Long Proofs. Michael Rathjen University of Leeds. RaTLoCC 2011 Ramsey Theory in Logic, Combinatorics and Complexity

Transcription

1 Long Proofs Michael Rathjen University of Leeds RaTLoCC 2011 Ramsey Theory in Logic, Combinatorics and Complexity Bertinoro International Center for Informatics May 26, 2011

2 Relevance of Gödel s incompleteness theorem to ordinary mathematical practice?

3 Relevance of Gödel s incompleteness theorem to ordinary mathematical practice? Paris-Harrington principle (1977) PHP: For all m, n, c N there exists K N so large that for every colouring of the m-subsets of {0, 1,..., K 1} with c colours, there is a monocromatic set H {0, 1,..., K 1} of size n such that H > min(h).

4 Relevance of Gödel s incompleteness theorem to ordinary mathematical practice? Paris-Harrington principle (1977) PHP: For all m, n, c N there exists K N so large that for every colouring of the m-subsets of {0, 1,..., K 1} with c colours, there is a monocromatic set H {0, 1,..., K 1} of size n such that H > min(h). Kruskal s Tree Theorem

5 Relevance of Gödel s incompleteness theorem to ordinary mathematical practice? Paris-Harrington principle (1977) PHP: For all m, n, c N there exists K N so large that for every colouring of the m-subsets of {0, 1,..., K 1} with c colours, there is a monocromatic set H {0, 1,..., K 1} of size n such that H > min(h). Kruskal s Tree Theorem Graph Minor Theorem

6 Relevance of Gödel s incompleteness theorem to ordinary mathematical practice? Paris-Harrington principle (1977) PHP: For all m, n, c N there exists K N so large that for every colouring of the m-subsets of {0, 1,..., K 1} with c colours, there is a monocromatic set H {0, 1,..., K 1} of size n such that H > min(h). Kruskal s Tree Theorem Graph Minor Theorem H. Friedman: Boolean relation theory, BRT.

7 Relevance of Gödel s incompleteness theorem to ordinary mathematical practice? Paris-Harrington principle (1977) PHP: For all m, n, c N there exists K N so large that for every colouring of the m-subsets of {0, 1,..., K 1} with c colours, there is a monocromatic set H {0, 1,..., K 1} of size n such that H > min(h). Kruskal s Tree Theorem Graph Minor Theorem H. Friedman: Boolean relation theory, BRT. BRT is concerned with the relationship between sets and their images under multivariate functions.

8 Relevance of Gödel s incompleteness theorem to ordinary mathematical practice? Paris-Harrington principle (1977) PHP: For all m, n, c N there exists K N so large that for every colouring of the m-subsets of {0, 1,..., K 1} with c colours, there is a monocromatic set H {0, 1,..., K 1} of size n such that H > min(h). Kruskal s Tree Theorem Graph Minor Theorem H. Friedman: Boolean relation theory, BRT. BRT is concerned with the relationship between sets and their images under multivariate functions. BRT encapsulates the proof-theoretic strength of certain large cardinals.

9 Relevance of Gödel s speed-up theorem to ordinary mathematical practice?

10 Relevance of Gödel s speed-up theorem to ordinary mathematical practice? Gödel 1936: Über die Länge von Beweisen.

11 Relevance of Gödel s speed-up theorem to ordinary mathematical practice? Gödel 1936: Über die Länge von Beweisen. A system of higher order logic S n+1 can prove theorems a system of lower order S n cannot prove.

12 Relevance of Gödel s speed-up theorem to ordinary mathematical practice? Gödel 1936: Über die Länge von Beweisen. A system of higher order logic S n+1 can prove theorems a system of lower order S n cannot prove. There are formulas φ that can be proved in both S n+1 and S n. But the proof of φ in S n+1 is much shorter than the shortest proof of φ in S n.

13 Relevance of Gödel s speed-up theorem to ordinary mathematical practice? Gödel 1936: Über die Länge von Beweisen. A system of higher order logic S n+1 can prove theorems a system of lower order S n cannot prove. There are formulas φ that can be proved in both S n+1 and S n. But the proof of φ in S n+1 is much shorter than the shortest proof of φ in S n. Gödel defined the length of a proof to be the number of lines in it.

14 Reducing a theory to another theory 1 Reduce a theory T to a theory S.

15 Reducing a theory to another theory 1 Reduce a theory T to a theory S. 2 T ϕ S ϕ for all ϕ belonging to a collection of formulae Φ (usually Φ contains all Π 0 2 formulae).

16 Reducing a theory to another theory 1 Reduce a theory T to a theory S. 2 T ϕ S ϕ for all ϕ belonging to a collection of formulae Φ (usually Φ contains all Π 0 2 formulae). 3 Happens all the time in mathematical logic.

17 Reducing a theory to another theory 1 Reduce a theory T to a theory S. 2 T ϕ S ϕ for all ϕ belonging to a collection of formulae Φ (usually Φ contains all Π 0 2 formulae). 3 Happens all the time in mathematical logic. 4 Tools used come from all areas of math. logic: model, recursion, set, and proof theory.

18 Reducing a theory to another theory: Examples

19 Reducing a theory to another theory: Examples 1 (Gödel, Gentzen 1933) PA ϕ HA ϕ for ϕ almost negative.

20 Reducing a theory to another theory: Examples 1 (Gödel, Gentzen 1933) PA ϕ HA ϕ for ϕ almost negative. 2 (Shoenfield 1961, Platek 1969) ZF + AC + GCH ϕ ZF ϕ for ϕ Π 1 4.

21 Reducing a theory to another theory: Examples 1 (Gödel, Gentzen 1933) PA ϕ HA ϕ for ϕ almost negative. 2 (Shoenfield 1961, Platek 1969) ZF + AC + GCH ϕ ZF ϕ for ϕ Π (Platek 1969, Silver, Kripke) ZFC + GCH ϕ ZFC ϕ for ϕ Π 1.

22 Reducing a theory to another theory: More examples

23 Reducing a theory to another theory: More examples (4) (Parsons 1970) For ϕ Π 0 2, IΣ 1 ϕ PRA ϕ.

24 Reducing a theory to another theory: More examples (4) (Parsons 1970) For ϕ Π 0 2, IΣ 1 ϕ PRA ϕ. (5) (Barwise, Schlipf 1976?) For arithmetical ϕ, Σ 1 1 -AC 0 ϕ PA ϕ.

25 Reducing a theory to another theory: More examples (4) (Parsons 1970) For ϕ Π 0 2, IΣ 1 ϕ PRA ϕ. (5) (Barwise, Schlipf 1976?) For arithmetical ϕ, Σ 1 1 -AC 0 ϕ PA ϕ. (6) (Friedman, Harrington 1977,...) For ϕ Π 1 1, WKL 0 ϕ RCA 0 ϕ.

