PROOF THEORY: From arithmetic to set theory

Transcription

1 PROOF THEORY: From arithmetic to set theory Michael Rathjen School of Mathematics University of Leeds Nordic Spring School in Logic, Nordfjordeid May 27-31, 2013

2 Plan of First and Second Talk The origins of Proof theory: Hilbert s Programme Gentzen s Result The General Form of Ordinal Analysis Gentzen s Hauptsatz: Cut Elimination A Brief History of Ordinal Representation Systems A Brief History of Ordinal Analyses Applications of Ordinal Analysis 1 Combinatorial Independence Results 2 Classification of Provable Functions 3 Equiconsistency Results

3 Plan of the Third and Fourth Talk PREDICATIVE PROOF THEORY IMPREDICATIVE PROOF THEORY Ordinal Analysis of Kripke-Platek Set Theory (sketch) Uniformity of Infinite Proofs Proof Theory of Much Stronger Theories

4 The Origins of Proof Theory (Beweistheorie) Hilbert s second problem (1900): Consistency of Analysis Hilbert s Programme (1922,1925)

5 Hilbert s Programme (1922,1925) I. Codify the whole of mathematical reasoning in a formal theory T.

6 Hilbert s Programme (1922,1925) I. Codify the whole of mathematical reasoning in a formal theory T. II. Prove the consistency of T by finitistic means.

7 Hilbert s Programme (1922,1925) I. Codify the whole of mathematical reasoning in a formal theory T. II. Prove the consistency of T by finitistic means. To carry out this task, Hilbert inaugurated a new mathematical discipline: Beweistheorie ( Proof Theory).

8 Hilbert s Programme (1922,1925) I. Codify the whole of mathematical reasoning in a formal theory T. II. Prove the consistency of T by finitistic means. To carry out this task, Hilbert inaugurated a new mathematical discipline: Beweistheorie ( Proof Theory). In Hilbert s Proof Theory, proofs become mathematical objects sui generis.

9 Ackermann s Dissertation 1925 Consistency proof for a second-order version of Primitive Recursive Arithmetic. Uses a finitistic version of transfinite induction up to the ordinal ω ωω.

10 Gentzen s Result Gerhard Gentzen showed that transfinite induction up to the ordinal ε 0 = sup{ω, ω ω, ω ωω,...} = least α. ω α = α suffices to prove the consistency of Peano Arithmetic, PA.

11 Gentzen s Result Gerhard Gentzen showed that transfinite induction up to the ordinal ε 0 = sup{ω, ω ω, ω ωω,...} = least α. ω α = α suffices to prove the consistency of Peano Arithmetic, PA. Gentzen s applied transfinite induction up to ε 0 solely to primitive recursive predicates and besides that his proof used only finitistically justified means.

12 Gentzen s Result in Detail F + PR-TI(ε 0 ) Con(PA), where F signifies a theory that is acceptable in finitism (e.g. F = PRA = Primitive Recursive Arithmetic) and PR-TI(ε 0 ) stands for transfinite induction up to ε 0 for primitive recursive predicates.

13 Gentzen s Result in Detail F + PR-TI(ε 0 ) Con(PA), where F signifies a theory that is acceptable in finitism (e.g. F = PRA = Primitive Recursive Arithmetic) and PR-TI(ε 0 ) stands for transfinite induction up to ε 0 for primitive recursive predicates. Gentzen also showed that his result is best possible: PA proves transfinite induction up to α for arithmetic predicates for any α < ε 0.

14 The Compelling Picture The non-finitist part of PA is encapsulated in PR-TI(ε 0 ) and therefore measured by ε 0, thereby tempting one to adopt the following definition of proof-theoretic ordinal of a theory T : T Con = least α. PRA + PR-TI(α) Con(T ).

15 The supremum of the provable ordinals A, is said to be provably wellordered in T if T WO(A, ).

16 The supremum of the provable ordinals A, is said to be provably wellordered in T if T WO(A, ). α is provably computable in T if there is a computable well ordering A, with order type α such that T WO(A, ) with A and being provably computable in T.

