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Nonstandard Analysis, Computability Theory, and metastability Dag - PowerPoint PPT Presentation

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Nonstandard Analysis, Computability Theory, and metastability Dag Normann & Sam Sanders CCC17, Nancy, June 2017 Motivation: G odel-Friedman-Tao Motivation: G odel-Friedman-Tao G odels famous incompleteness theorems imply that

  1. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Metastability: Tao, G¨ odel, and a dead horse Classically equivalent definitions behave differently in computable math. The ‘epsilon-delta’ definition of a Cauchy sequence: ( ∀ k 0 )( ∃ M 0 )( ∀ m , n ≥ M )( | x n − x m | < 1 / k ) (1) is classically equivalent to ‘metastability’ (Tao, G¨ odel, Kreisel, etc): ( ∀ k 0 , F 1 )( ∃ N 0 )( ∀ m , n ∈ [ N , F ( N )])( | x n − x m | < 1 / k ) (2) Computational behaviour: (for non-decreasing x ( · ) in [0 , 1]; MCT) For (1), there is NO computable upper bound for ( ∃ M 0 ). For (2), there is a highly uniform and elementary bound θ for ( ∃ N 0 ): ( ∀ k 0 , F 1 )( ∃ N 0 ∈ θ ( F , k ))( ∀ m , n ∈ [ N , F ( N )])( | x n − x m | < 1 / k ) (3)

  2. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Metastability: Tao, G¨ odel, and a dead horse Classically equivalent definitions behave differently in computable math. The ‘epsilon-delta’ definition of a Cauchy sequence: ( ∀ k 0 )( ∃ M 0 )( ∀ m , n ≥ M )( | x n − x m | < 1 / k ) (1) is classically equivalent to ‘metastability’ (Tao, G¨ odel, Kreisel, etc): ( ∀ k 0 , F 1 )( ∃ N 0 )( ∀ m , n ∈ [ N , F ( N )])( | x n − x m | < 1 / k ) (2) Computational behaviour: (for non-decreasing x ( · ) in [0 , 1]; MCT) For (1), there is NO computable upper bound for ( ∃ M 0 ). For (2), there is a highly uniform and elementary bound θ for ( ∃ N 0 ): ( ∀ k 0 , F 1 )( ∃ N 0 ∈ θ ( F , k ))( ∀ m , n ∈ [ N , F ( N )])( | x n − x m | < 1 / k ) (3) where θ ( k , F ) := � 0 , F (0) , F 2 (0) , . . . , F k +1 (0) � is independent of x ( · ) .

  3. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Metastability: Tao, G¨ odel, and a dead horse Classically equivalent definitions behave differently in computable math. The ‘epsilon-delta’ definition of a Cauchy sequence: ( ∀ k 0 )( ∃ M 0 )( ∀ m , n ≥ M )( | x n − x m | < 1 / k ) (1) is classically equivalent to ‘metastability’ (Tao, G¨ odel, Kreisel, etc): ( ∀ k 0 , F 1 )( ∃ N 0 )( ∀ m , n ∈ [ N , F ( N )])( | x n − x m | < 1 / k ) (2) Computational behaviour: (for non-decreasing x ( · ) in [0 , 1]; MCT) For (1), there is NO computable upper bound for ( ∃ M 0 ). For (2), there is a highly uniform and elementary bound θ for ( ∃ N 0 ): ( ∀ k 0 , F 1 )( ∃ N 0 ∈ θ ( F , k ))( ∀ m , n ∈ [ N , F ( N )])( | x n − x m | < 1 / k ) (3) where θ ( k , F ) := � 0 , F (0) , F 2 (0) , . . . , F k +1 (0) � is independent of x ( · ) . Metastability trade-off:finite domain [ N , F ( N )] yields uniform and comp θ

  4. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability The metastability trade-off in general Whereas in general noneffective proofs of convergence statements [. . . ] might not provide a (uniform) computable rate of convergence, highly uniform rates of metastability can under very general conditions always be extracted using tools from mathematical logic [. . . ] .

  5. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability The metastability trade-off in general Whereas in general noneffective proofs of convergence statements [. . . ] might not provide a (uniform) computable rate of convergence, highly uniform rates of metastability can under very general conditions always be extracted using tools from mathematical logic [. . . ] . Ulrich Kohlenbach and Angeliki Koutsoukou-Argyraki, Rates of convergence and metastability for abstract Cauchy problems generated by accretive operators , J. Math. Anal. Appl. 423 (2015), no. 2,1089-1112.

  6. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability The metastability trade-off in general Whereas in general noneffective proofs of convergence statements [. . . ] might not provide a (uniform) computable rate of convergence, highly uniform rates of metastability can under very general conditions always be extracted using tools from mathematical logic [. . . ] . Ulrich Kohlenbach and Angeliki Koutsoukou-Argyraki, Rates of convergence and metastability for abstract Cauchy problems generated by accretive operators , J. Math. Anal. Appl. 423 (2015), no. 2,1089-1112. This talk: How general is the metastability trade-off?

  7. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability The metastability trade-off in general Whereas in general noneffective proofs of convergence statements [. . . ] might not provide a (uniform) computable rate of convergence, highly uniform rates of metastability can under very general conditions always be extracted using tools from mathematical logic [. . . ] . Ulrich Kohlenbach and Angeliki Koutsoukou-Argyraki, Rates of convergence and metastability for abstract Cauchy problems generated by accretive operators , J. Math. Anal. Appl. 423 (2015), no. 2,1089-1112. This talk: How general is the metastability trade-off? Can we use it to obtain computable mathematics in general?

  8. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability The metastability trade-off in general Whereas in general noneffective proofs of convergence statements [. . . ] might not provide a (uniform) computable rate of convergence, highly uniform rates of metastability can under very general conditions always be extracted using tools from mathematical logic [. . . ] . Ulrich Kohlenbach and Angeliki Koutsoukou-Argyraki, Rates of convergence and metastability for abstract Cauchy problems generated by accretive operators , J. Math. Anal. Appl. 423 (2015), no. 2,1089-1112. This talk: How general is the metastability trade-off? Can we use it to obtain computable mathematics in general? Metastability trade-off: introducing a finite domain yields uniform and computable results .

