❇ ❇ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy A simple example Suppose we have proved ( ∃ n ∈ ◆ ) ϕ ( n ). ( ϕ is quantifier-free) Program ❆ to find n 0 ∈ ◆ s.t. ϕ ( n 0 ): 0) Define m := 0 . 1) Check ϕ ( m ) . 2) If ϕ ( m ) is TRUE, return m , otherwise define m := m + 1 and go to 1). The program ❆ will certainly halt. Nonstandard program ❇ to find n 0 ∈ ◆ s.t. ϕ ( n 0 ): 0) For i = 0 .. . . . ω do If ϕ ( i ) is TRUE, return i and halt.
❇ ❇ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy A simple example Suppose we have proved ( ∃ n ∈ ◆ ) ϕ ( n ). ( ϕ is quantifier-free) Program ❆ to find n 0 ∈ ◆ s.t. ϕ ( n 0 ): 0) Define m := 0 . 1) Check ϕ ( m ) . 2) If ϕ ( m ) is TRUE, return m , otherwise define m := m + 1 and go to 1). The program ❆ will certainly halt. Nonstandard program ❇ to find n 0 ∈ ◆ s.t. ϕ ( n 0 ): 0) For i = 0 .. . . . ω do If ϕ ( i ) is TRUE, return i and halt. Continue otherwise.
❇ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy A simple example Suppose we have proved ( ∃ n ∈ ◆ ) ϕ ( n ). ( ϕ is quantifier-free) Program ❆ to find n 0 ∈ ◆ s.t. ϕ ( n 0 ): 0) Define m := 0 . 1) Check ϕ ( m ) . 2) If ϕ ( m ) is TRUE, return m , otherwise define m := m + 1 and go to 1). The program ❆ will certainly halt. Nonstandard program ❇ to find n 0 ∈ ◆ s.t. ϕ ( n 0 ): 0) For i = 0 .. . . . ω do If ϕ ( i ) is TRUE, return i and halt. Continue otherwise. The program ❇ will certainly halt at some finite stage.
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy A simple example Suppose we have proved ( ∃ n ∈ ◆ ) ϕ ( n ). ( ϕ is quantifier-free) Program ❆ to find n 0 ∈ ◆ s.t. ϕ ( n 0 ): 0) Define m := 0 . 1) Check ϕ ( m ) . 2) If ϕ ( m ) is TRUE, return m , otherwise define m := m + 1 and go to 1). The program ❆ will certainly halt. Nonstandard program ❇ to find n 0 ∈ ◆ s.t. ϕ ( n 0 ): 0) For i = 0 .. . . . ω do If ϕ ( i ) is TRUE, return i and halt. Continue otherwise. The program ❇ will certainly halt at some finite stage. ❇ depends on the infinite ω , but not on the choice of ω .
◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Turing Computability We always assume that A ⊂ ◆ ⊂ ∗ ◆ and that ω is infinite.
◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Turing Computability We always assume that A ⊂ ◆ ⊂ ∗ ◆ and that ω is infinite. Definition The set A is ω -invariant if there is ψ ∈ ∆ 0 s.t. for all infinite ω ,
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Turing Computability We always assume that A ⊂ ◆ ⊂ ∗ ◆ and that ω is infinite. Definition The set A is ω -invariant if there is ψ ∈ ∆ 0 s.t. for all infinite ω , A = { k ∈ ◆ : ψ ( k , ω ) } .
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Turing Computability We always assume that A ⊂ ◆ ⊂ ∗ ◆ and that ω is infinite. Definition The set A is ω -invariant if there is ψ ∈ ∆ 0 s.t. for all infinite ω , A = { k ∈ ◆ : ψ ( k , ω ) } . The set A depends on ω , but not on the choice of ω .
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Turing Computability We always assume that A ⊂ ◆ ⊂ ∗ ◆ and that ω is infinite. Definition The set A is ω -invariant if there is ψ ∈ ∆ 0 s.t. for all infinite ω , A = { k ∈ ◆ : ψ ( k , ω ) } . The set A depends on ω , but not on the choice of ω . Theorem The ∆ 1 -sets ( =Turing computable ) are exactly the ω -invariant sets.
