0) LLPR : ( ∀ x ∈ R )( x ≥ 0 V x ≤ 0) � ⇚ ⇛ NIL ( ∀ x , y ∈ R )( xy = 0 → x = 0 ∨ y = 0) � IVT: Intermediate value theorem
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B II BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant LLPO LLPO For P , Q ∈ Σ 1 , ¬ ( P ∧ Q ) → ¬ P ∨ ¬ Q For P , Q ∈ Σ 1 , ∼ ( P ∧ Q ) ⇛ ∼ P V ∼ Q � ⇚ ⇛ LLPR: ( ∀ x ∈ R )( x ≥ 0 ∨ x ≤ 0) LLPR : ( ∀ x ∈ R )( x ≥ 0 V x ≤ 0) � ⇚ ⇛ NIL NIL ( ∀ x , y ∈ R )( xy = 0 → x = 0 ∨ y = 0) ( ∀ x , y ∈ R )( xy = 0 ⇛ x = 0 V y = 0) � ⇚ ⇛ IVT: Intermediate value theorem
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B II BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant LLPO LLPO For P , Q ∈ Σ 1 , ¬ ( P ∧ Q ) → ¬ P ∨ ¬ Q For P , Q ∈ Σ 1 , ∼ ( P ∧ Q ) ⇛ ∼ P V ∼ Q � ⇚ ⇛ LLPR: ( ∀ x ∈ R )( x ≥ 0 ∨ x ≤ 0) LLPR : ( ∀ x ∈ R )( x ≥ 0 V x ≤ 0) � ⇚ ⇛ NIL NIL ( ∀ x , y ∈ R )( xy = 0 → x = 0 ∨ y = 0) ( ∀ x , y ∈ R )( xy = 0 ⇛ x = 0 V y = 0) � ⇚ ⇛ IVT: Intermediate value theorem IVT : Intermediate value theorem
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B II BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant LLPO LLPO For P , Q ∈ Σ 1 , ¬ ( P ∧ Q ) → ¬ P ∨ ¬ Q For P , Q ∈ Σ 1 , ∼ ( P ∧ Q ) ⇛ ∼ P V ∼ Q � ⇚ ⇛ LLPR: ( ∀ x ∈ R )( x ≥ 0 ∨ x ≤ 0) LLPR : ( ∀ x ∈ R )( x ≥ 0 V x ≤ 0) � ⇚ ⇛ NIL NIL ( ∀ x , y ∈ R )( xy = 0 → x = 0 ∨ y = 0) ( ∀ x , y ∈ R )( xy = 0 ⇛ x = 0 V y = 0) � ⇚ ⇛ IVT: Intermediate value theorem IVT : Intermediate value theorem (int. value computed by algo)
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B II BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant LLPO LLPO For P , Q ∈ Σ 1 , ¬ ( P ∧ Q ) → ¬ P ∨ ¬ Q For P , Q ∈ Σ 1 , ∼ ( P ∧ Q ) ⇛ ∼ P V ∼ Q � ⇚ ⇛ LLPR: ( ∀ x ∈ R )( x ≥ 0 ∨ x ≤ 0) LLPR : ( ∀ x ∈ R )( x ≥ 0 V x ≤ 0) � ⇚ ⇛ NIL NIL ( ∀ x , y ∈ R )( xy = 0 → x = 0 ∨ y = 0) ( ∀ x , y ∈ R )( xy = 0 ⇛ x = 0 V y = 0) � ⇚ ⇛ IVT: Intermediate value theorem IVT : Intermediate value theorem (int. value computed by algo) ( int. value computed by Ω-inv. proc. )