  9. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability A slight variation of metastability

  10. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability A slight variation of metastability Metastability trade-off: finite domain [ N , F ( N )], but highly uniform θ . ( ∀ k 0 , F 1 )( ∃ N 0 ∈ θ ( F , k ))( ∀ m , n ∈ [ N , F ( N )])( | x n − x m | < 1 / k ) (4)

  11. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability A slight variation of metastability Metastability trade-off: finite domain [ N , F ( N )], but highly uniform θ . ( ∀ k 0 , F 1 )( ∃ N 0 ∈ θ ( F , k ))( ∀ m , n ∈ [ N , F ( N )])( | x n − x m | < 1 / k ) (4) The ‘epsilon-delta’ definition of limit with a modulus of convergence: ( ∃ x 1 , g 1 )( ∀ k 0 , n 0 )( n ≥ g ( k ) → | x − x n | < 1 / k ) (5)

  12. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability A slight variation of metastability Metastability trade-off: finite domain [ N , F ( N )], but highly uniform θ . ( ∀ k 0 , F 1 )( ∃ N 0 ∈ θ ( F , k ))( ∀ m , n ∈ [ N , F ( N )])( | x n − x m | < 1 / k ) (4) The ‘epsilon-delta’ definition of limit with a modulus of convergence: ( ∃ x 1 , g 1 )( ∀ k 0 , n 0 )( n ≥ g ( k ) → | x − x n | < 1 / k ) (5) is classically equivalent to the definition of ‘metastable limit’: ( ∀ G 2 )( ∃ x 1 , g 1 )( ∀ k 0 , n 0 ≤ G ( x , g ))( n ≥ g ( k ) → | x − x n | < 1 / k ) (6)

  13. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability A slight variation of metastability Metastability trade-off: finite domain [ N , F ( N )], but highly uniform θ . ( ∀ k 0 , F 1 )( ∃ N 0 ∈ θ ( F , k ))( ∀ m , n ∈ [ N , F ( N )])( | x n − x m | < 1 / k ) (4) The ‘epsilon-delta’ definition of limit with a modulus of convergence: ( ∃ x 1 , g 1 )( ∀ k 0 , n 0 )( n ≥ g ( k ) → | x − x n | < 1 / k ) (5) is classically equivalent to the definition of ‘metastable limit’: ( ∀ G 2 )( ∃ x 1 , g 1 )( ∀ k 0 , n 0 ≤ G ( x , g ))( n ≥ g ( k ) → | x − x n | < 1 / k ) (6) Upper bounds for x 1 , g 1 in (5) compute the Halting problem.

  14. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability A slight variation of metastability Metastability trade-off: finite domain [ N , F ( N )], but highly uniform θ . ( ∀ k 0 , F 1 )( ∃ N 0 ∈ θ ( F , k ))( ∀ m , n ∈ [ N , F ( N )])( | x n − x m | < 1 / k ) (4) The ‘epsilon-delta’ definition of limit with a modulus of convergence: ( ∃ x 1 , g 1 )( ∀ k 0 , n 0 )( n ≥ g ( k ) → | x − x n | < 1 / k ) (5) is classically equivalent to the definition of ‘metastable limit’: ( ∀ G 2 )( ∃ x 1 , g 1 )( ∀ k 0 , n 0 ≤ G ( x , g ))( n ≥ g ( k ) → | x − x n | < 1 / k ) (6) Upper bounds for x 1 , g 1 in (5) compute the Halting problem. In (4), replace ‘metastability’ by ‘metastable limit’ (6) and obtain:

  15. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability A slight variation of metastability Metastability trade-off: finite domain [ N , F ( N )], but highly uniform θ . ( ∀ k 0 , F 1 )( ∃ N 0 ∈ θ ( F , k ))( ∀ m , n ∈ [ N , F ( N )])( | x n − x m | < 1 / k ) (4) The ‘epsilon-delta’ definition of limit with a modulus of convergence: ( ∃ x 1 , g 1 )( ∀ k 0 , n 0 )( n ≥ g ( k ) → | x − x n | < 1 / k ) (5) is classically equivalent to the definition of ‘metastable limit’: ( ∀ G 2 )( ∃ x 1 , g 1 )( ∀ k 0 , n 0 ≤ G ( x , g ))( n ≥ g ( k ) → | x − x n | < 1 / k ) (6) Upper bounds for x 1 , g 1 in (5) compute the Halting problem. In (4), replace ‘metastability’ by ‘metastable limit’ (6) and obtain: ( ∀ G 2 )( ∃ x 1 , g 1 ∈ ζ ( G ))( ∀ k , n ≤ G ( x , g ))( n ≥ g ( k ) → | x − x n | < 1 k )

  16. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability A slight variation of metastability Metastability trade-off: finite domain [ N , F ( N )], but highly uniform θ . ( ∀ k 0 , F 1 )( ∃ N 0 ∈ θ ( F , k ))( ∀ m , n ∈ [ N , F ( N )])( | x n − x m | < 1 / k ) (4) The ‘epsilon-delta’ definition of limit with a modulus of convergence: ( ∃ x 1 , g 1 )( ∀ k 0 , n 0 )( n ≥ g ( k ) → | x − x n | < 1 / k ) (5) is classically equivalent to the definition of ‘metastable limit’: ( ∀ G 2 )( ∃ x 1 , g 1 )( ∀ k 0 , n 0 ≤ G ( x , g ))( n ≥ g ( k ) → | x − x n | < 1 / k ) (6) Upper bounds for x 1 , g 1 in (5) compute the Halting problem. In (4), replace ‘metastability’ by ‘metastable limit’ (6) and obtain: ( ∀ G 2 )( ∃ x 1 , g 1 ∈ ζ ( G ))( ∀ k , n ≤ G ( x , g ))( n ≥ g ( k ) → | x − x n | < 1 k ) (7) Metastability trade-off: finite domain [0 , G ( x , g )], but highly uniform ζ 3 .