26 Reducing a theory to another theory: More examples (4) (Parsons 1970) For ϕ Π 0 2, IΣ 1 ϕ PRA ϕ. (5) (Barwise, Schlipf 1976?) For arithmetical ϕ, Σ 1 1 -AC 0 ϕ PA ϕ. (6) (Friedman, Harrington 1977,...) For ϕ Π 1 1, WKL 0 ϕ RCA 0 ϕ. (7) (Goodman 1976) For arithmetic ϕ, HA ω + {AC στ all finite types σ, τ} ϕ HA ϕ.

27 Reducing a theory to another theory: More examples (4) (Parsons 1970) For ϕ Π 0 2, IΣ 1 ϕ PRA ϕ. (5) (Barwise, Schlipf 1976?) For arithmetical ϕ, Σ 1 1 -AC 0 ϕ PA ϕ. (6) (Friedman, Harrington 1977,...) For ϕ Π 1 1, WKL 0 ϕ RCA 0 ϕ. (7) (Goodman 1976) For arithmetic ϕ, HA ω + {AC στ all finite types σ, τ} ϕ HA ϕ. (8) (R 1994) For almost negative φ, Σ 1 2-AC + BI ϕ Martin-Löf type theory ϕ.

28 Speed-Up cont d

29 Speed-Up cont d Gödel used the number of lines, a more common measure is the total length of a proof, i.e. the total number of symbols in it.

30 Speed-Up cont d Gödel used the number of lines, a more common measure is the total length of a proof, i.e. the total number of symbols in it. Examples:

31 Speed-Up cont d Gödel used the number of lines, a more common measure is the total length of a proof, i.e. the total number of symbols in it. Examples: Pudlak,...: Non-elementary speed-up of Gödel-Bernays set theory over Zermelo-Fraenkel set theory.

32 Speed-Up cont d Gödel used the number of lines, a more common measure is the total length of a proof, i.e. the total number of symbols in it. Examples: Pudlak,...: Non-elementary speed-up of Gödel-Bernays set theory over Zermelo-Fraenkel set theory. Ignjatovic: IΣ 1 has non-elementary speed-up over PRA.

33 Speed-Up cont d Gödel used the number of lines, a more common measure is the total length of a proof, i.e. the total number of symbols in it. Examples: Pudlak,...: Non-elementary speed-up of Gödel-Bernays set theory over Zermelo-Fraenkel set theory. Ignjatovic: IΣ 1 has non-elementary speed-up over PRA. RCA 0 has at most polynomial speed-up over IΣ 1.

34 Speed-Up cont d Gödel used the number of lines, a more common measure is the total length of a proof, i.e. the total number of symbols in it. Examples: Pudlak,...: Non-elementary speed-up of Gödel-Bernays set theory over Zermelo-Fraenkel set theory. Ignjatovic: IΣ 1 has non-elementary speed-up over PRA. RCA 0 has at most polynomial speed-up over IΣ 1. Avigad; Hajek: WKL 0 has at most polynomial speed-up over IΣ 1.

35 Combinatorial Independence Results

36 Combinatorial Independence Results A finite tree is a finite partially ordered set such that: B = (B, )

37 Combinatorial Independence Results A finite tree is a finite partially ordered set B = (B, ) such that: (i) B has a smallest element (called the root of B);

38 Combinatorial Independence Results A finite tree is a finite partially ordered set B = (B, ) such that: (i) B has a smallest element (called the root of B); (ii) for each s B the set {t B : t s} is a totally ordered subset of B.

39 Combinatorial Independence Results A finite tree is a finite partially ordered set B = (B, ) such that: (i) B has a smallest element (called the root of B); (ii) for each s B the set {t B : t s} is a totally ordered subset of B. For finite trees B 1 and B 2, an embedding of B 1 into B 2 is a one-to-one mapping such that f : B 1 B 2 f (a b) = f (a) f (b) for all a, b B 1, where a b denotes the infimum of a and b.

40 Kruskal s ( Theorem. For every infinite sequence of trees Bk : k < ω ), there exist i and j such that i < j < ω and B i is embeddable into B j. (In particular, there is no infinite set of pairwise nonembeddable trees.)

41 Kruskal s ( Theorem. For every infinite sequence of trees Bk : k < ω ), there exist i and j such that i < j < ω and B i is embeddable into B j. (In particular, there is no infinite set of pairwise nonembeddable trees.) Theorem (H. Friedman, D. Schmidt) Kruskal s Theorem is not provable in ATR 0.

42 Kruskal s ( Theorem. For every infinite sequence of trees Bk : k < ω ), there exist i and j such that i < j < ω and B i is embeddable into B j. (In particular, there is no infinite set of pairwise nonembeddable trees.) Theorem (H. Friedman, D. Schmidt) Kruskal s Theorem is not provable in ATR 0. The proof utilizes that Kruskal s Theorem implies that Γ 0 is well-founded.

43 Miniaturization

44 Miniaturization Definition. SWQO(T ) is the following assertion: For any c there exists a constant k which is so large that, for any finite sequence T 0, T 1,..., T k of finite trees with T i c (i + 1) for all i k, there exist indices i 0 < j 0 k such that T i0 is embeddable into T j0.

45 Miniaturization Definition. SWQO(T ) is the following assertion: For any c there exists a constant k which is so large that, for any finite sequence T 0, T 1,..., T k of finite trees with T i c (i + 1) for all i k, there exist indices i 0 < j 0 k such that T i0 is embeddable into T j0. SWQO(T ) is a Π 0 2 statement.

46 Miniaturization Definition. SWQO(T ) is the following assertion: For any c there exists a constant k which is so large that, for any finite sequence T 0, T 1,..., T k of finite trees with T i c (i + 1) for all i k, there exist indices i 0 < j 0 k such that T i0 is embeddable into T j0. SWQO(T ) is a Π 0 2 statement. Theorem. (Kruskal) SWQO(T ) is provable in Π 1 1 -CA 0.

47 Miniaturization Definition. SWQO(T ) is the following assertion: For any c there exists a constant k which is so large that, for any finite sequence T 0, T 1,..., T k of finite trees with T i c (i + 1) for all i k, there exist indices i 0 < j 0 k such that T i0 is embeddable into T j0. SWQO(T ) is a Π 0 2 statement. Theorem. (Kruskal) SWQO(T ) is provable in Π 1 1 -CA 0. Theorem. (Friedman) SWQO(T ) is not provable in ATR 0.

48 Unfeasible "Proofs" of Σ01 statements

49 Unfeasible "Proofs" of Σ 0 1 statements Every true Σ 0 1 statement ϕ is provable in any theory that contains a modicum of arithmetic (Q).

50 Unfeasible "Proofs" of Σ 0 1 statements Every true Σ 0 1 statement ϕ is provable in any theory that contains a modicum of arithmetic (Q). But it can happen that ϕ has a short proof in a theory T whereas any proof of ϕ in another theory S would necessarily be totally unfeasible.