17 The supremum of the provable ordinals A, is said to be provably wellordered in T if T WO(A, ). α is provably computable in T if there is a computable well ordering A, with order type α such that T WO(A, ) with A and being provably computable in T. The supremum of the provable well-orderings of T: T sup := sup { α : α provably computable in T }.

18 Ordinal Structures We are interested in representing specific ordinals α as relations on N. Natural ordinal representation systems are frequently derived from structures of the form A = α, f 1,..., f n, < α where α is an ordinal, < α is the ordering of ordinals restricted to elements of α and the f i are functions for some natural number k i. f i : α } {{ α } α k i times

19 Ordinal Representation Systems A = A, g 1,..., g n, is a computable (or recursive) representation of A = α, f 1,..., f n, < α if the following conditions hold: 1 A N and A is a computable set.

20 Ordinal Representation Systems A = A, g 1,..., g n, is a computable (or recursive) representation of A = α, f 1,..., f n, < α if the following conditions hold: 1 A N and A is a computable set. 2 is a computable total ordering on A and the functions g i are computable.

21 Ordinal Representation Systems A = A, g 1,..., g n, is a computable (or recursive) representation of A = α, f 1,..., f n, < α if the following conditions hold: 1 A N and A is a computable set. 2 is a computable total ordering on A and the functions g i are computable. 3 A = A, i.e. the two structures are isomorphic.

22 Cantor s Representation of Ordinals Theorem (Cantor, 1897) For every ordinal β > 0 there exist unique ordinals β 0 β 1 β n such that β = ω β ω βn. (1) The representation of β in (1) is called the Cantor normal form. We shall write β = CNF β 0 β 1 β k. ω β 1 + ω βn to convey that

23 A Representation for ε 0 ε 0 denotes the least ordinal α > 0 such that β < α ω β < α.

24 A Representation for ε 0 ε 0 denotes the least ordinal α > 0 such that β < α ω β < α. ε 0 is the least ordinal α such that ω α = α.

25 A Representation for ε 0 ε 0 denotes the least ordinal α > 0 such that β < α ω β < α. ε 0 is the least ordinal α such that ω α = α. β < ε 0 has a Cantor normal form with exponents β i < β and these exponents have Cantor normal forms with yet again smaller exponents. As this process must terminate, ordinals < ε 0 can be coded by natural numbers.

26 Define a function by δ = Coding ε 0 in N. : ε 0 N { 0 if δ = 0 δ 1,..., δ n if δ = CNF ω δ 1 + ω δn where k 1,, k n := 2 k pn kn+1 with p i being the ith prime number (or any other coding of tuples). Further define A 0 := ran(. ) δ β : δ < β δ ˆ+ β := δ + β δ ˆ β := δ β ˆω δ := ω δ.

27 Coding ε 0 in N Then ε 0, +,, δ ω δ, < = A0, ˆ+,ˆ, x ˆω x,. A 0, ˆ+,ˆ, x ˆω x, are recursive, in point of fact, they are all elementary recursive.

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

29 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}.

30 The general form of ordinal analysis T framework for formalizing a certain part of mathematics. T should be a true theory which contains a modicum of arithmetic.

31 The general form of ordinal analysis T framework for formalizing a certain part of mathematics. T should be a true theory which contains a modicum of arithmetic. Every ordinal analysis of a classical or intuitionistic theory T that has ever appeared in the literature provides an EORS A,,... such that T is finitistically reducible to PA + α A TI(A α, α ).

32 The general form of ordinal analysis T framework for formalizing a certain part of mathematics. T should be a true theory which contains a modicum of arithmetic. Every ordinal analysis of a classical or intuitionistic theory T that has ever appeared in the literature provides an EORS A,,... such that T is finitistically reducible to PA + α A TI(A α, α ). T and HA + α A TI(A α, α ) prove the same Π 0 2 sentences.

33 The general form of ordinal analysis T framework for formalizing a certain part of mathematics. T should be a true theory which contains a modicum of arithmetic. Every ordinal analysis of a classical or intuitionistic theory T that has ever appeared in the literature provides an EORS A,,... such that T is finitistically reducible to PA + α A TI(A α, α ). T and HA + α A TI(A α, α ) prove the same Π 0 2 sentences. T sup =.