  17. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability A slight variation of metastability Metastability trade-off: finite domain [ N , F ( N )], but highly uniform θ . ( ∀ k 0 , F 1 )( ∃ N 0 ∈ θ ( F , k ))( ∀ m , n ∈ [ N , F ( N )])( | x n − x m | < 1 / k ) (4) The ‘epsilon-delta’ definition of limit with a modulus of convergence: ( ∃ x 1 , g 1 )( ∀ k 0 , n 0 )( n ≥ g ( k ) → | x − x n | < 1 / k ) (5) is classically equivalent to the definition of ‘metastable limit’: ( ∀ G 2 )( ∃ x 1 , g 1 )( ∀ k 0 , n 0 ≤ G ( x , g ))( n ≥ g ( k ) → | x − x n | < 1 / k ) (6) Upper bounds for x 1 , g 1 in (5) compute the Halting problem. In (4), replace ‘metastability’ by ‘metastable limit’ (6) and obtain: ( ∀ G 2 )( ∃ x 1 , g 1 ∈ ζ ( G ))( ∀ k , n ≤ G ( x , g ))( n ≥ g ( k ) → | x − x n | < 1 k ) (7) Metastability trade-off: finite domain [0 , G ( x , g )], but highly uniform ζ 3 . MCT meta ( ζ ) is (7) for any non-decreasing x ( · ) in [0 , 1].

  18. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability A slight variation of metastability Metastability trade-off: finite domain [ N , F ( N )], but highly uniform θ . ( ∀ k 0 , F 1 )( ∃ N 0 ∈ θ ( F , k ))( ∀ m , n ∈ [ N , F ( N )])( | x n − x m | < 1 / k ) (4) The ‘epsilon-delta’ definition of limit with a modulus of convergence: ( ∃ x 1 , g 1 )( ∀ k 0 , n 0 )( n ≥ g ( k ) → | x − x n | < 1 / k ) (5) is classically equivalent to the definition of ‘metastable limit’: ( ∀ G 2 )( ∃ x 1 , g 1 )( ∀ k 0 , n 0 ≤ G ( x , g ))( n ≥ g ( k ) → | x − x n | < 1 / k ) (6) Upper bounds for x 1 , g 1 in (5) compute the Halting problem. In (4), replace ‘metastability’ by ‘metastable limit’ (6) and obtain: ( ∀ G 2 )( ∃ x 1 , g 1 ∈ ζ ( G ))( ∀ k , n ≤ G ( x , g ))( n ≥ g ( k ) → | x − x n | < 1 k ) (7) Metastability trade-off: finite domain [0 , G ( x , g )], but highly uniform ζ 3 . MCT meta ( ζ ) is (7) for any non-decreasing x ( · ) in [0 , 1]. How hard is it to (S1-S9) compute ζ as in MCT meta ( ζ )?

  19. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability A slight variation of metastability Metastability trade-off: finite domain [ N , F ( N )], but highly uniform θ . ( ∀ k 0 , F 1 )( ∃ N 0 ∈ θ ( F , k ))( ∀ m , n ∈ [ N , F ( N )])( | x n − x m | < 1 / k ) (4) The ‘epsilon-delta’ definition of limit with a modulus of convergence: ( ∃ x 1 , g 1 )( ∀ k 0 , n 0 )( n ≥ g ( k ) → | x − x n | < 1 / k ) (5) is classically equivalent to the definition of ‘metastable limit’: ( ∀ G 2 )( ∃ x 1 , g 1 )( ∀ k 0 , n 0 ≤ G ( x , g ))( n ≥ g ( k ) → | x − x n | < 1 / k ) (6) Upper bounds for x 1 , g 1 in (5) compute the Halting problem. In (4), replace ‘metastability’ by ‘metastable limit’ (6) and obtain: ( ∀ G 2 )( ∃ x 1 , g 1 ∈ ζ ( G ))( ∀ k , n ≤ G ( x , g ))( n ≥ g ( k ) → | x − x n | < 1 k ) (7) Metastability trade-off: finite domain [0 , G ( x , g )], but highly uniform ζ 3 . MCT meta ( ζ ) is (7) for any non-decreasing x ( · ) in [0 , 1]. How hard is it to (S1-S9) compute ζ as in MCT meta ( ζ )? Full SOA is needed!

  20. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Metastable WKL and MCT Modulo the Halting problem, ζ as in MCT meta ( ζ ) is the the special fan functional; the latter originates as follows.

  21. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Metastable WKL and MCT Modulo the Halting problem, ζ as in MCT meta ( ζ ) is the the special fan functional; the latter originates as follows. Weak K¨ onig’s lemma WKL states that an infinite binary tree has a path: � ( ∀ n 0 )( ∃ β 0 )( | β | = n ∧ β ∈ T ) → ( ∃ α 1 ≤ 1 1)( ∀ n 0 )( α n ∈ T ) � ( ∀ T ≤ 1 1)

  22. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Metastable WKL and MCT Modulo the Halting problem, ζ as in MCT meta ( ζ ) is the the special fan functional; the latter originates as follows. Weak K¨ onig’s lemma WKL states that an infinite binary tree has a path: � ( ∀ n 0 )( ∃ β 0 )( | β | = n ∧ β ∈ T ) → ( ∃ α 1 ≤ 1 1)( ∀ n 0 )( α n ∈ T ) � ( ∀ T ≤ 1 1) Such a path α ≤ 1 1 cannot be computed from the tree T only.

  23. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Metastable WKL and MCT Modulo the Halting problem, ζ as in MCT meta ( ζ ) is the the special fan functional; the latter originates as follows. Weak K¨ onig’s lemma WKL states that an infinite binary tree has a path: � ( ∀ n 0 )( ∃ β 0 )( | β | = n ∧ β ∈ T ) → ( ∃ α 1 ≤ 1 1)( ∀ n 0 )( α n ∈ T ) � ( ∀ T ≤ 1 1) Such a path α ≤ 1 1 cannot be computed from the tree T only. The special fan functional Θ computes a ‘metastable path’: � ( ∀ n 0 )( ∃ β 0 )( | β | = n ∧ β ∈ T ) ( ∀ T ≤ 1 1) (SCF(Θ)) → ( ∀ G 2 )( ∃ α 1 ∈ Θ( G ))( ∀ n 0 ≤ G ( α ))( α n ∈ T ) �

  24. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Metastable WKL and MCT Modulo the Halting problem, ζ as in MCT meta ( ζ ) is the the special fan functional; the latter originates as follows. Weak K¨ onig’s lemma WKL states that an infinite binary tree has a path: � ( ∀ n 0 )( ∃ β 0 )( | β | = n ∧ β ∈ T ) → ( ∃ α 1 ≤ 1 1)( ∀ n 0 )( α n ∈ T ) � ( ∀ T ≤ 1 1) Such a path α ≤ 1 1 cannot be computed from the tree T only. The special fan functional Θ computes a ‘metastable path’: � ( ∀ n 0 )( ∃ β 0 )( | β | = n ∧ β ∈ T ) ( ∀ T ≤ 1 1) (SCF(Θ)) → ( ∀ G 2 )( ∃ α 1 ∈ Θ( G ))( ∀ n 0 ≤ G ( α ))( α n ∈ T ) � Metastability trade-off: finite domain [0 , G ( α )], but highly uniform Θ 3 .