51 Specific Example: Friedman s practical matters

52 Specific Example: Friedman s practical matters Consider labeled trees and embeddings between such trees which also respect the labels. Consider the following Σ 0 1 sentence Big:

53 Specific Example: Friedman s practical matters Consider labeled trees and embeddings between such trees which also respect the labels. Consider the following Σ 0 1 sentence Big: There exists n 1 such that whenever T 1,..., T n are finite trees whose vertices are labeled from {1,..., 6}, where for all i, T i i, then there exist i < j n such that T i is label preserving embeddable into T j.

54 Specific Example: Friedman s practical matters Consider labeled trees and embeddings between such trees which also respect the labels. Consider the following Σ 0 1 sentence Big: There exists n 1 such that whenever T 1,..., T n are finite trees whose vertices are labeled from {1,..., 6}, where for all i, T i i, then there exist i < j n such that T i is label preserving embeddable into T j. Big has a short proof in Π 1 1 -CA 0.

55 Specific Example: Friedman s practical matters Consider labeled trees and embeddings between such trees which also respect the labels. Consider the following Σ 0 1 sentence Big: There exists n 1 such that whenever T 1,..., T n are finite trees whose vertices are labeled from {1,..., 6}, where for all i, T i i, then there exist i < j n such that T i is label preserving embeddable into T j. Big has a short proof in Π 1 1 -CA 0. Any proof of Big in ATR 0 must have more than =: 2 [1000] symbols where the stack of powers of 2 has height 1000.

56 The strength of Kruskal s theorem M. Rathjen, A. Weiermann: Proof theoretic investigations on Kruskal s theorem (1993)

57 The strength of Kruskal s theorem M. Rathjen, A. Weiermann: Proof theoretic investigations on Kruskal s theorem (1993) Theorem 1. The proof-theoretic ordinal ordinal of Π 1 2 BI 0 and Π 1 2 BI 0 is the Ackermann ordinal θωω 0.

58 The strength of Kruskal s theorem M. Rathjen, A. Weiermann: Proof theoretic investigations on Kruskal s theorem (1993) Theorem 1. The proof-theoretic ordinal ordinal of Π 1 2 BI 0 and Π 1 2 BI 0 is the Ackermann ordinal θωω 0. Theorem 2. For every n, Π 1 2 BI 0 proves KT n, i.e. Kruskal s theorem for finite at most n branching trees. Π 1 2 BI 0 x KT x.

59 The strength of Kruskal s theorem M. Rathjen, A. Weiermann: Proof theoretic investigations on Kruskal s theorem (1993) Theorem 1. The proof-theoretic ordinal ordinal of Π 1 2 BI 0 and Π 1 2 BI 0 is the Ackermann ordinal θωω 0. Theorem 2. For every n, Π 1 2 BI 0 proves KT n, i.e. Kruskal s theorem for finite at most n branching trees. Π 1 2 BI 0 x KT x. Theorem 3. ACA 0 proves that Kruskal s theorem is equivalent to the uniform Π 1 1 -reflection for Π1 2 BI 0.

60 Specific Example continued

61 Specific Example continued Recall the Σ 0 1 sentence Big:

62 Specific Example continued Recall the Σ 0 1 sentence Big: There exists n 1 such that whenever T 1,..., T n are finite trees whose vertices are labeled from {1,..., 6}, where for all i, T i i, then there exist i < j n such that T i is label preserving embeddable into T j.

63 Specific Example continued Recall the Σ 0 1 sentence Big: There exists n 1 such that whenever T 1,..., T n are finite trees whose vertices are labeled from {1,..., 6}, where for all i, T i i, then there exist i < j n such that T i is label preserving embeddable into T j. Big has a short proof in Π 1 2 BI.

64 Specific Example continued Recall the Σ 0 1 sentence Big: There exists n 1 such that whenever T 1,..., T n are finite trees whose vertices are labeled from {1,..., 6}, where for all i, T i i, then there exist i < j n such that T i is label preserving embeddable into T j. Big has a short proof in Π 1 2 BI. Any proof of Big in Π 1 2 BI 0 must have more than =: 2 [1000] symbols where the stack of powers of 2 has height 1000.

65 Specific Example continued Recall the Σ 0 1 sentence Big: There exists n 1 such that whenever T 1,..., T n are finite trees whose vertices are labeled from {1,..., 6}, where for all i, T i i, then there exist i < j n such that T i is label preserving embeddable into T j. Big has a short proof in Π 1 2 BI. Any proof of Big in Π 1 2 BI 0 must have more than =: 2 [1000] symbols where the stack of powers of 2 has height Why?

66 An Example

67 An Example R.L. Smith: Consistency strength of some finite forms of the Higman and Kruskal theorems. In: Harvey Friedman s research on the foundations of mathematics. (1985)

68 An Example R.L. Smith: Consistency strength of some finite forms of the Higman and Kruskal theorems. In: Harvey Friedman s research on the foundations of mathematics. (1985) Example There exist labeled trees T 1,..., T q, q := 2 [1008], such that each T i uses labels from {1, 2, 3, 4, 5, 6}, T i i, and for no i < j is T i label preserving embeddable into T j.

69 From Gentzen to infinitary proof theory

70 From Gentzen to infinitary proof theory Gentzen 1936: Shows that an alleged proof of the empty sequent (a contradiction) in PA can always be reduced to another proof of but with a smaller ordinal (notation) assigned to it. Metatheory is finitistic.

71 From Gentzen to infinitary proof theory Gentzen 1936: Shows that an alleged proof of the empty sequent (a contradiction) in PA can always be reduced to another proof of but with a smaller ordinal (notation) assigned to it. Metatheory is finitistic. Consistency proof uses only PRWF(ε 0 ), i.e. that all primitive (elementary) recursive descending sequences below ε 0 are finite.

72 From Gentzen to infinitary proof theory Gentzen 1936: Shows that an alleged proof of the empty sequent (a contradiction) in PA can always be reduced to another proof of but with a smaller ordinal (notation) assigned to it. Metatheory is finitistic. Consistency proof uses only PRWF(ε 0 ), i.e. that all primitive (elementary) recursive descending sequences below ε 0 are finite. Schütte 1951: Calculi with the infinite ω-rule: Γ, F(0); Γ, F(1);... ; Γ, F(n);... Γ, x F (x) ωr

73 From Gentzen to infinitary proof theory Gentzen 1936: Shows that an alleged proof of the empty sequent (a contradiction) in PA can always be reduced to another proof of but with a smaller ordinal (notation) assigned to it. Metatheory is finitistic. Consistency proof uses only PRWF(ε 0 ), i.e. that all primitive (elementary) recursive descending sequences below ε 0 are finite. Schütte 1951: Calculi with the infinite ω-rule: Γ, F(0); Γ, F(1);... ; Γ, F(n);... Γ, x F (x) ωr F(0), Γ ; F(1), Γ ;... ; F(n), Γ ;... x F (x), Γ ωl

74 From Gentzen to... cont d

75 From Gentzen to... cont d Schütte 1952: Infinitary Calculi for Ramified Analysis

76 From Gentzen to... cont d Schütte 1952: Infinitary Calculi for Ramified Analysis Tait 1968: Cut elimination in L ω1,ω.