34 Ordinally Informative Proof Theory The two main strands of research are: Cut Elimination (and Proof Collapsing Techniques)

35 Ordinally Informative Proof Theory The two main strands of research are: Cut Elimination (and Proof Collapsing Techniques) Development of ever stronger Ordinal Representation Systems

36 The Sequent Calculus SEQUENTS A sequent is an expression Γ where Γ and are finite sequences of formulae A 1,..., A n and B 1,..., B m, respectively.

37 The Sequent Calculus SEQUENTS A sequent is an expression Γ where Γ and are finite sequences of formulae A 1,..., A n and B 1,..., B m, respectively. Γ is read, informally, as Γ yields or, rather, the conjunction of the A i yields the disjunction of the B j.

38 The Sequent Calculus LOGICAL INFERENCES I Negation Γ, A A, Γ L B, Γ R Γ, B Implication Γ, A B, Λ Θ L A B, Γ, Λ, Θ A, Γ, B R Γ, A B

39 Conjunction A, Γ L1 A B, Γ B, Γ L2 A B, Γ Disjunction Γ, A Γ, B R Γ, A B A, Γ B, Γ L A B, Γ Γ, A Γ, A B R1 Γ, B Γ, A B R2

40 The Sequent Calculus LOGICAL INFERENCES II Quantifiers F(t), Γ x F (x), Γ L Γ, F(a) Γ, x F (x) R F(a), Γ x F (x), Γ L Γ, F(t) Γ, x F (x) R In L and R, t is an arbitrary term. The variable a in R and L is an eigenvariable of the respective inference, i.e. a is not to occur in the lower sequent.

41 The Sequent Calculus AXIOMS Identity Axiom where A is any formula. A A One could limit this axiom to the case of atomic formulae A

42 The Sequent Calculus CUTS CUT Γ, A A, Λ Θ Cut Γ, Λ, Θ A is called the cut formula of the inference. Example B A A C Cut B C

43 The Sequent Calculus STRUCTURAL RULES Structural Rules Exchange, Weakening, Contraction Γ, A, B, Λ Xl Γ, B, A, Λ Γ, A, B, Λ Xr Γ, B, A, Λ Γ Γ, A Wl Γ Γ, A Wr Γ, A, A Cl Γ, A Γ, A, A Cr Γ, A

44 The INTUITIONISTIC case

45 The INTUITIONISTIC case The intuitionistic sequent calculus is obtained by requiring that all sequents be intuitionistic.

46 The INTUITIONISTIC case The intuitionistic sequent calculus is obtained by requiring that all sequents be intuitionistic. A sequent Γ is said to be intuitionistic if consists of at most one formula.

47 The INTUITIONISTIC case The intuitionistic sequent calculus is obtained by requiring that all sequents be intuitionistic. A sequent Γ is said to be intuitionistic if consists of at most one formula. Specifically, in the intuitionistic sequent calculus there are no inferences corresponding to contraction right or exchange right.

48 Classical Example

49 Classical Example Our first example is a deduction of the law of excluded middle.

50 Classical Example Our first example is a deduction of the law of excluded middle. A A R A, A R A, A A Xr A A, A R A A, A A Cr A A

51 Classical Example Our first example is a deduction of the law of excluded middle. A A R A, A R A, A A Xr A A, A R A A, A A Cr A A Notice that the above proof is not intuitionistic since it involves sequents that are not intuitionistic.