◆ ◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy The Limit Lemma
◆ ◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy The Limit Lemma Theorem (Limit lemma) f ≤ T 0 ′ ⇐ ⇒ f ∈ ∆ 2 ⇐ ⇒ f = lim n →∞ f n ( f n is computable)
◆ ◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy The Limit Lemma Theorem (Limit lemma) f ≤ T 0 ′ ⇐ ⇒ f ∈ ∆ 2 ⇐ ⇒ f = lim n →∞ f n ( f n is computable) 0 ′ is a decision procedure for Σ 1 -formulas called ‘halting problem’.
◆ ◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy The Limit Lemma Theorem (Limit lemma) f ≤ T 0 ′ ⇐ ⇒ f ∈ ∆ 2 ⇐ ⇒ f = lim n →∞ f n ( f n is computable) 0 ′ is a decision procedure for Σ 1 -formulas called ‘halting problem’. Theorem (Hyperlimit Lemma) f ≤ T Π 1 ⇐ ⇒ f ∈ ∆ 2 ⇐ ⇒ f = f ω ( f n is computable)
◆ ◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy The Limit Lemma Theorem (Limit lemma) f ≤ T 0 ′ ⇐ ⇒ f ∈ ∆ 2 ⇐ ⇒ f = lim n →∞ f n ( f n is computable) 0 ′ is a decision procedure for Σ 1 -formulas called ‘halting problem’. Theorem (Hyperlimit Lemma) f ≤ T Π 1 ⇐ ⇒ f ∈ ∆ 2 ⇐ ⇒ f = f ω ( f n is computable) Π 1 is a decision procedure for Σ 1 -formulas, given by:
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy The Limit Lemma Theorem (Limit lemma) f ≤ T 0 ′ ⇐ ⇒ f ∈ ∆ 2 ⇐ ⇒ f = lim n →∞ f n ( f n is computable) 0 ′ is a decision procedure for Σ 1 -formulas called ‘halting problem’. Theorem (Hyperlimit Lemma) f ≤ T Π 1 ⇐ ⇒ f ∈ ∆ 2 ⇐ ⇒ f = f ω ( f n is computable) Π 1 is a decision procedure for Σ 1 -formulas, given by: Theorem ( Π 1 ) For every ϕ ∈ ∆ 0 , we have ( ∀ n ∈ ◆ ) ϕ ( n ) → ( ∀ n ∈ ∗ ◆ ) ϕ ( n ) .
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy The Limit Lemma Theorem (Limit lemma) f ≤ T 0 ′ ⇐ ⇒ f ∈ ∆ 2 ⇐ ⇒ f = lim n →∞ f n ( f n is computable) 0 ′ is a decision procedure for Σ 1 -formulas called ‘halting problem’. Theorem (Hyperlimit Lemma) f ≤ T Π 1 ⇐ ⇒ f ∈ ∆ 2 ⇐ ⇒ f = f ω ( f n is computable) Π 1 is a decision procedure for Σ 1 -formulas, given by: Theorem ( Π 1 ) For every ϕ ∈ ∆ 0 , we have ( ∀ n ∈ ◆ ) ϕ ( n ) → ( ∀ n ∈ ∗ ◆ ) ϕ ( n ) . Also called ‘Transfer principle for Π 1 -formulas’ or ‘Π 1 -transfer’.
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Origins in RM
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Origins in RM ERNA = Nonstandard Analysis in I ∆ 0 + exp.
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Origins in RM ERNA = Nonstandard Analysis in I ∆ 0 + exp. RCA 0 defines exactly the ∆ 1 -sets.
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Origins in RM ERNA = Nonstandard Analysis in I ∆ 0 + exp. RCA 0 defines exactly the ∆ 1 -sets. “Theorem”
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Origins in RM ERNA = Nonstandard Analysis in I ∆ 0 + exp. RCA 0 defines exactly the ∆ 1 -sets. “Theorem” 1 . If RCA 0 proves T (=) , then ERNA proves T ( ≈ ) .
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Origins in RM ERNA = Nonstandard Analysis in I ∆ 0 + exp. RCA 0 defines exactly the ∆ 1 -sets. “Theorem” 1 . If RCA 0 proves T (=) , then ERNA proves T ( ≈ ) . 2 . If RCA 0 proves [ T (=) ⇔ WKL 0 ] ,
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Origins in RM ERNA = Nonstandard Analysis in I ∆ 0 + exp. RCA 0 defines exactly the ∆ 1 -sets. “Theorem” 1 . If RCA 0 proves T (=) , then ERNA proves T ( ≈ ) . 2 . If RCA 0 proves [ T (=) ⇔ WKL 0 ] , then ERNA proves [ T ( ≈ ) ⇔ Π 1 -TRANS] .