77 From Gentzen to... cont d Schütte 1952: Infinitary Calculi for Ramified Analysis Tait 1968: Cut elimination in L ω1,ω. Buchholz 1977: Ω µ -rules.

78 From Gentzen to... cont d Schütte 1952: Infinitary Calculi for Ramified Analysis Tait 1968: Cut elimination in L ω1,ω. Buchholz 1977: Ω µ -rules. 1979, Pohlers, Jäger, Buchholz, Rathjen,... : Infinitary Calculi for set theory

79 Kreisel (1958): Is it possible to obtain the same information from the infinitary treatment of proof theory as one can obtain from the Hilbert Ackermann substitution method?

80 Kreisel (1958): Is it possible to obtain the same information from the infinitary treatment of proof theory as one can obtain from the Hilbert Ackermann substitution method? Friedman:

81 Kreisel (1958): Is it possible to obtain the same information from the infinitary treatment of proof theory as one can obtain from the Hilbert Ackermann substitution method? Friedman: 1 Proof theory (almost?) always is establishing a conservative extension result that says that a given formal system T is a conservative extension of a system of arithmetic with transfinite induction on a notation system, for Σ 0 1 sentences, or even Π0 2 sentences.

82 Kreisel (1958): Is it possible to obtain the same information from the infinitary treatment of proof theory as one can obtain from the Hilbert Ackermann substitution method? Friedman: 1 Proof theory (almost?) always is establishing a conservative extension result that says that a given formal system T is a conservative extension of a system of arithmetic with transfinite induction on a notation system, for Σ 0 1 sentences, or even Π0 2 sentences. 2 The conservative extension statement itself is a Π 0 2 sentence. All I really need is that this Π 0 2 sentence has a reasonable Skolem function. E.g., a "reasonable" primitive recursive function will do. By "reasonable" I mean, e.g., that its presentation in the primitive recursion calculus of Kleene uses at most, say, symbols.

83 Proof-Theoretic Reductions 1 Reduce a theory T to a theory S.

84 Proof-Theoretic Reductions 1 Reduce a theory T to a theory S. 2 ( ) T ϕ S ϕ for all ϕ belonging to a collection of formulae Φ (usually Φ contains all Π 0 2 formulae).

85 Proof-Theoretic Reductions 1 Reduce a theory T to a theory S. 2 ( ) T ϕ S ϕ for all ϕ belonging to a collection of formulae Φ (usually Φ contains all Π 0 2 formulae). 3 The statement ( ) is itself Π 0 2.

86 Proof-Theoretic Reductions 1 Reduce a theory T to a theory S. 2 ( ) T ϕ S ϕ for all ϕ belonging to a collection of formulae Φ (usually Φ contains all Π 0 2 formulae). 3 The statement ( ) is itself Π What metatheory is sufficient for proving ( )?

87 Proof-Theoretic Reductions 1 Reduce a theory T to a theory S. 2 ( ) T ϕ S ϕ for all ϕ belonging to a collection of formulae Φ (usually Φ contains all Π 0 2 formulae). 3 The statement ( ) is itself Π What metatheory is sufficient for proving ( )? 5 What is the complexity of the Skolem function f pertaining to ( )? ( ) D ϕ D T-proof f (D) ϕ and f (D) S-proof.

88 For a theory T define χ T (n) to be the least integer k such that if x A(x) is any Σ 0 1 statement provable in T using n symbols, then A(m) is true for some m k.

89 For a theory T define χ T (n) to be the least integer k such that if x A(x) is any Σ 0 1 statement provable in T using n symbols, then A(m) is true for some m k. Let χ λ (n) be the least integer k such that if x A(x) is any Σ 0 1 statement provable in PA + {TI(α) α < λ} using n symbols, then A(m) is true for some m k.

90 Main Step

91 Main Step Ordinal analysis reduces a theory T which is a subsystem of second order arithmetic or set theory to PA + {TI(α) α < λ}, where λ is the proof-theoretic ordinal of T.

92 Main Step Ordinal analysis reduces a theory T which is a subsystem of second order arithmetic or set theory to PA + {TI(α) α < λ}, where λ is the proof-theoretic ordinal of T. χ λ is well understood. The above reduction should establish a (tight) relationship between χ λ and χ T.

93 Extracting computational information from infinite derivations without coding

94 Extracting computational information from infinite derivations without coding Recall that the goal is to reduce a theory T to PA + α<λ TI(α). and that this ordinal analysis uses infinite derivations.

95 Extracting computational information from infinite derivations without coding Recall that the goal is to reduce a theory T to PA + α<λ TI(α). and that this ordinal analysis uses infinite derivations. A possible solution: Buy axiomatic freedom by switching to intuitionistic logic.

96 Extracting computational information from infinite derivations without coding Recall that the goal is to reduce a theory T to PA + α<λ TI(α). and that this ordinal analysis uses infinite derivations. A possible solution: Buy axiomatic freedom by switching to intuitionistic logic. Kreisel (1968): Church s thesis: a kind of reducibility axiom for constructive mathematics.

97 Extracting computational information from infinite derivations without coding Recall that the goal is to reduce a theory T to PA + α<λ TI(α). and that this ordinal analysis uses infinite derivations. A possible solution: Buy axiomatic freedom by switching to intuitionistic logic. Kreisel (1968): Church s thesis: a kind of reducibility axiom for constructive mathematics. Work in an intuitionistic theory which is conservative over HA and allows for unfettered quantification over Baire space N N and also has the axiom of choice AC-NF: n f N N ϕ(n, f ) g N N n ϕ(n, (g) n ).

98 Extracting computational information from infinite derivations without coding Recall that the goal is to reduce a theory T to PA + α<λ TI(α). and that this ordinal analysis uses infinite derivations. A possible solution: Buy axiomatic freedom by switching to intuitionistic logic. Kreisel (1968): Church s thesis: a kind of reducibility axiom for constructive mathematics. Work in an intuitionistic theory which is conservative over HA and allows for unfettered quantification over Baire space N N and also has the axiom of choice AC-NF: n f N N ϕ(n, f ) g N N n ϕ(n, (g) n ). This is possible. But there is a better solution.

99 An intuitionistic fixed point theory Wilfried Buchholz: An intuitionistic fixed point theory Archive Math Logic (1997)

100 An intuitionistic fixed point theory Wilfried Buchholz: An intuitionistic fixed point theory Archive Math Logic (1997) The strongly positive formulas are built up from formulas P(t) and atomic formulas of HA by means of,,,.

101 An intuitionistic fixed point theory Wilfried Buchholz: An intuitionistic fixed point theory Archive Math Logic (1997) The strongly positive formulas are built up from formulas P(t) and atomic formulas of HA by means of,,,. ID i1 is obtained from HA by adding for each strongly positive operator form Φ(P, x) a new predicate symbol I Φ and the axiom (FP Φ ) x [Φ(I Φ, x) I Φ (x)]. Moreover, the induction schema is extended to the new language.

