Introduction NSA and Constructive Analysis Philosophical implications Feature 3: Stratified Nonstandard Analysis The usual picture of ∗ N : In NSA , ∗ N has extra structure: countably many levels of infinity N ⊂ N 1 ⊂ . . . N k ⊂ N k +1 ⊂ · · · ⊂ ∗ N ∗ N , the hypernatural numbers � �� � 0 1 . . . ω 1 . . . ω 2 . . . ω k . . . ✲ � �� � � �� � N , the finite numbers Ω = ∗ N \ N , the infinite numbers
Introduction NSA and Constructive Analysis Philosophical implications Feature 3: Stratified Nonstandard Analysis The usual picture of ∗ N : In NSA , ∗ N has extra structure: countably many levels of infinity N ⊂ N 1 ⊂ . . . N k ⊂ N k +1 ⊂ · · · ⊂ ∗ N ∗ N , the hypernatural numbers � �� � N 1 , the 1-finite numbers Ω 1 = ∗ N \ N 1 , the 1-infinite numbers � �� � � �� � 0 1 . . . ω 1 . . . ω 2 . . . ω k . . . ✲ � �� � � �� � N , the finite numbers Ω = ∗ N \ N , the infinite numbers
Introduction NSA and Constructive Analysis Philosophical implications Feature 3: Stratified Nonstandard Analysis The usual picture of ∗ N : In NSA , ∗ N has extra structure: countably many levels of infinity N ⊂ N 1 ⊂ . . . N k ⊂ N k +1 ⊂ · · · ⊂ ∗ N ∗ N , the hypernatural numbers � �� � N 2 , the 2-finite numbers Ω 2 = ∗ N \ N 2 , the 2-infinite numbers � �� � � �� � 0 1 . . . ω 1 . . . ω 2 . . . ω k . . . ✲ � �� � � �� � N , the finite numbers Ω = ∗ N \ N , the infinite numbers
Introduction NSA and Constructive Analysis Philosophical implications Feature 3: Stratified Nonstandard Analysis The usual picture of ∗ N : In NSA , ∗ N has extra structure: countably many levels of infinity N ⊂ N 1 ⊂ . . . N k ⊂ N k +1 ⊂ · · · ⊂ ∗ N ∗ N , the hypernatural numbers � �� � N k , the k -finite numbers Ω k = ∗ N \ N k , the k -infinite numbers � �� � � �� � 0 1 . . . ω 1 . . . ω 2 . . . ω k . . . ✲ � �� � � �� � N , the finite numbers Ω = ∗ N \ N , the infinite numbers
Introduction NSA and Constructive Analysis Philosophical implications Feature 2: Ω-invariance
Introduction NSA and Constructive Analysis Philosophical implications Feature 2: Ω-invariance Ω-invariance ≈ algorithm ≈ finite procedure ≈ explicit computation.
Introduction NSA and Constructive Analysis Philosophical implications Feature 2: Ω-invariance Ω-invariance ≈ algorithm ≈ finite procedure ≈ explicit computation. Definition (Ω-invariance) A set A ⊂ N is Ω -invariant
Introduction NSA and Constructive Analysis Philosophical implications Feature 2: Ω-invariance Ω-invariance ≈ algorithm ≈ finite procedure ≈ explicit computation. Definition (Ω-invariance) A set A ⊂ N is Ω -invariant if there is a quantifier-free formula ψ such that for all ω ∈ Ω ,
Introduction NSA and Constructive Analysis Philosophical implications Feature 2: Ω-invariance Ω-invariance ≈ algorithm ≈ finite procedure ≈ explicit computation. Definition (Ω-invariance) A set A ⊂ N is Ω -invariant if there is a quantifier-free formula ψ such that for all ω ∈ Ω , A = { k ∈ N : ψ ( k , ω ) } .
Introduction NSA and Constructive Analysis Philosophical implications Feature 2: Ω-invariance Ω-invariance ≈ algorithm ≈ finite procedure ≈ explicit computation. Definition (Ω-invariance) A set A ⊂ N is Ω -invariant if there is a quantifier-free formula ψ such that for all ω ∈ Ω , A = { k ∈ N : ψ ( k , ω ) } . Note that A depends on ω ∈ Ω, but not on the choice of ω ∈ Ω.
Introduction NSA and Constructive Analysis Philosophical implications Feature 2: Ω-invariance Theorem (Finiteness) For every Ω -invariant A ⊂ N and k ∈ N , there is M ∈ N , such that
Introduction NSA and Constructive Analysis Philosophical implications Feature 2: Ω-invariance Theorem (Finiteness) For every Ω -invariant A ⊂ N and k ∈ N , there is M ∈ N , such that k ∈ A ⇐ ⇒ ψ ( k , ω ) ⇐ ⇒ ψ ( k , M ) .
Introduction NSA and Constructive Analysis Philosophical implications Feature 2: Ω-invariance Theorem (Finiteness) For every Ω -invariant A ⊂ N and k ∈ N , there is M ∈ N , such that k ∈ A ⇐ ⇒ ψ ( k , ω ) ⇐ ⇒ ψ ( k , M ) . Thus, to verify whether k ∈ A , we only need to perform finitely many operations (i.e. determine if ψ ( k , M )).