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Origins in RM ERNA = Nonstandard Analysis in I ∆ 0 + exp. RCA 0 defines exactly the ∆ 1 -sets. “Theorem” 1 . If RCA 0 proves T (=) , then ERNA proves T ( ≈ ) . 2 . If RCA 0 proves [ T (=) ⇔ WKL 0 ] , then ERNA proves [ T ( ≈ ) ⇔ Π 1 -TRANS] . Here, T (=) is a theorem of ordinary Mathematics.
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Origins in RM ERNA = Nonstandard Analysis in I ∆ 0 + exp. RCA 0 defines exactly the ∆ 1 -sets. “Theorem” 1 . If RCA 0 proves T (=) , then ERNA proves T ( ≈ ) . 2 . If RCA 0 proves [ T (=) ⇔ WKL 0 ] , then ERNA proves [ T ( ≈ ) ⇔ Π 1 -TRANS] . Here, T (=) is a theorem of ordinary Mathematics. Example of 1: Intermediate Value Theorem.
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Origins in RM ERNA = Nonstandard Analysis in I ∆ 0 + exp. RCA 0 defines exactly the ∆ 1 -sets. “Theorem” 1 . If RCA 0 proves T (=) , then ERNA proves T ( ≈ ) . 2 . If RCA 0 proves [ T (=) ⇔ WKL 0 ] , then ERNA proves [ T ( ≈ ) ⇔ Π 1 -TRANS] . Here, T (=) is a theorem of ordinary Mathematics. Example of 1: Intermediate Value Theorem. Example of 2: Peano’s theorem for diff. eq. y ′ = f ( x , y ).
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Origins in RM ERNA = Nonstandard Analysis in I ∆ 0 + exp. RCA 0 defines exactly the ∆ 1 -sets. “Theorem” 1 . If RCA 0 proves T (=) , then ERNA proves T ( ≈ ) . 2 . If RCA 0 proves [ T (=) ⇔ WKL 0 ] , then ERNA proves [ T ( ≈ ) ⇔ Π 1 -TRANS] . Here, T (=) is a theorem of ordinary Mathematics. Example of 1: Intermediate Value Theorem. Example of 2: Peano’s theorem for diff. eq. y ′ = f ( x , y ). But RCA 0 and WKL 0 are recursion theoretic!
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Origins in RM ERNA = Nonstandard Analysis in I ∆ 0 + exp. RCA 0 defines exactly the ∆ 1 -sets. “Theorem” 1 . If RCA 0 proves T (=) , then ERNA proves T ( ≈ ) . 2 . If RCA 0 proves [ T (=) ⇔ WKL 0 ] , then ERNA proves [ T ( ≈ ) ⇔ Π 1 -TRANS] . Here, T (=) is a theorem of ordinary Mathematics. Example of 1: Intermediate Value Theorem. Example of 2: Peano’s theorem for diff. eq. y ′ = f ( x , y ). But RCA 0 and WKL 0 are recursion theoretic! How about ERNA and Π 1 -TRANS?
◆ ◆ ◆ ◆ ◆ ◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Constructive Reverse Mathematics CRM = RM in Bishop’s ‘constructive analysis’.
◆ ◆ ◆ ◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Constructive Reverse Mathematics CRM = RM in Bishop’s ‘constructive analysis’. An important principle is: Principle (Σ 1 -excluded middle or LPO) For every q.f. formula ϕ , we have ( ∃ n ∈ ◆ ) ϕ ( n ) ∨ ( ∀ n ∈ ◆ ) ¬ ϕ ( n ) .
◆ ◆ ◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Constructive Reverse Mathematics CRM = RM in Bishop’s ‘constructive analysis’. An important principle is: Principle (Σ 1 -excluded middle or LPO) For every q.f. formula ϕ , we have ( ∃ n ∈ ◆ ) ϕ ( n ) ∨ ( ∀ n ∈ ◆ ) ¬ ϕ ( n ) . The previous principle states: There is a finite procedure that decides whether ( ∃ n ∈ ◆ ) ϕ ( n ) or not.
◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Constructive Reverse Mathematics CRM = RM in Bishop’s ‘constructive analysis’. An important principle is: Principle (Σ 1 -excluded middle or LPO) For every q.f. formula ϕ , we have ( ∃ n ∈ ◆ ) ϕ ( n ) ∨ ( ∀ n ∈ ◆ ) ¬ ϕ ( n ) . The previous principle states: There is a finite procedure that decides whether ( ∃ n ∈ ◆ ) ϕ ( n ) or not. Principle ( Π 1 -Transfer) For every q.f. formula ϕ , we have ( ∃ n ∈ ◆ ) ϕ ( n ) ∨ ( ∀ n ∈ ∗ ◆ ) ¬ ϕ ( n ) .
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Constructive Reverse Mathematics CRM = RM in Bishop’s ‘constructive analysis’. An important principle is: Principle (Σ 1 -excluded middle or LPO) For every q.f. formula ϕ , we have ( ∃ n ∈ ◆ ) ϕ ( n ) ∨ ( ∀ n ∈ ◆ ) ¬ ϕ ( n ) . The previous principle states: There is a finite procedure that decides whether ( ∃ n ∈ ◆ ) ϕ ( n ) or not. Principle ( Π 1 -Transfer) For every q.f. formula ϕ , we have ( ∃ n ∈ ◆ ) ϕ ( n ) ∨ ( ∀ n ∈ ∗ ◆ ) ¬ ϕ ( n ) . The previous principle is equivalent to: There is an ω -invariant procedure that decides whether ( ∃ n ∈ ◆ ) ϕ ( n ) or not.
❘ ❘ ❘ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Constructive Reverse Mathematics
❘ ❘ ❘ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Constructive Reverse Mathematics In CRM, LPO is equivalent to MCT and to
❘ ❘ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Constructive Reverse Mathematics In CRM, LPO is equivalent to MCT and to Principle ( ∀ x ∈ ❘ )( x > 0 ∨ ¬ ( x > 0))
❘ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Constructive Reverse Mathematics In CRM, LPO is equivalent to MCT and to Principle ( ∀ x ∈ ❘ )( x > 0 ∨ ¬ ( x > 0)) The previous principle should be read: For x ∈ ❘ , there is a finite procedure that decides if x > 0.
❘ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Constructive Reverse Mathematics In CRM, LPO is equivalent to MCT and to Principle ( ∀ x ∈ ❘ )( x > 0 ∨ ¬ ( x > 0)) The previous principle should be read: For x ∈ ❘ , there is a finite procedure that decides if x > 0. In NSA, Π 1 -TRANS is equivalent to MCT( ≈ ) and to
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Constructive Reverse Mathematics In CRM, LPO is equivalent to MCT and to Principle ( ∀ x ∈ ❘ )( x > 0 ∨ ¬ ( x > 0)) The previous principle should be read: For x ∈ ❘ , there is a finite procedure that decides if x > 0. In NSA, Π 1 -TRANS is equivalent to MCT( ≈ ) and to Principle For x ∈ ❘ , there is an ω -invariant procedure that decides if x > 0 .
◆ ◆ ◆ ◆ ◆ ◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Lost in translation
◆ ◆ ◆ ◆ ◆ ◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Lost in translation Hypernegation provides a translation between NSA and CRM:
◆ ◆ ◆ ◆ ◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Lost in translation Hypernegation provides a translation between NSA and CRM: Definition (Hypernegation) ∼ [( ∃ n ∈ ◆ ) ϕ ( n )]
◆ ◆ ◆ ◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Lost in translation Hypernegation provides a translation between NSA and CRM: Definition (Hypernegation) ∼ [( ∃ n ∈ ◆ ) ϕ ( n )] ≡ ( ∀ n ∈ ∗ ◆ ) ¬ ϕ ( n ).
◆ ◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Lost in translation Hypernegation provides a translation between NSA and CRM: Definition (Hypernegation) ∼ [( ∃ n ∈ ◆ ) ϕ ( n )] ≡ ( ∀ n ∈ ∗ ◆ ) ¬ ϕ ( n ). ∼ [( ∀ n ∈ ◆ ) ϕ ( n )] ≡ ( ∃ n ∈ ∗ ◆ ) ¬ ϕ ( n ).
◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Lost in translation Hypernegation provides a translation between NSA and CRM: Definition (Hypernegation) ∼ [( ∃ n ∈ ◆ ) ϕ ( n )] ≡ ( ∀ n ∈ ∗ ◆ ) ¬ ϕ ( n ). ∼ [( ∀ n ∈ ◆ ) ϕ ( n )] ≡ ( ∃ n ∈ ∗ ◆ ) ¬ ϕ ( n ). ∼ [( ∀ n ∈ ∗ ◆ ) ϕ ( n )]
◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Lost in translation Hypernegation provides a translation between NSA and CRM: Definition (Hypernegation) ∼ [( ∃ n ∈ ◆ ) ϕ ( n )] ≡ ( ∀ n ∈ ∗ ◆ ) ¬ ϕ ( n ). ∼ [( ∀ n ∈ ◆ ) ϕ ( n )] ≡ ( ∃ n ∈ ∗ ◆ ) ¬ ϕ ( n ). ∼ [( ∀ n ∈ ∗ ◆ ) ϕ ( n )] ≡ ( ∃ n ≤ ω ) ¬ ϕ ( n ).
◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Lost in translation Hypernegation provides a translation between NSA and CRM: Definition (Hypernegation) ∼ [( ∃ n ∈ ◆ ) ϕ ( n )] ≡ ( ∀ n ∈ ∗ ◆ ) ¬ ϕ ( n ). ∼ [( ∀ n ∈ ◆ ) ϕ ( n )] ≡ ( ∃ n ∈ ∗ ◆ ) ¬ ϕ ( n ). ∼ [( ∀ n ∈ ∗ ◆ ) ϕ ( n )] ≡ ( ∃ n ≤ ω ) ¬ ϕ ( n ). ( ω is independent of parameters in ϕ )
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Lost in translation Hypernegation provides a translation between NSA and CRM: Definition (Hypernegation) ∼ [( ∃ n ∈ ◆ ) ϕ ( n )] ≡ ( ∀ n ∈ ∗ ◆ ) ¬ ϕ ( n ). ∼ [( ∀ n ∈ ◆ ) ϕ ( n )] ≡ ( ∃ n ∈ ∗ ◆ ) ¬ ϕ ( n ). ∼ [( ∀ n ∈ ∗ ◆ ) ϕ ( n )] ≡ ( ∃ n ≤ ω ) ¬ ϕ ( n ). ( ω is independent of parameters in ϕ ) ∼ [( ∃ n ∈ ∗ ◆ ) ϕ ( n )] ≡ ( ∀ n ≤ ω ) ¬ ϕ ( n ).
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Lost in translation Hypernegation provides a translation between NSA and CRM: Definition (Hypernegation) ∼ [( ∃ n ∈ ◆ ) ϕ ( n )] ≡ ( ∀ n ∈ ∗ ◆ ) ¬ ϕ ( n ). ∼ [( ∀ n ∈ ◆ ) ϕ ( n )] ≡ ( ∃ n ∈ ∗ ◆ ) ¬ ϕ ( n ). ∼ [( ∀ n ∈ ∗ ◆ ) ϕ ( n )] ≡ ( ∃ n ≤ ω ) ¬ ϕ ( n ). ( ω is independent of parameters in ϕ ) ∼ [( ∃ n ∈ ∗ ◆ ) ϕ ( n )] ≡ ( ∀ n ≤ ω ) ¬ ϕ ( n ). With the hypernegation ∼ , we get the usual results from CRM:
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Lost in translation Hypernegation provides a translation between NSA and CRM: Definition (Hypernegation) ∼ [( ∃ n ∈ ◆ ) ϕ ( n )] ≡ ( ∀ n ∈ ∗ ◆ ) ¬ ϕ ( n ). ∼ [( ∀ n ∈ ◆ ) ϕ ( n )] ≡ ( ∃ n ∈ ∗ ◆ ) ¬ ϕ ( n ). ∼ [( ∀ n ∈ ∗ ◆ ) ϕ ( n )] ≡ ( ∃ n ≤ ω ) ¬ ϕ ( n ). ( ω is independent of parameters in ϕ ) ∼ [( ∃ n ∈ ∗ ◆ ) ϕ ( n )] ≡ ( ∀ n ≤ ω ) ¬ ϕ ( n ). With the hypernegation ∼ , we get the usual results from CRM: Theorem In NSA, LPO is equivalent to MP plus LLPO