102 Let CT 0 be Church s thesis, i.e. the schema x y B(x, y) e x B(x, {e}(x)).

103 Let CT 0 be Church s thesis, i.e. the schema x y B(x, y) e x B(x, {e}(x)). Theorem 1 For each strongly positive operator form Φ there is an arithmetical formula A Φ (x) such that HA + CT 0 x[φ(a Φ, x) A Φ (x)].

104 Theorem 2 ID i 1 is conservative over HA w.r.t. almost negative formulas.

105 Theorem 2 ID i 1 is conservative over HA w.r.t. almost negative formulas. Proof: For each formula B of ID i 1 let B be the result of replacing each subformula I Φ (t) by A Φ (t).

106 Theorem 2 ID i 1 is conservative over HA w.r.t. almost negative formulas. Proof: For each formula B of ID i 1 let B be the result of replacing each subformula I Φ (t) by A Φ (t). 1 ID i1 B HA + CT 0 B (Theorem 1)

107 Theorem 2 ID i 1 is conservative over HA w.r.t. almost negative formulas. Proof: For each formula B of ID i 1 let B be the result of replacing each subformula I Φ (t) by A Φ (t). 1 ID i1 B HA + CT 0 B (Theorem 1) 2 HA + CT 0 C HA e (erc).

108 Theorem 2 ID i 1 is conservative over HA w.r.t. almost negative formulas. Proof: For each formula B of ID i 1 let B be the result of replacing each subformula I Φ (t) by A Φ (t). 1 ID i1 B HA + CT 0 B (Theorem 1) 2 HA + CT 0 C HA e (erc). 3 HA e (erc) C for almost negative C.

109 Theorem 2 ID i 1 is conservative over HA w.r.t. almost negative formulas. Proof: For each formula B of ID i 1 let B be the result of replacing each subformula I Φ (t) by A Φ (t). 1 ID i1 B HA + CT 0 B (Theorem 1) 2 HA + CT 0 C HA e (erc). 3 HA e (erc) C for almost negative C. 4 Corresponding results hold for ID i 1 + α<λ TI(α).

110 ID i 1 + α<λ TI(α) as a metatheory for ordinal analysis A semi-formal system à la Schütte is given by a derivability predicate D(α, ρ, Γ) meaning Γ is derivable with order α and cut-rank ρ defined by transfinite recursion on α as follows:

111 ID i 1 + α<λ TI(α) as a metatheory for ordinal analysis A semi-formal system à la Schütte is given by a derivability predicate D(α, ρ, Γ) meaning Γ is derivable with order α and cut-rank ρ defined by transfinite recursion on α as follows: ( ) D(α, ρ, Γ) α < λ, and either Γ contains an axiom or Γ is the conclusion of an inference with premisses (Γ i ) i I such that for every i I there exists β i < α with D(β i, ρ, Γ i ), and if the inference is a cut it has rank < ρ.

112 ID i 1 + α<λ TI(α) as a metatheory for ordinal analysis A semi-formal system à la Schütte is given by a derivability predicate D(α, ρ, Γ) meaning Γ is derivable with order α and cut-rank ρ defined by transfinite recursion on α as follows: ( ) D(α, ρ, Γ) α < λ, and either Γ contains an axiom or Γ is the conclusion of an inference with premisses (Γ i ) i I such that for every i I there exists β i < α with D(β i, ρ, Γ i ), and if the inference is a cut it has rank < ρ. ( ) can be viewed as a fixed-point axiom which together with α<λ TI(α) defines D implicitly, whence the metatheory ID i 1 + α<λ TI(α) suffices.

113 Buchholz s Ω-rule Ω-rule particularly suited to deal with Bar induction, BI.

114 Buchholz s Ω-rule Ω-rule particularly suited to deal with Bar induction, BI. BI is equivalent (over RCA 0 to the schema ( -Inst) X A(X) A(F) where A(X) is an arithmetic formula and F(u) is an arbitrary formula of second order arithmetic. A(F) results from A(X) by replacing every subformula t X by F(t).

115 Buchholz s Ω-rule Ω-rule particularly suited to deal with Bar induction, BI. BI is equivalent (over RCA 0 to the schema ( -Inst) X A(X) A(F) where A(X) is an arithmetic formula and F(u) is an arbitrary formula of second order arithmetic. A(F) results from A(X) by replacing every subformula t X by F(t). 1 Crude motivation for the Ω-rule: An intuitionistic proof of an implication B C is a method which transforms a proof of B into a proof of C.

116 Buchholz s Ω-rule Ω-rule particularly suited to deal with Bar induction, BI. BI is equivalent (over RCA 0 to the schema ( -Inst) X A(X) A(F) where A(X) is an arithmetic formula and F(u) is an arbitrary formula of second order arithmetic. A(F) results from A(X) by replacing every subformula t X by F(t). 1 Crude motivation for the Ω-rule: An intuitionistic proof of an implication B C is a method which transforms a proof of B into a proof of C. 2 How to transform a proof of X A(X) into a proof of A(F)?

117 Buchholz s Ω-rule Ω-rule particularly suited to deal with Bar induction, BI. BI is equivalent (over RCA 0 to the schema ( -Inst) X A(X) A(F) where A(X) is an arithmetic formula and F(u) is an arbitrary formula of second order arithmetic. A(F) results from A(X) by replacing every subformula t X by F(t). 1 Crude motivation for the Ω-rule: An intuitionistic proof of an implication B C is a method which transforms a proof of B into a proof of C. 2 How to transform a proof of X A(X) into a proof of A(F)? 3 Easy if the proof of X A(X) is cut-free: substitution.

118 Buchholz s Ω-rule cont d Weak formulas are formulas that are arithmetic or Π 1 1.

119 Buchholz s Ω-rule cont d Weak formulas are formulas that are arithmetic or Π 1 1. Inductive definition of T ϱ < ω + ω. α ϱ Γ for α OT (ψ) and

120 Buchholz s Ω-rule cont d Weak formulas are formulas that are arithmetic or Π 1 1. Inductive definition of T ϱ < ω + ω. α ϱ Γ for α OT (ψ) and (Ω-rule). Let f be a fundamental function satisfying

121 Buchholz s Ω-rule cont d Weak formulas are formulas that are arithmetic or Π 1 1. Inductive definition of T ϱ < ω + ω. α ϱ Γ for α OT (ψ) and (Ω-rule). Let f be a fundamental function satisfying 1 Ω dom(f ) and f (Ω) α,

122 Buchholz s Ω-rule cont d Weak formulas are formulas that are arithmetic or Π 1 1. Inductive definition of T ϱ < ω + ω. α ϱ Γ for α OT (ψ) and (Ω-rule). Let f be a fundamental function satisfying 1 Ω dom(f ) and f (Ω) α, 2 T f (0) ϱ Γ, XF (X), where XF(X) Π 1 1, and