52 Intuitionistic Example

53 Intuitionistic Example The second example is an intuitionistic deduction.

54 Intuitionistic Example The second example is an intuitionistic deduction. F (a) F(a) R F(a) x F (x) L x F (x), F(a) Xl F(a), x F (x) L xf(x) F(a) R x F (x) x F(x) R x F (x) x F(x)

55 Gentzen s Hauptsatz (1934) Cut Elimination If a sequent Γ is provable, then it is provable without cuts.

56 Cut Elimination EXAMPLE Here is an example of how to eliminate cuts of a special form: A, Γ, B Λ Θ, A B, Ξ Φ R L Γ, A B A B, Λ, Ξ Θ, Φ Cut Γ, Λ, Ξ, Θ, Φ is replaced by Λ Θ, A A, Γ, B Cut Λ, Γ Θ,, B B, Ξ Φ Cut Γ, Λ, Ξ, Θ, Φ

57 The Subformula Property The Hauptsatz has an important corollary:

58 The Subformula Property The Hauptsatz has an important corollary: The Subformula Property If a sequent Γ is provable, then it has a deduction all of whose formulae are subformulae of the formulae in Γ and.

59 The Subformula Property The Hauptsatz has an important corollary: The Subformula Property If a sequent Γ is provable, then it has a deduction all of whose formulae are subformulae of the formulae in Γ and. Corollary A contradiction, i.e. the empty sequent, is not deducible.

60 Applications of the Haupsatz

61 Applications of the Haupsatz Herbrand s Theorem in LK (classical): xr(x) implies R(t 1 )... R(t n ) some t i (R quantifier-free).

62 Applications of the Haupsatz Herbrand s Theorem in LK (classical): xr(x) implies R(t 1 )... R(t n ) some t i (R quantifier-free). Extended Herbrand s Theorem in LK : Γ xr(x) implies Γ R(t 1 )... R(t n ) some t i (R quantifier-free, Γ purely universal).

63 Applications of the Haupsatz Herbrand s Theorem in LK (classical): xr(x) implies R(t 1 )... R(t n ) some t i (R quantifier-free). Extended Herbrand s Theorem in LK : Γ xr(x) implies Γ R(t 1 )... R(t n ) some t i (R quantifier-free, Γ purely universal). In LJ (intuitionistic): for some term t. xr(x) implies R(t)

64 Applications of the Haupsatz Herbrand s Theorem in LK (classical): xr(x) implies R(t 1 )... R(t n ) some t i (R quantifier-free). Extended Herbrand s Theorem in LK : Γ xr(x) implies Γ R(t 1 )... R(t n ) some t i (R quantifier-free, Γ purely universal). In LJ (intuitionistic): xr(x) implies R(t) for some term t. Hilbert-Ackermann Consistency

65 Applications of the Haupsatz Herbrand s Theorem in LK (classical): xr(x) implies R(t 1 )... R(t n ) some t i (R quantifier-free). Extended Herbrand s Theorem in LK : Γ xr(x) implies Γ R(t 1 )... R(t n ) some t i (R quantifier-free, Γ purely universal). In LJ (intuitionistic): xr(x) implies R(t) for some term t. Hilbert-Ackermann Consistency If T is a geometric theory and T classically proves a geometric implication A then T intuitionistically proves A.

66 Theories and Cut Elimination What happens when we try to apply the procedure of cut elimination to theories?

67 Theories and Cut Elimination What happens when we try to apply the procedure of cut elimination to theories? Axioms are detrimental to this procedure. It breaks down because the symmetry of the sequent calculus is lost. In general, we cannot remove cuts from deductions in a theory T when the cut formula is an axiom of T.

68 Theories and Cut Elimination What happens when we try to apply the procedure of cut elimination to theories? Axioms are detrimental to this procedure. It breaks down because the symmetry of the sequent calculus is lost. In general, we cannot remove cuts from deductions in a theory T when the cut formula is an axiom of T. However, sometimes the axioms of a theory are of bounded syntactic complexity. Then the procedure applies partially in that one can remove all cuts that exceed the complexity of the axioms of T.

69 Partial Cut Elimination Gives rise to partial cut elimination.

70 Partial Cut Elimination Gives rise to partial cut elimination. This is a very important tool in proof theory. For example, it works very well if the axioms of a theory can be presented as atomic intuitionistic sequents (also called Horn clauses), yielding the completeness of Robinsons resolution method.

71 Partial cut elimination also pays off in the case of fragments of PA and set theory with restricted induction schemes, be it induction on natural numbers or sets. This method can be used to extract bounds from proofs of Π 0 2 statements in such fragments.