Introduction NSA and Constructive Analysis Philosophical implications Feature 2: Ω-invariance Theorem (Finiteness) For every Ω -invariant A ⊂ N and k ∈ N , there is M ∈ N , such that k ∈ A ⇐ ⇒ ψ ( k , ω ) ⇐ ⇒ ψ ( k , M ) . Thus, to verify whether k ∈ A , we only need to perform finitely many operations (i.e. determine if ψ ( k , M )). NSA has Ω-CA instead of ∆ 1 -CA. Principle (Ω-CA) All Ω -invariant sets exist.
Introduction NSA and Constructive Analysis Philosophical implications Lost in translation
Introduction NSA and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL)
Introduction NSA and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL) NSA (based on CL)
Introduction NSA and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL) NSA (based on CL) Central: algorithm and proof
Introduction NSA and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL) NSA (based on CL) Central: algorithm and proof A ∨ B : an algo yields a proof of A or of B
Introduction NSA and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL) 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 and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL) 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 [ A ∨ B ] ∧ [ A → A ∈ T ] ∧ [ B → B ∈ T ]
Introduction NSA and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL) 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 [ A ∨ B ] ∧ [ A → A ∈ T ] ∧ [ B → B ∈ T ] A → B : an algo converts a proof of A to a proof of B
Introduction NSA and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL) 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 [ A ∨ B ] ∧ [ A → A ∈ T ] ∧ [ B → B ∈ 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 and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL) 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 [ A ∨ B ] ∧ [ A → A ∈ T ] ∧ [ B → B ∈ 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 and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL) 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 [ A ∨ B ] ∧ [ A → A ∈ T ] ∧ [ B → B ∈ 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 and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL) 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 [ A ∨ B ] ∧ [ A → A ∈ T ] ∧ [ B → B ∈ 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 and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL) 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 [ A ∨ B ] ∧ [ A → A ∈ T ] ∧ [ B → B ∈ 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 and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL) 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 [ A ∨ B ] ∧ [ A → A ∈ T ] ∧ [ B → B ∈ 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 and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL) 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 [ A ∨ B ] ∧ [ A → A ∈ T ] ∧ [ B → B ∈ 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 and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL) 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 [ A ∨ B ] ∧ [ A → A ∈ T ] ∧ [ B → B ∈ T ] ≈ “an algo decides if A or if B ” 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 and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL) 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 [ A ∨ B ] ∧ [ A → A ∈ T ] ∧ [ B → B ∈ T ] ≈ “an algo decides if A or if B ” ≈ “an Ω-inv.proc. decides if A or if B ” 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 and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL) 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 [ A ∨ B ] ∧ [ A → A ∈ T ] ∧ [ B → B ∈ T ] ≈ “an algo decides if A or if B ” ≈ “an Ω-inv.proc. decides if A or if B ” 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 and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL) 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 [ A ∨ B ] ∧ [ A → A ∈ T ] ∧ [ B → B ∈ T ] ≈ “an algo decides if A or if B ” ≈ “an Ω-inv.proc. decides if A or if B ” 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 and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL) 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 [ A ∨ B ] ∧ [ A → A ∈ T ] ∧ [ B → B ∈ T ] ≈ “an algo decides if A or if B ” ≈ “an Ω-inv.proc. decides if A or if B ” 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 and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL) 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 [ A ∨ B ] ∧ [ A → A ∈ T ] ∧ [ B → B ∈ T ] ≈ “an algo decides if A or if B ” ≈ “an Ω-inv.proc. decides if A or if B ” 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 and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL) 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 [ A ∨ B ] ∧ [ A → A ∈ T ] ∧ [ B → B ∈ T ] ≈ “an algo decides if A or if B ” ≈ “an Ω-inv.proc. decides if A or if B ” 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 and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL) 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 [ A ∨ B ] ∧ [ A → A ∈ T ] ∧ [ B → B ∈ T ] ≈ “an algo decides if A or if B ” ≈ “an Ω-inv.proc. decides if A or if B ” 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 ) WHY is this a good/faithful/reasonable/. . . translation?
Introduction NSA and Constructive Analysis Philosophical implications Lost in translation BISH (based on IL) 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 [ A ∨ B ] ∧ [ A → A ∈ T ] ∧ [ B → B ∈ T ] ≈ “an algo decides if A or if B ” ≈ “an Ω-inv.proc. decides if A or if B ” 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 ) WHY is this a good/faithful/reasonable/. . . translation? BECAUSE the non-algorithmic/non-constructive principles behave the same!