123 Buchholz s Ω-rule cont d Weak formulas are formulas that are arithmetic or Π 1 1. Inductive definition of T ϱ < ω + ω. α ϱ Γ for α OT (ψ) and (Ω-rule). Let f be a fundamental function satisfying 1 Ω dom(f ) and f (Ω) α, 2 T f (0) ϱ 3 T β 0 Γ, XF (X), where XF(X) Π 1 1, and f (β) Ξ, XF (X) implies T ϱ weak formulas Ξ and β < Ω. Ξ, Γ for every set of

124 Buchholz s Ω-rule cont d Weak formulas are formulas that are arithmetic or Π 1 1. Inductive definition of T ϱ < ω + ω. α ϱ Γ for α OT (ψ) and (Ω-rule). Let f be a fundamental function satisfying 1 Ω dom(f ) and f (Ω) α, 2 T f (0) ϱ Γ, XF (X), where XF(X) Π 1 1, and 3 T β f (β) Ξ, XF (X) implies T 0 ϱ weak formulas Ξ and β < Ω. Then T α ϱ Γ holds. Ξ, Γ for every set of

125 Metatheory for the Ω-rule? The derivability notion T inductive definition. α ϱ Γ seems to require an iterated

126 Metatheory for the Ω-rule? The derivability notion T inductive definition. α ϱ Γ seems to require an iterated 1 First inductively defined set, T : Infinitary (cut-free) proofs without Ω-rule.

127 Metatheory for the Ω-rule? The derivability notion T inductive definition. α ϱ Γ seems to require an iterated 1 First inductively defined set, T : Infinitary (cut-free) proofs without Ω-rule. 2 Inductive definition of T α ϱ Γ involves T negatively.

128 Metatheory for the Ω-rule? The derivability notion T inductive definition. α ϱ Γ seems to require an iterated 1 First inductively defined set, T : Infinitary (cut-free) proofs without Ω-rule. 2 Inductive definition of T α ϱ Γ involves T negatively. Buchholz s result can be extended to finitely iterated inductive definitions.

129 Metatheory for the Ω-rule? The derivability notion T inductive definition. α ϱ Γ seems to require an iterated 1 First inductively defined set, T : Infinitary (cut-free) proofs without Ω-rule. 2 Inductive definition of T α ϱ Γ involves T negatively. Buchholz s result can be extended to finitely iterated inductive definitions. T. Arai: Some results on cut-elimination, provable well-orderings, induction and reflection (1998).

130 Metatheory for the Ω-rule? The derivability notion T inductive definition. α ϱ Γ seems to require an iterated 1 First inductively defined set, T : Infinitary (cut-free) proofs without Ω-rule. 2 Inductive definition of T α ϱ Γ involves T negatively. Buchholz s result can be extended to finitely iterated inductive definitions. T. Arai: Some results on cut-elimination, provable well-orderings, induction and reflection (1998). ID in(strong) can be interpreted in intuitionistic analysis EL + AC-NF basically by the same proof as the classical second recursion theorem. EL + AC-NF is conservative over HA by Goodman s theorem.

131 Problem: Metatheory for the Ω-rule cont d

132 Metatheory for the Ω-rule cont d Problem: (2) ID in is formulated for strongly positive operator forms. But the iterated inductive definition of T α ϱ Γ seems to require a strictly positive iterated inductive definition.

133 Metatheory for the Ω-rule cont d Problem: (2) ID in is formulated for strongly positive operator forms. But the iterated inductive definition of T α ϱ Γ seems to require a strictly positive iterated inductive definition. The strictly positive (with respect to P) formulas of L 1 (P, Q) formulas are closed under the following clause: If A is an L 1 (Q) formula and B is strictly positive, then A B is strictly positive.

134 Metatheory for the Ω-rule cont d C. Rüede, T. Strahm: Intuitionistic fixed point theories for strictly positive operators. (MLQ 2002)

135 Metatheory for the Ω-rule cont d C. Rüede, T. Strahm: Intuitionistic fixed point theories for strictly positive operators. (MLQ 2002) i Theorem. ID n(strict) is conservative over HA w.r.t. Π 0 2 sentences.

136 Metatheory for the Ω-rule cont d C. Rüede, T. Strahm: Intuitionistic fixed point theories for strictly positive operators. (MLQ 2002) i Theorem. ID n(strict) is conservative over HA w.r.t. Π 0 2 sentences. Proof uses a realizability interpretation of ID i n(strict) into ID i n(acc) (preserves almost negative formulas).

137 Metatheory for the Ω-rule cont d C. Rüede, T. Strahm: Intuitionistic fixed point theories for strictly positive operators. (MLQ 2002) i Theorem. ID n(strict) is conservative over HA w.r.t. Π 0 2 sentences. Proof uses a realizability interpretation of ID i n(strict) into ID i n(acc) (preserves almost negative formulas). Proof can be easily made fully formal.

138 Metatheory for the Ω-rule cont d C. Rüede, T. Strahm: Intuitionistic fixed point theories for strictly positive operators. (MLQ 2002) i Theorem. ID n(strict) is conservative over HA w.r.t. Π 0 2 sentences. Proof uses a realizability interpretation of ID i n(strict) into ID i n(acc) (preserves almost negative formulas). Proof can be easily made fully formal. An accessibility operator form A(P, Q, x, y) is of the form A(x, y) z[b(x, y, z) P(z)], where A, B belong to L 1 (Q).

139 Summary

140 Summary Elementary speedup of ID i 2(strict) over Π 1 2 BI 0.

141 Summary Elementary speedup of ID i 2(strict) over Π 1 2 BI 0. Elementary speedup of PA + α<θω ω 0 TI(α) over ID i 2(strict).

142 Summary Elementary speedup of ID i 2(strict) over Π 1 2 BI 0. Elementary speedup of PA + α<θω ω 0 TI(α) over ID i 2(strict). χ Π 1 2 BI 0 elementary in χ θω ω.

143 The End

144 Thank you! The End

145 The Steps of Ordinal Analysis Let T be a theory we want to analyze with proof-theoretic ordinal λ. Let S λ := ID i 1 + α<λ TI(α). Moreover, let (λ n ) n N be a canonical fundamental sequence associated with λ. In particular sup λ n = λ.

146 The Steps of Ordinal Analysis Let T be a theory we want to analyze with proof-theoretic ordinal λ. Let S λ := ID i 1 + α<λ TI(α). Moreover, let (λ n ) n N be a canonical fundamental sequence associated with λ. In particular sup λ n = λ. Theorem: (Embedding) There is an elementary function f e such that if d is a T-proof of a sentence φ then f e (d) is an S λ -proof of D(α, ρ, φ ) for some α, ρ. Of course, α, ρ depend on d.

147 The Steps of Ordinal Analysis cont d

148 The Steps of Ordinal Analysis cont d Theorem: (Cut-Elimination) There is an elementary function f c such that if φ is an arithmetic sentence and d is an S λ -proof of D(α, ρ, φ ) then f c (d) is an S λ -proof of D(α, 0, φ ) for some α.