72 Going Infinite Full arithmetic, i.e. PA, does not even allow for partial cut elimination since the induction axioms have unbounded complexity.

73 Going Infinite Full arithmetic, i.e. PA, does not even allow for partial cut elimination since the induction axioms have unbounded complexity. However, one can remove the obstacle against cut elimination in a drastic way by going infinite. The so-called ω-rule consists of the two types of infinitary inferences:

74 Going Infinite Full arithmetic, i.e. PA, does not even allow for partial cut elimination since the induction axioms have unbounded complexity. However, one can remove the obstacle against cut elimination in a drastic way by going infinite. The so-called ω-rule consists of the two types of infinitary inferences: Γ, F(0); Γ, F(1);... ; Γ, F(n);... Γ, x F (x) ωr

75 Going Infinite Full arithmetic, i.e. PA, does not even allow for partial cut elimination since the induction axioms have unbounded complexity. However, one can remove the obstacle against cut elimination in a drastic way by going infinite. The so-called ω-rule consists of the two types of infinitary inferences: Γ, F(0); Γ, F(1);... ; Γ, F(n);... Γ, x F (x) F(0), Γ ; F(1), Γ ;... ; F(n), Γ ;... x F (x), Γ ωr ωl

76 Going Infinite Full arithmetic, i.e. PA, does not even allow for partial cut elimination since the induction axioms have unbounded complexity. However, one can remove the obstacle against cut elimination in a drastic way by going infinite. The so-called ω-rule consists of the two types of infinitary inferences: Γ, F(0); Γ, F(1);... ; Γ, F(n);... Γ, x F (x) F(0), Γ ; F(1), Γ ;... ; F(n), Γ ;... x F (x), Γ ωr ωl The price to pay will be that deductions become infinite.

77 With the aid of the ω-rule, induction becomes logically deducible in infinitary logic. Theorem For every n there is a finite deduction D n of the sequent F(0), x [F(x) F(Sx)] F(n). Proof. Since B, Γ B is deducible for every formula B and sequence Γ, we obtain D 0. Let := F(0), x [F(x) F(Sx)]. From D n we obtain D n+1 : D n D F(n). F(Sn), F(Sn) L F(n) F(Sn), F(S(n)) L x [F(x) F(Sx)], F(S(n)) Struc F(0), x [F (x) F(Sx)] F (S(n))

78 Embedding PA Embedding Theorem If then for some m, k < ω. PA ω PA Γ ω+m k Γ

79 Reduction Lemma If PA ω with k = A, then α k Γ, A and PA ω β k A, Λ Θ PA ω α#β k Γ, Λ, Θ.

80 Cut Elimination for PA ω Theorem If PA ω α k+1 Γ, then PA ω ω α k Γ. Cut Elimination Theorem If PA ω α n Γ, then PA ω ω ω...ωα 0 Γ. ωα ω} ω.. {{}. n times

81 Infinitary Calculi for Set Theory To achieve (partial) cut elimination for set theory, one needs infinitary rules similar to the ω-rule. These rules enable one to get cut-free deductions of -induction. β-logic x [[ y x A(y)] A(x)] x A(x)

82 A brief history of ordinal representation systems

83 A brief history of ordinal representation systems Hardy (1904) wanted to construct a subset of R of size ℵ 1.

84 A brief history of ordinal representation systems Hardy (1904) wanted to construct a subset of R of size ℵ 1. Hardy gives explicit representations for all ordinals < ω 2.