Introduction NSA and Constructive Analysis Philosophical implications Constructive Reverse Mathematics
Introduction NSA and Constructive Analysis Philosophical implications Constructive Reverse Mathematics BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic
Introduction NSA and Constructive Analysis Philosophical implications Constructive Reverse Mathematics BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ 1 , P ∨ ¬ P �
Introduction NSA and Constructive Analysis Philosophical implications Constructive Reverse Mathematics BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ 1 , P ∨ ¬ P � LPR: ( ∀ x ∈ R )( x > 0 ∨ ¬ ( x > 0)) �
Introduction NSA and Constructive Analysis Philosophical implications Constructive Reverse Mathematics BISH (based on IL) 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 and Constructive Analysis Philosophical implications Constructive Reverse Mathematics BISH (based on IL) 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 and Constructive Analysis Philosophical implications Constructive Reverse Mathematics BISH (based on IL) 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 and Constructive Analysis Philosophical implications Constructive Reverse Mathematics BISH (based on IL) 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 and Constructive Analysis Philosophical implications Constructive Reverse Mathematics BISH (based on IL) 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 and Constructive Analysis Philosophical implications Constructive Reverse Mathematics BISH (based on IL) 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 and Constructive Analysis Philosophical implications Constructive Reverse Mathematics BISH (based on IL) 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 and Constructive Analysis Philosophical implications Constructive Reverse Mathematics BISH (based on IL) 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 and Constructive Analysis Philosophical implications Constructive Reverse Mathematics BISH (based on IL) 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 and Constructive Analysis Philosophical implications Constructive Reverse Mathematics BISH (based on IL) 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 and Constructive Analysis Philosophical implications Constructive Reverse Mathematics BISH (based on IL) 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 and Constructive Analysis Philosophical implications Constructive Reverse Mathematics BISH (based on IL) 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 ( ∀ n ∈ N ) ϕ ( n ) → ( ∀ n ∈ ∗ N ) ϕ ( n )
Introduction NSA and Constructive Analysis Philosophical implications Constructive Reverse Mathematics II
Introduction NSA and Constructive Analysis Philosophical implications Constructive Reverse Mathematics II BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic
Introduction NSA and Constructive Analysis Philosophical implications Constructive Reverse Mathematics II BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic LLPO For P , Q ∈ Σ 1 , ¬ ( P ∧ Q ) → ¬ P ∨ ¬ Q �
Introduction NSA and Constructive Analysis Philosophical implications Constructive Reverse Mathematics II BISH (based on IL) 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 and Constructive Analysis Philosophical implications Constructive Reverse Mathematics II BISH (based on IL) 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 and Constructive Analysis Philosophical implications Constructive Reverse Mathematics II BISH (based on IL) 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 and Constructive Analysis Philosophical implications Constructive Reverse Mathematics II BISH (based on IL) 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 and Constructive Analysis Philosophical implications Constructive Reverse Mathematics II BISH (based on IL) 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 and Constructive Analysis Philosophical implications Constructive Reverse Mathematics II BISH (based on IL) 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 and Constructive Analysis Philosophical implications Constructive Reverse Mathematics II BISH (based on IL) 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 and Constructive Analysis Philosophical implications Constructive Reverse Mathematics II BISH (based on IL) 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 and Constructive Analysis Philosophical implications Constructive Reverse Mathematics II BISH (based on IL) 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 and Constructive Analysis Philosophical implications Constructive Reverse Mathematics II BISH (based on IL) 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 and Constructive Analysis Philosophical implications Constructive Reverse Mathematics II BISH (based on IL) 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. ) Axioms of R : ¬ ( x > 0 ∧ x < 0)
Introduction NSA and Constructive Analysis Philosophical implications Constructive Reverse Mathematics II BISH (based on IL) 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. ) Axioms of R : ¬ ( x > 0 ∧ x < 0) Axioms of R : ∼ ( x > 0 ∧ x < 0)
Introduction NSA and Constructive Analysis Philosophical implications Constructive Reverse Mathematics III
Introduction NSA and Constructive Analysis Philosophical implications Constructive Reverse Mathematics III BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic
Introduction NSA and Constructive Analysis Philosophical implications Constructive Reverse Mathematics III BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ 1 , ¬¬ P → P �
Introduction NSA and Constructive Analysis Philosophical implications Constructive Reverse Mathematics III BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ 1 , ¬¬ P → P � MPR: ( ∀ x ∈ R )( ¬¬ ( x > 0) → x > 0) �
Introduction NSA and Constructive Analysis Philosophical implications Constructive Reverse Mathematics III BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ 1 , ¬¬ P → P � MPR: ( ∀ x ∈ R )( ¬¬ ( x > 0) → x > 0) � EXT: the extensionality theorem
Introduction NSA and Constructive Analysis Philosophical implications Constructive Reverse Mathematics III BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant MP: For P ∈ Σ 1 , ¬¬ P → P � MPR: ( ∀ x ∈ R )( ¬¬ ( x > 0) → x > 0) � EXT: the extensionality theorem
Introduction NSA and Constructive Analysis Philosophical implications Constructive Reverse Mathematics III BISH (based on IL) NSA (based on CL) non-constructive/non-algorithmic non-Ω-invariant MP: For P ∈ Σ 1 , ¬¬ P → P MP : For P ∈ Σ 1 , ∼∼ P ⇛ P � � MPR: ( ∀ x ∈ R )( ¬¬ ( x > 0) → x > 0) � EXT: the extensionality theorem
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