  25. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Metastable WKL and MCT Modulo the Halting problem, ζ as in MCT meta ( ζ ) is the the special fan functional; the latter originates as follows. Weak K¨ onig’s lemma WKL states that an infinite binary tree has a path: � ( ∀ n 0 )( ∃ β 0 )( | β | = n ∧ β ∈ T ) → ( ∃ α 1 ≤ 1 1)( ∀ n 0 )( α n ∈ T ) � ( ∀ T ≤ 1 1) Such a path α ≤ 1 1 cannot be computed from the tree T only. The special fan functional Θ computes a ‘metastable path’: � ( ∀ n 0 )( ∃ β 0 )( | β | = n ∧ β ∈ T ) ( ∀ T ≤ 1 1) (SCF(Θ)) → ( ∀ G 2 )( ∃ α 1 ∈ Θ( G ))( ∀ n 0 ≤ G ( α ))( α n ∈ T ) � Metastability trade-off: finite domain [0 , G ( α )], but highly uniform Θ 3 . Modulo the Halting problem, Θ computes ζ from MCT meta and vice versa

  26. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability The higher-order picture

  27. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability The higher-order picture ✻ Π 1 ∞ -CA 0 Π 1 1 -CA 0 ATR 0 ACA 0 WKL 0 RCA 0

  28. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability The higher-order picture ✻ � � Π 1 ∞ -CA 0 ( E 2 ) : ( ∃E 3 )( ∀ ϕ 2 ) ( ∃ f 1 )( ϕ ( f ) = 0) ↔ E ( ϕ ) = 0 Π 1 1 -CA 0 ATR 0 ACA 0 WKL 0 RCA 0

  29. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability The higher-order picture ✻ � � Π 1 ∞ -CA 0 ( E 2 ) : ( ∃E 3 )( ∀ ϕ 2 ) ( ∃ f 1 )( ϕ ( f ) = 0) ↔ E ( ϕ ) = 0 � � Π 1 ( S 2 ) : ( ∃ S 2 )( ∀ f 1 ) ( ∃ g 1 )( ∀ n 0 )( f ( gn ) = 0) ↔ S ( f ) = 0 1 -CA 0 ATR 0 ACA 0 WKL 0 RCA 0

  30. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability The higher-order picture ✻ � � Π 1 ∞ -CA 0 ( E 2 ) : ( ∃E 3 )( ∀ ϕ 2 ) ( ∃ f 1 )( ϕ ( f ) = 0) ↔ E ( ϕ ) = 0 � � Π 1 ( S 2 ) : ( ∃ S 2 )( ∀ f 1 ) ( ∃ g 1 )( ∀ n 0 )( f ( gn ) = 0) ↔ S ( f ) = 0 1 -CA 0 ATR 0 UATR 0 : ‘there is a realiser for transfinite recursion’ ACA 0 WKL 0 RCA 0

  31. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability The higher-order picture ✻ � � Π 1 ∞ -CA 0 ( E 2 ) : ( ∃E 3 )( ∀ ϕ 2 ) ( ∃ f 1 )( ϕ ( f ) = 0) ↔ E ( ϕ ) = 0 � � Π 1 ( S 2 ) : ( ∃ S 2 )( ∀ f 1 ) ( ∃ g 1 )( ∀ n 0 )( f ( gn ) = 0) ↔ S ( f ) = 0 1 -CA 0 ATR 0 UATR 0 : ‘there is a realiser for transfinite recursion’ � � ( µ 2 ) : ( ∃ µ 2 )( ∀ f 1 ) ( ∃ n 0 )( f ( n ) = 0) → f ( µ ( f )) = 0 ACA 0 WKL 0 RCA 0

  32. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability The higher-order picture ✻ � � Π 1 ∞ -CA 0 ( E 2 ) : ( ∃E 3 )( ∀ ϕ 2 ) ( ∃ f 1 )( ϕ ( f ) = 0) ↔ E ( ϕ ) = 0 � � Π 1 ( S 2 ) : ( ∃ S 2 )( ∀ f 1 ) ( ∃ g 1 )( ∀ n 0 )( f ( gn ) = 0) ↔ S ( f ) = 0 1 -CA 0 ATR 0 UATR 0 : ‘there is a realiser for transfinite recursion’ � � ( µ 2 ) : ( ∃ µ 2 )( ∀ f 1 ) ( ∃ n 0 )( f ( n ) = 0) → f ( µ ( f )) = 0 ACA 0 MUC : ( ∃ Ω 3 )( ∀ Y 2 )( ∀ f 1 , g 1 ≤ 1 1) WKL 0 ( f Ω( Y ) = g Ω( Y ) → Y ( f ) = Y ( g )) RCA 0

  33. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability The higher-order picture ✻ � � Π 1 ∞ -CA 0 ( E 2 ) : ( ∃E 3 )( ∀ ϕ 2 ) ( ∃ f 1 )( ϕ ( f ) = 0) ↔ E ( ϕ ) = 0 � � Π 1 ( S 2 ) : ( ∃ S 2 )( ∀ f 1 ) ( ∃ g 1 )( ∀ n 0 )( f ( gn ) = 0) ↔ S ( f ) = 0 1 -CA 0 ATR 0 UATR 0 : ‘there is a realiser for transfinite recursion’ � � ( µ 2 ) : ( ∃ µ 2 )( ∀ f 1 ) ( ∃ n 0 )( f ( n ) = 0) → f ( µ ( f )) = 0 ACA 0 MUC : ( ∃ Ω 3 )( ∀ Y 2 )( ∀ f 1 , g 1 ≤ 1 1) WKL 0 ( f Ω( Y ) = g Ω( Y ) → Y ( f ) = Y ( g )) The special fan functional Θ RCA 0