85 O. Veblen, 1908 Veblen extended the initial segment of the countable for which fundamental sequences can be given effectively. He applied two new operations to continuous increasing functions on ordinals:

86 O. Veblen, 1908 Veblen extended the initial segment of the countable for which fundamental sequences can be given effectively. He applied two new operations to continuous increasing functions on ordinals: Derivation

87 O. Veblen, 1908 Veblen extended the initial segment of the countable for which fundamental sequences can be given effectively. He applied two new operations to continuous increasing functions on ordinals: Derivation Transfinite Iteration

88 O. Veblen, 1908 Veblen extended the initial segment of the countable for which fundamental sequences can be given effectively. He applied two new operations to continuous increasing functions on ordinals: Derivation Transfinite Iteration Let ON be the class of ordinals. A (class) function f : ON ON is said to be increasing if α < β implies f (α) < f (β) and continuous (in the order topology on ON) if f (lim α ξ ) = lim f (α ξ ) ξ<λ ξ<λ holds for every limit ordinal λ and increasing sequence (α ξ ) ξ<λ.

89 Derivations f is called normal if it is increasing and continuous.

90 Derivations f is called normal if it is increasing and continuous. The function β ω + β is normal while β β + ω is not continuous at ω since lim ξ<ω (ξ + ω) = ω but (lim ξ<ω ξ) + ω = ω + ω.

91 Derivations f is called normal if it is increasing and continuous. The function β ω + β is normal while β β + ω is not continuous at ω since lim ξ<ω (ξ + ω) = ω but (lim ξ<ω ξ) + ω = ω + ω. The derivative f of a function f : ON ON is the function which enumerates in increasing order the solutions of the equation f (α) = α, also called the fixed points of f.

92 Derivations f is called normal if it is increasing and continuous. The function β ω + β is normal while β β + ω is not continuous at ω since lim ξ<ω (ξ + ω) = ω but (lim ξ<ω ξ) + ω = ω + ω. The derivative f of a function f : ON ON is the function which enumerates in increasing order the solutions of the equation f (α) = α, also called the fixed points of f. If f is a normal function, {α : f (α) = α} is a proper class and f will be a normal function, too.

93 A Hierarchy of Ordinal Functions Given a normal function f : ON ON, define a hierarchy of normal functions as follows:

94 A Hierarchy of Ordinal Functions Given a normal function f : ON ON, define a hierarchy of normal functions as follows: f 0 = f

95 A Hierarchy of Ordinal Functions Given a normal function f : ON ON, define a hierarchy of normal functions as follows: f 0 = f f α+1 = f α

96 A Hierarchy of Ordinal Functions Given a normal function f : ON ON, define a hierarchy of normal functions as follows: f 0 = f f α+1 = f α f λ (ξ) = ξ th element of {Fixed points of f α } α<λ for λ limit.

97 The Feferman-Schütte Ordinal Γ 0 From the normal function f we get a two-place function, ϕ f (α, β) := f α (β). Veblen then discusses the hierarchy when f = l, l(α) = ω α.

98 The Feferman-Schütte Ordinal Γ 0 From the normal function f we get a two-place function, ϕ f (α, β) := f α (β). Veblen then discusses the hierarchy when f = l, l(α) = ω α. The least ordinal γ > 0 closed under ϕ l, i.e. the least ordinal > 0 satisfying ( α, β < γ) ϕ l (α, β) < γ is the famous ordinal Γ 0 which Feferman and Schütte determined to be the least ordinal unreachable by predicative means.

99 The Big Veblen Number Veblen extended this idea first to arbitrary finite numbers of arguments, but then also to transfinite numbers of arguments, with the proviso that in, for example Φ f (α 0, α 1,..., α η ), only a finite number of the arguments may be non-zero. α ν

100 The Big Veblen Number Veblen extended this idea first to arbitrary finite numbers of arguments, but then also to transfinite numbers of arguments, with the proviso that in, for example Φ f (α 0, α 1,..., α η ), only a finite number of the arguments may be non-zero. α ν Veblen singled out the ordinal E(0), where E(0) is the least ordinal δ > 0 which cannot be named in terms of functions with η < δ, and each α γ < δ. Φ l (α 0, α 1,..., α η )

101 The Big Leap: H. Bachmann 1950 Bachmann s novel idea: Use uncountable ordinals to keep track of the functions defined by diagonalization.

102 The Big Leap: H. Bachmann 1950 Bachmann s novel idea: Use uncountable ordinals to keep track of the functions defined by diagonalization. Define a set of ordinals B closed under successor such that with each limit λ B is associated an increasing sequence λ[ξ] : ξ < τ λ of ordinals λ[ξ] B of length τ λ B and lim ξ<τλ λ[ξ] = λ.