149 The Steps of Ordinal Analysis cont d Theorem: (Cut-Elimination) There is an elementary function f c such that if φ is an arithmetic sentence and d is an S λ -proof of D(α, ρ, φ ) then f c (d) is an S λ -proof of D(α, 0, φ ) for some α. Theorem: (Truth) There is an elementary function f t such that for every arithmetic sentence φ and ordinal α < λ,, f t ( φ, α) is an S λ -proof of D(α, 0, φ ) φ g where φ g denotes the Gödel-Gentzen translation of φ.

150 The Steps of Ordinal Analysis cont d

151 The Steps of Ordinal Analysis cont d Theorem: (Realizability) There is an elementary function f rel such that if φ an (almost negative) arithmetic sentence and d is an S λ -proof of φ then f rel (d) is an HA + α<λ TI(α) proof of φ.

152 The Steps of Ordinal Analysis cont d Theorem: (Realizability) There is an elementary function f rel such that if φ an (almost negative) arithmetic sentence and d is an S λ -proof of φ then f rel (d) is an HA + α<λ TI(α) proof of φ. Theorem: (Upshot classically) There is an elem. function g such that for every T c -proof d of an arith. sentence φ, g(d) is a PA + α<λ TI(α) proof of φ.

153 The Steps of Ordinal Analysis cont d Theorem: (Realizability) There is an elementary function f rel such that if φ an (almost negative) arithmetic sentence and d is an S λ -proof of φ then f rel (d) is an HA + α<λ TI(α) proof of φ. Theorem: (Upshot classically) There is an elem. function g such that for every T c -proof d of an arith. sentence φ, g(d) is a PA + α<λ TI(α) proof of φ. Theorem: (Upshot intuitionistically) There is an elem. function g such that for every T i -proof d of an arith. sentence φ, g(d) is an HA + α<λ TI(α) proof of φ.

154 The other rules 1 If A is a true constant prime formula or negated prime formula and A Γ, then T α ϱ Γ. 2 If Γ contains formulas A(s 1,..., s n ) and A(t 1,..., t n ) of grade 0 or ω, where s i and t i (1 i n) are equivalent terms, then T α ϱ Γ. 3 If T β ϱ Γ i and β α hold for every premiss Γ i of an inference ( ), ( ), ( 1 ), ( 2 ) or (Cut) with a cut formula having grade < ϱ, and conclusion Γ, then T α ϱ Γ. 4 If T α 0 ϱ Γ, F(U) holds for some α 0 α and a non-arithmetic formula F(U), then T α ϱ Γ, XF(X).

155 Elementary Arithmetic Elementary arithmetic, EA, is a weak system of number theory, in a language with 0, 1, +,, E (exponentiation), <, whose axioms are:

156 Elementary Arithmetic Elementary arithmetic, EA, is a weak system of number theory, in a language with 0, 1, +,, E (exponentiation), <, whose axioms are: the usual recursion axioms for +,, E, <. induction on 0 -formulae with free variables. EA s provably computable functions are exactly the Kalmar elementary functions, i.e. the class of functions which contains the successor, projection, zero, addition, multiplication, and modified subtraction functions and is closed under composition and bounded sums and products.

157 Friedman s impractical matters

158 Friedman s impractical matters R.L. Smith: Consistency strength of some finite forms of the Higman and Kruskal theorems. In: Harvey Friedman s research on the foundations of mathematics. (1985)

159 Friedman s impractical matters R.L. Smith: Consistency strength of some finite forms of the Higman and Kruskal theorems. In: Harvey Friedman s research on the foundations of mathematics. (1985) Section 4: Practical Matters

160 Friedman s impractical matters R.L. Smith: Consistency strength of some finite forms of the Higman and Kruskal theorems. In: Harvey Friedman s research on the foundations of mathematics. (1985) Section 4: Practical Matters Let Q be the well-quasi ordering of all finite trees with 6 labels under label preserving homoemorphisms.

161 Friedman s impractical matters R.L. Smith: Consistency strength of some finite forms of the Higman and Kruskal theorems. In: Harvey Friedman s research on the foundations of mathematics. (1985) Section 4: Practical Matters Let Q be the well-quasi ordering of all finite trees with 6 labels under label preserving homoemorphisms. Let SWQ(Q) be the statement that for any c there exists a number k which is so large that, for any sequence T 0,..., T k of trees in Q with T i c (i + 1) for all i k, there exist indices i < j k such that T i is homoemorphically embeddable into T j.

162 Friedman s impractical matters R.L. Smith: Consistency strength of some finite forms of the Higman and Kruskal theorems. In: Harvey Friedman s research on the foundations of mathematics. (1985) Section 4: Practical Matters Let Q be the well-quasi ordering of all finite trees with 6 labels under label preserving homoemorphisms. Let SWQ(Q) be the statement that for any c there exists a number k which is so large that, for any sequence T 0,..., T k of trees in Q with T i c (i + 1) for all i k, there exist indices i < j k such that T i is homoemorphically embeddable into T j. Let Ψ Q (c) be the smallest such k.

163 Impractical matters cont d Theorem. SWQ(Q) is provable in Π 1 2 -BI but not in Π1 2 -BI 0.

164 Impractical matters cont d Theorem. SWQ(Q) is provable in Π 1 2 -BI but not in Π1 2 -BI 0. When c is specialized we obtain a Σ 0 1 statement SWQ c(q).

165 Impractical matters cont d Theorem. SWQ(Q) is provable in Π 1 2 -BI but not in Π1 2 -BI 0. When c is specialized we obtain a Σ 0 1 statement SWQ c(q). Theorem. SWQ 1 (Q) is provable in Π 1 2 -BI 0 (of course!) but any proof requires at least 2 [900] symbols. 2 [0] := 1, 2 [n+1] := 2 2[n].

166 Impractical matters cont d Theorem. SWQ(Q) is provable in Π 1 2 -BI but not in Π1 2 -BI 0. When c is specialized we obtain a Σ 0 1 statement SWQ c(q). Theorem. SWQ 1 (Q) is provable in Π 1 2 -BI 0 (of course!) but any proof requires at least 2 [900] symbols. 2 [0] := 1, 2 [n+1] := 2 2[n]. Why?

167 Impractical matters cont d Theorem. SWQ(Q) is provable in Π 1 2 -BI but not in Π1 2 -BI 0. When c is specialized we obtain a Σ 0 1 statement SWQ c(q). Theorem. SWQ 1 (Q) is provable in Π 1 2 -BI 0 (of course!) but any proof requires at least 2 [900] symbols. 2 [0] := 1, 2 [n+1] := 2 2[n]. Why? > atoms in the visible universe

168 Impractical matters cont d For a theory T define χ T (n) to be the least integer k such that if x A(x) is any Σ 0 1 statement provable in T using n symbols, then A(m) is true for some m k.

169 Impractical matters cont d For a theory T define χ T (n) to be the least integer k such that if x A(x) is any Σ 0 1 statement provable in T using n symbols, then A(m) is true for some m k. Let χ λ (n) be the least integer k such that if x A(x) is any Σ 0 1 statement provable in PA + α<λ TI(α) using n symbols, then A(m) is true for some m k.