  34. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability The higher-order picture ✻ � � Π 1 ∞ -CA 0 ( E 2 ) : ( ∃E 3 )( ∀ ϕ 2 ) ( ∃ f 1 )( ϕ ( f ) = 0) ↔ E ( ϕ ) = 0 � � Π 1 ( S 2 ) : ( ∃ S 2 )( ∀ f 1 ) ( ∃ g 1 )( ∀ n 0 )( f ( gn ) = 0) ↔ S ( f ) = 0 1 -CA 0 ATR 0 UATR 0 : ‘there is a realiser for transfinite recursion’ ❅ � � ( µ 2 ) : ( ∃ µ 2 )( ∀ f 1 ) ( ∃ n 0 )( f ( n ) = 0) → f ( µ ( f )) = 0 ACA 0 ❅ MUC : ( ∃ Ω 3 )( ∀ Y 2 )( ∀ f 1 , g 1 ≤ 1 1) WKL 0 ( f Ω( Y ) = g Ω( Y ) → Y ( f ) = Y ( g )) ❄ ❄ ❄ ❄ The special fan functional Θ RCA 0 Red arrows denote relative (non-)computability (Kleene S1-S9)

  35. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability The higher-order picture ✻ � � Π 1 ∞ -CA 0 ( E 2 ) : ( ∃E 3 )( ∀ ϕ 2 ) ( ∃ f 1 )( ϕ ( f ) = 0) ↔ E ( ϕ ) = 0 � � Π 1 ( S 2 ) : ( ∃ S 2 )( ∀ f 1 ) ( ∃ g 1 )( ∀ n 0 )( f ( gn ) = 0) ↔ S ( f ) = 0 1 -CA 0 ATR 0 UATR 0 : ‘there is a realiser for transfinite recursion’ ❅ � � ( µ 2 ) : ( ∃ µ 2 )( ∀ f 1 ) ( ∃ n 0 )( f ( n ) = 0) → f ( µ ( f )) = 0 ACA 0 ❅ MUC : ( ∃ Ω 3 )( ∀ Y 2 )( ∀ f 1 , g 1 ≤ 1 1) WKL 0 ( f Ω( Y ) = g Ω( Y ) → Y ( f ) = Y ( g )) ❄ ❄ ❄ ❄ The special fan functional Θ RCA 0 Red arrows denote relative (non-)computability (Kleene S1-S9) NO type two functional computes Θ (Same for ζ as in MCT meta ( ζ ))

  36. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Some strange Reverse Mathematics By itself, the special fan functional Θ is finitistically reducible:

  37. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Some strange Reverse Mathematics By itself, the special fan functional Θ is finitistically reducible: RCA ω 0 + ( ∃ Θ)SCF(Θ) is conservative over WKL 0

  38. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Some strange Reverse Mathematics By itself, the special fan functional Θ is finitistically reducible: RCA ω 0 + ( ∃ Θ)SCF(Θ) is conservative over WKL 0 Together with ( µ 2 ) or ( S 2 ) (which imply WKL 0 ), Θ is powerful: 0 + ( ∃ Θ)SCF(Θ) + ( µ 2 ) proves ATR 0 RCA ω

  39. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Some strange Reverse Mathematics By itself, the special fan functional Θ is finitistically reducible: RCA ω 0 + ( ∃ Θ)SCF(Θ) is conservative over WKL 0 Together with ( µ 2 ) or ( S 2 ) (which imply WKL 0 ), Θ is powerful: 0 + ( ∃ Θ)SCF(Θ) + ( µ 2 ) proves ATR 0 RCA ω 0 + ( ∃ Θ)SCF(Θ) + ( S 2 ) proves Π 1 RCA ω 2 -CA 0

  40. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Some strange Reverse Mathematics By itself, the special fan functional Θ is finitistically reducible: RCA ω 0 + ( ∃ Θ)SCF(Θ) is conservative over WKL 0 Together with ( µ 2 ) or ( S 2 ) (which imply WKL 0 ), Θ is powerful: 0 + ( ∃ Θ)SCF(Θ) + ( µ 2 ) proves ATR 0 RCA ω 0 + ( ∃ Θ)SCF(Θ) + ( S 2 ) proves Π 1 RCA ω 2 -CA 0 And some strange Reverse Mathematics: 0 + ( ∃ Θ)SCF(Θ) proves [( µ 2 ) ↔ UATR 0 ] RCA ω

  41. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Some strange Reverse Mathematics By itself, the special fan functional Θ is finitistically reducible: RCA ω 0 + ( ∃ Θ)SCF(Θ) is conservative over WKL 0 Together with ( µ 2 ) or ( S 2 ) (which imply WKL 0 ), Θ is powerful: 0 + ( ∃ Θ)SCF(Θ) + ( µ 2 ) proves ATR 0 RCA ω 0 + ( ∃ Θ)SCF(Θ) + ( S 2 ) proves Π 1 RCA ω 2 -CA 0 And some strange Reverse Mathematics: 0 + ( ∃ Θ)SCF(Θ) proves [( µ 2 ) ↔ UATR 0 ] RCA ω but WKL 0 cannot prove this equivalence!

  42. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Θ is everywhere: metastable IVT The intermediate value theorem IVT is the prototypical non-constructive theorem.

  43. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Θ is everywhere: metastable IVT The intermediate value theorem IVT is the prototypical non-constructive theorem. Constructive versions of IVT involve ‘approximate’ intermediate values like ( ∀ k 0 )( ∃ q 0 ∈ I )( | f ( q ) | < 1 k ).