103 The Big Leap: H. Bachmann 1950 Bachmann s novel idea: Use uncountable ordinals to keep track of the functions defined by diagonalization. Define a set of ordinals B closed under successor such that with each limit λ B is associated an increasing sequence λ[ξ] : ξ < τ λ of ordinals λ[ξ] B of length τ λ B and lim ξ<τλ λ[ξ] = λ. Let Ω be the first uncountable ordinal. A hierarchy of functions (ϕ B α) α B is then obtained as follows: ϕ B 0 (β) = 1 + β ϕb α+1 (ϕ = B α ϕ B λ enumerates (Range of ϕ B λ[ξ] ) ξ<τ λ ) λ limit, τ λ < Ω ϕ B λ enumerates {β < Ω : ϕb λ[β] (0) = β} λ limit, τ λ = Ω.

104 After Bachmann, the story of ordinal representation systems becomes very complicated. Isles, Bridge, Gerber, Pfeiffer, Schütte extended Bachmann s approach. Drawback: Horrendous computations.

105 After Bachmann, the story of ordinal representation systems becomes very complicated. Isles, Bridge, Gerber, Pfeiffer, Schütte extended Bachmann s approach. Drawback: Horrendous computations. Aczel and Weyhrauch combined Bachmann s approach with uses of higher type functionals.

106 After Bachmann, the story of ordinal representation systems becomes very complicated. Isles, Bridge, Gerber, Pfeiffer, Schütte extended Bachmann s approach. Drawback: Horrendous computations. Aczel and Weyhrauch combined Bachmann s approach with uses of higher type functionals. Feferman s new proposal: Bachmann-type hierarchy without fundamental sequences.

107 After Bachmann, the story of ordinal representation systems becomes very complicated. Isles, Bridge, Gerber, Pfeiffer, Schütte extended Bachmann s approach. Drawback: Horrendous computations. Aczel and Weyhrauch combined Bachmann s approach with uses of higher type functionals. Feferman s new proposal: Bachmann-type hierarchy without fundamental sequences. Bridge and Buchholz showed computability of systems obtained by Feferman s approach.

108 Natural well-orderings Set-theoretical (Cantor, Veblen, Gentzen, Bachmann, Schütte, Feferman, Pfeiffer, Isles, Bridge, Buchholz, Pohlers, Jäger, Rathjen) Define hierarchies of functions on the ordinals. Build up terms from function symbols for those functions. The ordering on the values of terms induces an ordering on the terms. Reductions in proof figures (Takeuti, Yasugi, Kino, Arai) Ordinal diagrams; formal terms endowed with an inductively defined ordering on them.

109 Natural well-orderings Patterns of elementary substructurehood (Carlson) Finite structures with Σ n -elementary substructure relations. Category-theoretical (Aczel, Girard, Jervell, Vauzeilles) Functors on the category of ordinals (with strictly increasing functions) respecting direct limits and pull-backs. Representation systems from below (Setzer)

110 Second order arithmetic; Z 2 aka Analysis Z 2 is a two sorted formal system. Extends PA. Variables n, m,... range over natural numbers. Variables X, Y, Z,... range over sets of natural numbers. Relation symbols =, <,. Function symbols +,,...

111 Second order arithmetic; Z 2 aka Analysis Z 2 is a two sorted formal system. Extends PA. Variables n, m,... range over natural numbers. Variables X, Y, Z,... range over sets of natural numbers. Relation symbols =, <,. Function symbols +,,... Comprehension Principle/Axiom: For any property P definable in the language of Z 2, is a set; or more formally {n N P(n)} (CA) X n [n X A(x)] for any formula A(x) of Z 2.

112 Stratification of Comprehension A Π 1 k -formula (Σ1 k -formula) is a formula of Z 2 of the form X 1... QX k A(X 1,..., X k ) ( X 1... QX k A(X 1,..., X k )) with X 1... QX k ( X 1... QX k ) a string of k alternating set quantifiers, beginning with a universal quantifier (existential quantifier), followed by a formula A(X 1,..., X k ) without set quantifiers.