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Lost in translation Hypernegation provides a translation between NSA and CRM: Definition (Hypernegation) ∼ [( ∃ n ∈ ◆ ) ϕ ( n )] ≡ ( ∀ n ∈ ∗ ◆ ) ¬ ϕ ( n ). ∼ [( ∀ n ∈ ◆ ) ϕ ( n )] ≡ ( ∃ n ∈ ∗ ◆ ) ¬ ϕ ( n ). ∼ [( ∀ n ∈ ∗ ◆ ) ϕ ( n )] ≡ ( ∃ n ≤ ω ) ¬ ϕ ( n ). ( ω is independent of parameters in ϕ ) ∼ [( ∃ n ∈ ∗ ◆ ) ϕ ( n )] ≡ ( ∀ n ≤ ω ) ¬ ϕ ( n ). With the hypernegation ∼ , we get the usual results from CRM: Theorem In NSA, LPO is equivalent to MP plus LLPO LPO: P ∨ ∼ P , MP: ∼∼ P → P , LLPO: ∼ ( P ∧ Q ) → ∼ P ∨ ∼ Q ( P , Q ∈ Σ 1 )
❘ ❘ ◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Lost in translation Theorem In NSA, TFAE 1 LLPO
❘ ◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Lost in translation Theorem In NSA, TFAE 1 LLPO 2 ( ∀ x ∈ ❘ )( ∼ ( x > 0) ∨ ∼ ( x < 0))
◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Lost in translation Theorem In NSA, TFAE 1 LLPO 2 ( ∀ x ∈ ❘ )( ∼ ( x > 0) ∨ ∼ ( x < 0)) 3 ( ∀ x , y ∈ ❘ )( xy = 0 → x = 0 ∨ y = 0)
◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Lost in translation Theorem In NSA, TFAE 1 LLPO 2 ( ∀ x ∈ ❘ )( ∼ ( x > 0) ∨ ∼ ( x < 0)) 3 ( ∀ x , y ∈ ❘ )( xy = 0 → x = 0 ∨ y = 0) LLPO: ∼ ( P ∧ Q ) → ∼ P ∨ ∼ Q ( P , Q ∈ Σ 1 )
◆ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Lost in translation Theorem In NSA, TFAE 1 LLPO 2 ( ∀ x ∈ ❘ )( ∼ ( x > 0) ∨ ∼ ( x < 0)) 3 ( ∀ x , y ∈ ❘ )( xy = 0 → x = 0 ∨ y = 0) LLPO: ∼ ( P ∧ Q ) → ∼ P ∨ ∼ Q ( P , Q ∈ Σ 1 ) Why does this connection exist?
Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy Lost in translation Theorem In NSA, TFAE 1 LLPO 2 ( ∀ x ∈ ❘ )( ∼ ( x > 0) ∨ ∼ ( x < 0)) 3 ( ∀ x , y ∈ ❘ )( xy = 0 → x = 0 ∨ y = 0) LLPO: ∼ ( P ∧ Q ) → ∼ P ∨ ∼ Q ( P , Q ∈ Σ 1 ) Why does this connection exist? Compare ◆ and N .
◆ ❆ ◆ ✷ ✷ ✶ ✶ ✶ ✶ ✷ ✸ ◆ ✵ ◆ ❆ ✵ ✶ ✷ ✸ ❆ ✷ ✶ ✸ ✶ ✷ ✸ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy The nonstandard Turing hierarchy In classical NSA, a number is either finite or infinite.
◆ ❆ ◆ ✷ ✷ ✶ ✶ ✶ ✶ ✷ ✸ ◆ ✵ ◆ ❆ ✵ ✶ ✷ ✸ ❆ ✷ ✶ ✸ ✶ ✷ ✸ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy The nonstandard Turing hierarchy In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity.