  44. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Θ is everywhere: metastable IVT The intermediate value theorem IVT is the prototypical non-constructive theorem. Constructive versions of IVT involve ‘approximate’ intermediate values like ( ∀ k 0 )( ∃ q 0 ∈ I )( | f ( q ) | < 1 k ). What about a metastable zero? ( ∃ x ∈ I )( f ( x ) = R 0) ↔ ( ∀ G 2 )( ∃ x ∈ I )( | f ( x ) | < 1 G ( x ) ) (8)

  45. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Θ is everywhere: metastable IVT The intermediate value theorem IVT is the prototypical non-constructive theorem. Constructive versions of IVT involve ‘approximate’ intermediate values like ( ∀ k 0 )( ∃ q 0 ∈ I )( | f ( q ) | < 1 k ). What about a metastable zero? ( ∃ x ∈ I )( f ( x ) = R 0) ↔ ( ∀ G 2 )( ∃ x ∈ I )( | f ( x ) | < 1 G ( x ) ) (8) Equivalence in (8) only holds classically; constructively, we still have:

  46. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Θ is everywhere: metastable IVT The intermediate value theorem IVT is the prototypical non-constructive theorem. Constructive versions of IVT involve ‘approximate’ intermediate values like ( ∀ k 0 )( ∃ q 0 ∈ I )( | f ( q ) | < 1 k ). What about a metastable zero? ( ∃ x ∈ I )( f ( x ) = R 0) ↔ ( ∀ G 2 )( ∃ x ∈ I )( | f ( x ) | < 1 G ( x ) ) (8) Equivalence in (8) only holds classically; constructively, we still have: ( ∃ x ∈ I )( f ( x ) = 0) → ( ∀ G 2 )( ∃ x ∈ I )( | f ( x ) | < 1 G ( x ) ) → ( ∀ k 0 )( ∃ q 0 ∈ I )( | f ( q ) | < 1 k ) . (9)

  47. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Θ is everywhere: metastable IVT The intermediate value theorem IVT is the prototypical non-constructive theorem. Constructive versions of IVT involve ‘approximate’ intermediate values like ( ∀ k 0 )( ∃ q 0 ∈ I )( | f ( q ) | < 1 k ). What about a metastable zero? ( ∃ x ∈ I )( f ( x ) = R 0) ↔ ( ∀ G 2 )( ∃ x ∈ I )( | f ( x ) | < 1 G ( x ) ) (8) Equivalence in (8) only holds classically; constructively, we still have: ( ∃ x ∈ I )( f ( x ) = 0) → ( ∀ G 2 )( ∃ x ∈ I )( | f ( x ) | < 1 G ( x ) ) → ( ∀ k 0 )( ∃ q 0 ∈ I )( | f ( q ) | < 1 k ) . (9) Principle (IVT meta ) There is Ψ 2 → 1 ∗ such that for f : [0 , 1] → R with modulus of continuity � � H 2 and f (0) f (1) < 0 , ( ∀ G 2 )( ∃ x ∈ Ψ( G , H )) 1 x ∈ I ∧ | f ( x ) | < . G ( x )

  48. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Θ is everywhere: metastable IVT The intermediate value theorem IVT is the prototypical non-constructive theorem. Constructive versions of IVT involve ‘approximate’ intermediate values like ( ∀ k 0 )( ∃ q 0 ∈ I )( | f ( q ) | < 1 k ). What about a metastable zero? ( ∃ x ∈ I )( f ( x ) = R 0) ↔ ( ∀ G 2 )( ∃ x ∈ I )( | f ( x ) | < 1 G ( x ) ) (8) Equivalence in (8) only holds classically; constructively, we still have: ( ∃ x ∈ I )( f ( x ) = 0) → ( ∀ G 2 )( ∃ x ∈ I )( | f ( x ) | < 1 G ( x ) ) → ( ∀ k 0 )( ∃ q 0 ∈ I )( | f ( q ) | < 1 k ) . (9) Principle (IVT meta ) There is Ψ 2 → 1 ∗ such that for f : [0 , 1] → R with modulus of continuity � � H 2 and f (0) f (1) < 0 , ( ∀ G 2 )( ∃ x ∈ Ψ( G , H )) 1 x ∈ I ∧ | f ( x ) | < . G ( x ) Metastability trade-off: finite domain yields uniform results, i.e. Ψ is independent of f like for (9).

  49. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Θ is everywhere: metastable IVT The intermediate value theorem IVT is the prototypical non-constructive theorem. Constructive versions of IVT involve ‘approximate’ intermediate values like ( ∀ k 0 )( ∃ q 0 ∈ I )( | f ( q ) | < 1 k ). What about a metastable zero? ( ∃ x ∈ I )( f ( x ) = R 0) ↔ ( ∀ G 2 )( ∃ x ∈ I )( | f ( x ) | < 1 G ( x ) ) (8) Equivalence in (8) only holds classically; constructively, we still have: ( ∃ x ∈ I )( f ( x ) = 0) → ( ∀ G 2 )( ∃ x ∈ I )( | f ( x ) | < 1 G ( x ) ) → ( ∀ k 0 )( ∃ q 0 ∈ I )( | f ( q ) | < 1 k ) . (9) Principle (IVT meta ) There is Ψ 2 → 1 ∗ such that for f : [0 , 1] → R with modulus of continuity � � H 2 and f (0) f (1) < 0 , ( ∀ G 2 )( ∃ x ∈ Ψ( G , H )) 1 x ∈ I ∧ | f ( x ) | < . G ( x ) Metastability trade-off: finite domain yields uniform results, i.e. Ψ is independent of f like for (9). However: Ψ computes Θ, and vice versa.

  50. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Θ is everywhere: metastable theorems Metastable versions of the following theorems give rise to Θ.

  51. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Θ is everywhere: metastable theorems Metastable versions of the following theorems give rise to Θ. BD-N, extreme value theorem, Riemann permutation theorem, . . .

  52. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Θ is everywhere: metastable theorems Metastable versions of the following theorems give rise to Θ. BD-N, extreme value theorem, Riemann permutation theorem, . . . But constructive theorems also give rise to Θ:

  53. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Θ is everywhere: metastable theorems Metastable versions of the following theorems give rise to Θ. BD-N, extreme value theorem, Riemann permutation theorem, . . . But constructive theorems also give rise to Θ: Theorem (Metastable Riemann integration) There is Ψ such that for f : [0 , 1] → R with modulus of uniform continuity h, and S n ( f ) := � 2 n 2 n ) 1 i =0 f ( i 2 n , we have ( ∀ G 2 )( ∃ x 1 , g 1 ∈ Ψ( G , h ))( ∀ k , n ≤ G ( x , g ))( n ≥ g ( k ) → | S n ( f ) − x | < 1 k ) .

  54. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Θ is everywhere: metastable theorems Metastable versions of the following theorems give rise to Θ. BD-N, extreme value theorem, Riemann permutation theorem, . . . But constructive theorems also give rise to Θ: Theorem (Metastable Riemann integration) There is Ψ such that for f : [0 , 1] → R with modulus of uniform continuity h, and S n ( f ) := � 2 n 2 n ) 1 i =0 f ( i 2 n , we have ( ∀ G 2 )( ∃ x 1 , g 1 ∈ Ψ( G , h ))( ∀ k , n ≤ G ( x , g ))( n ≥ g ( k ) → | S n ( f ) − x | < 1 k ) . Metastability trade-off: finite domain yields uniform results, i.e. Ψ is independent of f .