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B II BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant LLPO LLPO For P , Q ∈ Σ 1 , ¬ ( P ∧ Q ) → ¬ P ∨ ¬ Q For P , Q ∈ Σ 1 , ∼ ( P ∧ Q ) ⇛ ∼ P V ∼ Q � ⇚ ⇛ LLPR: ( ∀ x ∈ R )( x ≥ 0 ∨ x ≤ 0) LLPR : ( ∀ x ∈ R )( x ≥ 0 V x ≤ 0) � ⇚ ⇛ NIL NIL ( ∀ x , y ∈ R )( xy = 0 → x = 0 ∨ y = 0) ( ∀ x , y ∈ R )( xy = 0 ⇛ x = 0 V y = 0) � ⇚ ⇛ IVT: Intermediate value theorem IVT : Intermediate value theorem (int. value computed by algo) ( int. value computed by Ω-inv. proc. ) � WKL ⇛ WKL ⇚
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B II BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant LLPO LLPO For P , Q ∈ Σ 1 , ¬ ( P ∧ Q ) → ¬ P ∨ ¬ Q For P , Q ∈ Σ 1 , ∼ ( P ∧ Q ) ⇛ ∼ P V ∼ Q � ⇚ ⇛ LLPR: ( ∀ x ∈ R )( x ≥ 0 ∨ x ≤ 0) LLPR : ( ∀ x ∈ R )( x ≥ 0 V x ≤ 0) � ⇚ ⇛ NIL NIL ( ∀ x , y ∈ R )( xy = 0 → x = 0 ∨ y = 0) ( ∀ x , y ∈ R )( xy = 0 ⇛ x = 0 V y = 0) � ⇚ ⇛ IVT: Intermediate value theorem IVT : Intermediate value theorem (int. value computed by algo) ( int. value computed by Ω-inv. proc. ) � WKL ⇛ ∨ -Transfer ⇛ WKL ⇚ ⇚
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B II BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant LLPO LLPO For P , Q ∈ Σ 1 , ¬ ( P ∧ Q ) → ¬ P ∨ ¬ Q For P , Q ∈ Σ 1 , ∼ ( P ∧ Q ) ⇛ ∼ P V ∼ Q � ⇚ ⇛ LLPR: ( ∀ x ∈ R )( x ≥ 0 ∨ x ≤ 0) LLPR : ( ∀ x ∈ R )( x ≥ 0 V x ≤ 0) � ⇚ ⇛ NIL NIL ( ∀ x , y ∈ R )( xy = 0 → x = 0 ∨ y = 0) ( ∀ x , y ∈ R )( xy = 0 ⇛ x = 0 V y = 0) � ⇚ ⇛ IVT: Intermediate value theorem IVT : Intermediate value theorem (int. value computed by algo) ( int. value computed by Ω-inv. proc. ) � WKL ⇛ ∨ -Transfer ⇛ WKL ⇚ ⇚ Axioms of R : ¬ ( x > 0 ∧ x < 0)
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B II BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant LLPO LLPO For P , Q ∈ Σ 1 , ¬ ( P ∧ Q ) → ¬ P ∨ ¬ Q For P , Q ∈ Σ 1 , ∼ ( P ∧ Q ) ⇛ ∼ P V ∼ Q � ⇚ ⇛ LLPR: ( ∀ x ∈ R )( x ≥ 0 ∨ x ≤ 0) LLPR : ( ∀ x ∈ R )( x ≥ 0 V x ≤ 0) � ⇚ ⇛ NIL NIL ( ∀ x , y ∈ R )( xy = 0 → x = 0 ∨ y = 0) ( ∀ x , y ∈ R )( xy = 0 ⇛ x = 0 V y = 0) � ⇚ ⇛ IVT: Intermediate value theorem IVT : Intermediate value theorem (int. value computed by algo) ( int. value computed by Ω-inv. proc. ) � WKL ⇛ ∨ -Transfer ⇛ WKL ⇚ ⇚ Axioms of R : ¬ ( x > 0 ∧ x < 0) Axioms of R : ∼ ( x > 0 ∧ x < 0)
Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion Constructive Reverse Mathematics under B III
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