✷ ✷ ✶ ✶ ✶ ◆ ✵ ❆ ✵ ✶ ✷ ✸ ❆ ✷ ✶ ✸ ✶ ✷ ✸ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy The nonstandard Turing hierarchy In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆ ❆ , an extension of ∗ ◆ : 0 1 . . . ω ✶ . . . ω ✷ . . . ω ✸ . . . ✲ � �� � ◆
✷ ✷ ✶ ✶ ✶ ◆ ✵ ❆ ✷ ✶ ✸ ✶ ✷ ✸ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy The nonstandard Turing hierarchy In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆ ❆ , an extension of ∗ ◆ : 0 1 . . . ω ✶ . . . ω ✷ . . . ω ✸ . . . ✲ � �� � ◆ where ❆ = { ✵ , ✶ , ✷ , ✸ , . . . }
✷ ✷ ✶ ✶ ✶ ◆ ✵ ❆ ✷ ✶ ✸ ✶ ✷ ✸ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy The nonstandard Turing hierarchy In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆ ❆ , an extension of ∗ ◆ : 0 1 . . . ω ✶ . . . ω ✷ . . . ω ✸ . . . ✲ � �� � ◆ where ❆ = { ✵ , ✶ , ✷ , ✸ , . . . }
✷ ✷ ✶ ✶ ✶ ❆ ✷ ✶ ✸ ✶ ✷ ✸ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy The nonstandard Turing hierarchy In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆ ❆ , an extension of ∗ ◆ : 0 1 . . . ω ✶ . . . ω ✷ . . . ω ✸ . . . ✲ � �� � ◆ , ✵ -finite where ❆ = { ✵ , ✶ , ✷ , ✸ , . . . }
✷ ✷ ✶ ✶ ✶ ❆ ✷ ✶ ✸ ✶ ✷ ✸ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy The nonstandard Turing hierarchy In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆ ❆ , an extension of ∗ ◆ : 0 1 . . . ω ✶ . . . ω ✷ . . . ω ✸ . . . ✲ � �� � � �� ◆ , ✵ -finite ✵ -infinite where ❆ = { ✵ , ✶ , ✷ , ✸ , . . . }
✷ ✷ ✶ ✶ ✶ ❆ ✷ ✶ ✸ ✶ ✷ ✸ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy The nonstandard Turing hierarchy In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆ ❆ , an extension of ∗ ◆ : 0 1 . . . ω ✶ . . . ω ✷ . . . ω ✸ . . . ✲ � �� � � �� ◆ , finite infinite where ❆ = { ✵ , ✶ , ✷ , ✸ , . . . }
✷ ✷ ✶ ✶ ❆ ✷ ✶ ✸ ✶ ✷ ✸ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy The nonstandard Turing hierarchy In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆ ❆ , an extension of ∗ ◆ : ✶ -finite � �� � 0 1 . . . ω ✶ . . . ω ✷ . . . ω ✸ . . . ✲ � �� � � �� ◆ , finite infinite where ❆ = { ✵ , ✶ , ✷ , ✸ , . . . }
✷ ✷ ✶ ❆ ✷ ✶ ✸ ✶ ✷ ✸ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy The nonstandard Turing hierarchy In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆ ❆ , an extension of ∗ ◆ : ✶ -finite ✶ -infinite � �� � � �� 0 1 . . . ω ✶ . . . ω ✷ . . . ω ✸ . . . ✲ � �� � � �� ◆ , finite infinite where ❆ = { ✵ , ✶ , ✷ , ✸ , . . . }
✷ ✷ ✶ ❆ ✷ ✶ ✸ ✶ ✷ ✸ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy The nonstandard Turing hierarchy In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆ ❆ , an extension of ∗ ◆ : Fuzzy border: ✶ -finite ✶ -infinite � �� � � �� 0 1 . . . ω ✶ . . . ω ✷ . . . ω ✸ . . . ✲ � �� � � �� ◆ , finite infinite where ❆ = { ✵ , ✶ , ✷ , ✸ , . . . }
✷ ✷ ❆ ✷ ✶ ✸ ✶ ✷ ✸ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy The nonstandard Turing hierarchy In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆ ❆ , an extension of ∗ ◆ : Fuzzy border: no least ✶ -infinite number ✶ -finite ✶ -infinite � �� � � �� 0 1 . . . ω ✶ . . . ω ✷ . . . ω ✸ . . . ✲ � �� � � �� ◆ , finite infinite where ❆ = { ✵ , ✶ , ✷ , ✸ , . . . }
✷ ✷ ❆ ✷ ✶ ✸ ✶ ✷ ✸ Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy The nonstandard Turing hierarchy In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆ ❆ , an extension of ∗ ◆ : ✶ -finite ✶ -infinite � �� � � �� 0 1 . . . ω ✶ . . . ω ✷ . . . ω ✸ . . . ✲ � �� � � �� ◆ , finite infinite where ❆ = { ✵ , ✶ , ✷ , ✸ , . . . }
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