  55. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Θ is everywhere: metastable theorems Metastable versions of the following theorems give rise to Θ. BD-N, extreme value theorem, Riemann permutation theorem, . . . But constructive theorems also give rise to Θ: Theorem (Metastable Riemann integration) There is Ψ such that for f : [0 , 1] → R with modulus of uniform continuity h, and S n ( f ) := � 2 n 2 n ) 1 i =0 f ( i 2 n , we have ( ∀ G 2 )( ∃ x 1 , g 1 ∈ Ψ( G , h ))( ∀ k , n ≤ G ( x , g ))( n ≥ g ( k ) → | S n ( f ) − x | < 1 k ) . Metastability trade-off: finite domain yields uniform results, i.e. Ψ is independent of f . However: Ψ computes Θ, and vice versa.

  56. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Normann-Sanders I Many more results in https://arxiv.org/abs/1702.06556 : E 2 Z 2 S 2 Π 1 1 -CA 0 µ 2 + Θ ATR 0 µ 2 + Λ µ 2 ACA 0 Θ 3 WKL 0 Λ 3 WWKL 0

  57. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Normann-Sanders I Many more results in https://arxiv.org/abs/1702.06556 : E 2 Z 2 S 2 Π 1 1 -CA 0 µ 2 + Θ ATR 0 µ 2 + Λ µ 2 ACA 0 Θ 3 WKL 0 Λ 3 WWKL 0 Λ is the functional from ‘metastable WWKL’, where the latter is weak weak K¨ onig’s lemma.

  58. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Normann-Sanders I Many more results in https://arxiv.org/abs/1702.06556 : E 2 Z 2 S 2 Π 1 1 -CA 0 µ 2 + Θ ATR 0 µ 2 + Λ µ 2 ACA 0 Θ 3 WKL 0 Λ 3 WWKL 0 Λ is the functional from ‘metastable WWKL’, where the latter is weak weak K¨ onig’s lemma. Where do Θ and Λ come from?

  59. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Nonstandard Analysis: it all started with . . .

  60. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Nonstandard Analysis: it all started with . . . Archimedes, Euler, Newton, Stevin, Leibniz, etc informally used infinitely small quantities (fluxions/infinitesimals) to obtain Weierstraß-style limit results avant la lettre ,

  61. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Nonstandard Analysis: it all started with . . . Archimedes, Euler, Newton, Stevin, Leibniz, etc informally used infinitely small quantities (fluxions/infinitesimals) to obtain Weierstraß-style limit results avant la lettre , and the same for modern physics.

  62. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Nonstandard Analysis: it all started with . . . Archimedes, Euler, Newton, Stevin, Leibniz, etc informally used infinitely small quantities (fluxions/infinitesimals) to obtain Weierstraß-style limit results avant la lettre , and the same for modern physics.

  63. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Nonstandard Analysis: it all started with . . . Archimedes, Euler, Newton, Stevin, Leibniz, etc informally used infinitely small quantities (fluxions/infinitesimals) to obtain Weierstraß-style limit results avant la lettre , and the same for modern physics. Robinson: Non-standard Analysis (1965): Formalisation of the intuitive infinitesimal calculus via nonstandard models.

  64. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Nonstandard Analysis: it all started with . . . Archimedes, Euler, Newton, Stevin, Leibniz, etc informally used infinitely small quantities (fluxions/infinitesimals) to obtain Weierstraß-style limit results avant la lettre , and the same for modern physics. Robinson: Non-standard Analysis (1965): Formalisation of the intuitive infinitesimal calculus via nonstandard models. Nelson: Internal Set Theory (1977): well-known axiomatic approach to Nonstandard Analysis based on ZFC set theory.

  65. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Introducing Nonstandard Analysis Robinson’s semantic approach (1965):

  66. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Introducing Nonstandard Analysis Robinson’s semantic approach (1965): For a given structure M , build ∗ M � M , a nonstandard model of M (using free ultrafilter).

  67. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Introducing Nonstandard Analysis Robinson’s semantic approach (1965): For a given structure M , build ∗ M � M , a nonstandard model of M (using free ultrafilter). ∗ M M

  68. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Introducing Nonstandard Analysis Robinson’s semantic approach (1965): For a given structure M , build ∗ M � M , a nonstandard model of M (using free ultrafilter). ∗ M M N = { 0 , 1 , 2 , . . . }

  69. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Introducing Nonstandard Analysis Robinson’s semantic approach (1965): For a given structure M , build ∗ M � M , a nonstandard model of M (using free ultrafilter). ∗ M ∗ N = { 0 , 1 , 2 , . . . . . . , ω, ω + 1 , . . . } M N = { 0 , 1 , 2 , . . . }

  70. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Introducing Nonstandard Analysis Robinson’s semantic approach (1965): For a given structure M , build ∗ M � M , a nonstandard model of M (using free ultrafilter). ∗ M ∗ N = { 0 , 1 , 2 , . . . . . . , ω, ω + 1 , ω + 2 , ω + 3 , . . . } � �� � nonstandard objects not in N M N = { 0 , 1 , 2 , . . . }

  71. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Introducing Nonstandard Analysis Robinson’s semantic approach (1965): For a given structure M , build ∗ M � M , a nonstandard model of M (using free ultrafilter). ∗ M ∗ N = { 0 , 1 , 2 , . . . . . . , ω, ω + 1 , ω + 2 , ω + 3 , . . . } � �� � nonstandard objects not in N M ✲ X ∈ M ∗ X ∈ ∗ M N = { 0 , 1 , 2 , . . . }

  72. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Introducing Nonstandard Analysis Robinson’s semantic approach (1965): For a given structure M , build ∗ M � M , a nonstandard model of M (using free ultrafilter). ∗ M ∗ N = { 0 , 1 , 2 , . . . . . . , ω, ω + 1 , ω + 2 , ω + 3 , . . . } � �� � nonstandard objects not in N M star morphism ✲ X ∈ M ∗ X ∈ ∗ M N = { 0 , 1 , 2 , . . . }