170 Impractical matters finale 1 Theorem. The proof in [RW] of the 1-consistency of T := Π 1 2 -BI 0 in PA + TI(θΩ ω 0) can be carried out by a proof shorter than symbols.

171 Impractical matters finale 1 Theorem. The proof in [RW] of the 1-consistency of T := Π 1 2 -BI 0 in PA + TI(θΩ ω 0) can be carried out by a proof shorter than symbols. 2 Corollary. χ T (n) χ θω ω 0 (2 2(n+1) 1000 )

172 Impractical matters finale 1 Theorem. The proof in [RW] of the 1-consistency of T := Π 1 2 -BI 0 in PA + TI(θΩ ω 0) can be carried out by a proof shorter than symbols. 2 Corollary. χ T (n) χ θω ω 0 (2 2(n+1) 1000 ) 3 (Friedman: in [Smith] Lemma 22) χ θω ω 0 (2 [1000] ) Ψ Q (1).

173 Impractical matters finale 1 Theorem. The proof in [RW] of the 1-consistency of T := Π 1 2 -BI 0 in PA + TI(θΩ ω 0) can be carried out by a proof shorter than symbols. 2 Corollary. χ T (n) χ θω ω 0 (2 2(n+1) 1000 ) 3 (Friedman: in [Smith] Lemma 22) χ θω ω 0 (2 [1000] ) Ψ Q (1). 4 χ T (n) χ θω ω 0 (2 2(n+1) 1000 )

174 Impractical matters finale 1 Theorem. The proof in [RW] of the 1-consistency of T := Π 1 2 -BI 0 in PA + TI(θΩ ω 0) can be carried out by a proof shorter than symbols. 2 Corollary. χ T (n) χ θω ω 0 (2 2(n+1) 1000 ) 3 (Friedman: in [Smith] Lemma 22) χ θω ω 0 (2 [1000] ) Ψ Q (1). 4 χ T (n) χ θω ω 0 (2 2(n+1) 1000 ) 5 χ T (2 [900] ) χ θω ω 0 (2 [1000] ) Ψ Q (1)

175 The End

176 Thank you! The End

177 Elementary Arithmetic Elementary arithmetic, EA, is a weak system of number theory, in a language with 0, 1, +,, E (exponentiation), <, whose axioms are:

178 Elementary Arithmetic Elementary arithmetic, EA, is a weak system of number theory, in a language with 0, 1, +,, E (exponentiation), <, whose axioms are: the usual recursion axioms for +,, E, <. induction on 0 -formulae with free variables. EA s provably computable functions are exactly the Kalmar elementary functions, i.e. the class of functions which contains the successor, projection, zero, addition, multiplication, and modified subtraction functions and is closed under composition and bounded sums and products.

179 Elementary Ordinal Representation Systems An elementary ordinal representation system (EORS) for a limit ordinal λ is a structure A,, n λ n, +,, u ω u such that everything" is provably elementary in EA.

180 Transfinite Induction TI(A, ) is the schema n A [ k n P(k) P(n)] n A P(n) with P arithmetical.

181 Transfinite Induction TI(A, ) is the schema n A [ k n P(k) P(n)] n A P(n) with P arithmetical. For α A let α be restricted to A α := {β A β α}.

182 Descent Arithmetic For each α A, EWF(, α) is the schema expressing that: All elementary -descending chains below α are finite.

183 Descent Arithmetic For each α A, EWF(, α) is the schema expressing that: All elementary -descending chains below α are finite. DA A, (Descent Arithmetic) is the theory whose axioms are EA + α A EWF(, α).

184 The general form of Ordinal Analysis refined DA A, and PA + α A TI(A α, α ) prove the same Π 0 2 sentences.

185 The general form of Ordinal Analysis refined DA A, and PA + α A TI(A α, α ) prove the same Π 0 2 sentences. Suppose an ordinal analysis of the formal system T has been attained using an EORS A,,....

186 The general form of Ordinal Analysis refined DA A, and PA + α A TI(A α, α ) prove the same Π 0 2 sentences. Suppose an ordinal analysis of the formal system T has been attained using an EORS A,,.... Then the provably computable functions of T are the descent computable functions over A.

187 The Descent Computable Functions The provably computable functions of DA A, are all functions f of the form f ( m) = g( m, least n.h( m, n) h( m, n + 1)) (1) where g and h are elementary functions and for some α A. EA xy h( x, y) A α The above class of functions is called the descent computable functions over A.

188 Notions of Proof-Theoretical Reduction

189 Notions of Proof-Theoretical Reduction All theories T considered in the following are assumed to contain a modicum of arithmetic. For definiteness let this mean that the system PRA of Primitive Recursive Arithmetic is contained in T, either directly or by translation.

190 Notions of Proof-Theoretical Reduction All theories T considered in the following are assumed to contain a modicum of arithmetic. For definiteness let this mean that the system PRA of Primitive Recursive Arithmetic is contained in T, either directly or by translation. Definition (Feferman: Hilbert s program relativized: Proof-theoretical and foundational reductions, J. Symbolic Logic 53 (1988) ) Let T 1, T 2 be a pair of theories with languages L 1 and L 2, respectively, and let Φ be a (primitive recursive) collection of formulae common to both languages. Furthermore, Φ should contain the closed equations of the language of PRA.

191 Notions of Proof-Theoretical Reduction cont ed We then say that T 1 is proof-theoretically Φ-reducible to T 2, written T 1 Φ T 2, if there exists a T 2 -recursive function f such that T 2 φ Φ x [Proof T1 (x, φ) Proof T2 (f (x), φ)]. (2)

192 Notions of Proof-Theoretical Reduction cont ed We then say that T 1 is proof-theoretically Φ-reducible to T 2, written T 1 Φ T 2, if there exists a T 2 -recursive function f such that T 2 φ Φ x [Proof T1 (x, φ) Proof T2 (f (x), φ)]. (2) In The realm of ordinal analysis, S.B. Cooper and J.K. Truss (eds.): Sets and Proofs, (Cambridge University Press, 1999) , I required instead that there exists a primitive recursive function f such that PRA φ Φ x [Proof T1 (x, φ) Proof T2 (f (x), φ)]. (3)

193 Extracting computational information from infinite derivations without coding

194 Extracting computational information from infinite derivations without coding Buchholz and Wainer 1987

195 Extracting computational information from infinite derivations without coding Buchholz and Wainer 1987 Buchholz 1991, 1999

196 Extracting computational information from infinite derivations without coding Buchholz and Wainer 1987 Buchholz 1991, 1999 Weiermann 1996

197 Extracting computational information from infinite derivations without coding Buchholz and Wainer 1987 Buchholz 1991, 1999 Weiermann 1996 Blankertz 1998

198 Extracting computational information from infinite derivations without coding Buchholz and Wainer 1987 Buchholz 1991, 1999 Weiermann 1996 Blankertz 1998 Michelbrink 2001