  73. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Introducing Nonstandard Analysis Robinson’s semantic approach (1965): For a given structure M , build ∗ M � M , a nonstandard model of M (using free ultrafilter). ∗ M ∗ N = { 0 , 1 , 2 , . . . . . . , ω, ω + 1 , ω + 2 , ω + 3 , . . . } � �� � nonstandard objects not in N M star morphism ✲ X ∈ M ∗ X ∈ ∗ M X contains the standard objects N = { 0 , 1 , 2 , . . . }

  74. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Introducing Nonstandard Analysis Robinson’s semantic approach (1965): For a given structure M , build ∗ M � M , a nonstandard model of M (using free ultrafilter). ∗ M ∗ N = { 0 , 1 , 2 , . . . . . . , ω, ω + 1 , ω + 2 , ω + 3 , . . . } � �� � nonstandard objects not in N M star morphism ✲ X ∈ M ∗ X ∈ ∗ M X contains the standard objects ∗ X \ X contains the nonstandard objects N = { 0 , 1 , 2 , . . . }

  75. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Introducing Nonstandard Analysis Robinson’s semantic approach (1965): For a given structure M , build ∗ M � M , a nonstandard model of M (using free ultrafilter). ∗ M ∗ N = { 0 , 1 , 2 , . . . . . . , ω, ω + 1 , ω + 2 , ω + 3 , . . . } � �� � nonstandard objects not in N M star morphism ✲ X ∈ M ∗ X ∈ ∗ M X contains the standard objects ∗ X \ X contains the nonstandard objects N = { 0 , 1 , 2 , . . . } Three important properties connecting M and ∗ M :

  76. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Introducing Nonstandard Analysis Robinson’s semantic approach (1965): For a given structure M , build ∗ M � M , a nonstandard model of M (using free ultrafilter). ∗ M ∗ N = { 0 , 1 , 2 , . . . . . . , ω, ω + 1 , ω + 2 , ω + 3 , . . . } � �� � nonstandard objects not in N M star morphism ✲ X ∈ M ∗ X ∈ ∗ M X contains the standard objects ∗ X \ X contains the nonstandard objects N = { 0 , 1 , 2 , . . . } Three important properties connecting M and ∗ M : 1) Transfer M � ϕ ↔ ∗ M � ∗ ϕ ( ϕ ∈ L ZFC )

  77. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Introducing Nonstandard Analysis Robinson’s semantic approach (1965): For a given structure M , build ∗ M � M , a nonstandard model of M (using free ultrafilter). ∗ M ∗ N = { 0 , 1 , 2 , . . . . . . , ω, ω + 1 , ω + 2 , ω + 3 , . . . } � �� � nonstandard objects not in N M star morphism ✲ X ∈ M ∗ X ∈ ∗ M X contains the standard objects ∗ X \ X contains the nonstandard objects N = { 0 , 1 , 2 , . . . } Three important properties connecting M and ∗ M : 1) Transfer M � ϕ ↔ ∗ M � ∗ ϕ ( ϕ ∈ L ZFC ) 2) Standard Part ( ∀ x ∈ ∗ M )( ∃ y ∈ M )( ∀ z ∈ M )( z ∈ x ↔ z ∈ y )

  78. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Introducing Nonstandard Analysis Robinson’s semantic approach (1965): For a given structure M , build ∗ M � M , a nonstandard model of M (using free ultrafilter). ∗ M ∗ N = { 0 , 1 , 2 , . . . . . . , ω, ω + 1 , ω + 2 , ω + 3 , . . . } � �� � nonstandard objects not in N M star morphism ✲ X ∈ M ∗ X ∈ ∗ M X contains the standard objects ∗ X \ X contains the nonstandard objects N = { 0 , 1 , 2 , . . . } Three important properties connecting M and ∗ M : 1) Transfer M � ϕ ↔ ∗ M � ∗ ϕ ( ϕ ∈ L ZFC ) 2) Standard Part ( ∀ x ∈ ∗ M )( ∃ y ∈ M )( ∀ z ∈ M )( z ∈ x ↔ z ∈ y ) (reverse of ∗ )

  79. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Introducing Nonstandard Analysis Robinson’s semantic approach (1965): For a given structure M , build ∗ M � M , a nonstandard model of M (using free ultrafilter). ∗ M ∗ N = { 0 , 1 , 2 , . . . . . . , ω, ω + 1 , ω + 2 , ω + 3 , . . . } � �� � nonstandard objects not in N M star morphism ✲ X ∈ M ∗ X ∈ ∗ M X contains the standard objects ∗ X \ X contains the nonstandard objects N = { 0 , 1 , 2 , . . . } Three important properties connecting M and ∗ M : 1) Transfer M � ϕ ↔ ∗ M � ∗ ϕ ( ϕ ∈ L ZFC ) 2) Standard Part ( ∀ x ∈ ∗ M )( ∃ y ∈ M )( ∀ z ∈ M )( z ∈ x ↔ z ∈ y ) (reverse of ∗ ) 3) Idealization/Saturation . . .

  80. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Introducing Nonstandard Analysis Nelson’s Internal Set Theory (IST) is a syntactic approach to Nonstandard Analysis based on ZFC set theory.

  81. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Introducing Nonstandard Analysis Nelson’s Internal Set Theory (IST) is a syntactic approach to Nonstandard Analysis based on ZFC set theory. Add a new predicate st( x ) read x is standard to L ZFC .

  82. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Introducing Nonstandard Analysis Nelson’s Internal Set Theory (IST) is a syntactic approach to Nonstandard Analysis based on ZFC set theory. Add a new predicate st( x ) read x is standard to L ZFC . We write ( ∃ st x ) and ( ∀ st y ) for ( ∃ x )(st( x ) ∧ . . . ) and ( ∀ y )(st( y ) → . . . ).

  83. Introduction Metastability and the special fan functional Nonstandard Analysis and metastability Introducing Nonstandard Analysis Nelson’s Internal Set Theory (IST) is a syntactic approach to Nonstandard Analysis based on ZFC set theory. Add a new predicate st( x ) read x is standard to L ZFC . We write ( ∃ st x ) and ( ∀ st y ) for ( ∃ x )(st( x ) ∧ . . . ) and ( ∀ y )(st( y ) → . . . ). A formula is internal if it does not contain st; external otherwise

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