113 Stratification of Comprehension A Π 1 k -formula (Σ1 k -formula) is a formula of Z 2 of the form X 1... QX k A(X 1,..., X k ) ( X 1... QX k A(X 1,..., X k )) with X 1... QX k ( X 1... QX k ) a string of k alternating set quantifiers, beginning with a universal quantifier (existential quantifier), followed by a formula A(X 1,..., X k ) without set quantifiers. Π 1 k -comprehension (Σ1 k-comprehension) is the scheme with A(x) Π 1 k (Σ 1 k ). X n [n X A(x)]

114 Subsystems of Z 2 Basic arithmetical axioms in all subtheories of Z 2 are: defining axioms for 0, 1, +,, E, < (as for PA) and the induction axiom X [ 0 X n(n X n + 1 X) n (n X)].

115 Subsystems of Z 2 Basic arithmetical axioms in all subtheories of Z 2 are: defining axioms for 0, 1, +,, E, < (as for PA) and the induction axiom X [ 0 X n(n X n + 1 X) n (n X)]. For each axiom scheme Ax, (Ax) 0 denotes the theory consisting of the basic arithmetical axioms plus the scheme Ax.

116 Subsystems of Z 2 Basic arithmetical axioms in all subtheories of Z 2 are: defining axioms for 0, 1, +,, E, < (as for PA) and the induction axiom X [ 0 X n(n X n + 1 X) n (n X)]. For each axiom scheme Ax, (Ax) 0 denotes the theory consisting of the basic arithmetical axioms plus the scheme Ax. (Ax) stands for the theory (Ax) 0 augmented by the scheme of induction for all L 2 -formulae.

117 Subsystems of Z 2 Basic arithmetical axioms in all subtheories of Z 2 are: defining axioms for 0, 1, +,, E, < (as for PA) and the induction axiom X [ 0 X n(n X n + 1 X) n (n X)]. For each axiom scheme Ax, (Ax) 0 denotes the theory consisting of the basic arithmetical axioms plus the scheme Ax. (Ax) stands for the theory (Ax) 0 augmented by the scheme of induction for all L 2 -formulae. Let F be a collection of formulae of Z 2. Another important axiom scheme for formulae F in C is C AC n YF(n, Y ) Y nf (x, Y n ), where Y n := {m : 2 n 3 m Y }.

118 How much of Z 2 is needed? Hermann Weyl 1918 Das Kontinuum" Predicative Analysis.

119 How much of Z 2 is needed? Hermann Weyl 1918 Das Kontinuum" Predicative Analysis. Hilbert, Bernays 1938: Z 2 sufficient for Ordinary Mathematics"

120 How much of Z 2 is needed? Hermann Weyl 1918 Das Kontinuum" Predicative Analysis. Hilbert, Bernays 1938: Z 2 sufficient for Ordinary Mathematics" Minimal foundational frameworks for Ordinary Mathematics: Feferman, Lorenzen, Takeuti... Reverse Mathematics, early 1970s-now H. Friedman, S. Simpson,... Given a specific theorem τ of ordinary mathematics, which set existence axioms are needed in order to prove τ?

121 Five Systems For many mathematical theorems τ, there is a weakest natural subsystem S(τ) of Z 2 such that S(τ) proves τ. Moreover, it has turned out that S(τ) often belongs to a small list of specific subsystems of Z 2. Reverse Mathematics has singled out five subsystems of Z 2 : RCA 0 Recursive Comprehension

122 Five Systems For many mathematical theorems τ, there is a weakest natural subsystem S(τ) of Z 2 such that S(τ) proves τ. Moreover, it has turned out that S(τ) often belongs to a small list of specific subsystems of Z 2. Reverse Mathematics has singled out five subsystems of Z 2 : RCA 0 WKL 0 Recursive Comprehension Weak König s Lemma

123 Five Systems For many mathematical theorems τ, there is a weakest natural subsystem S(τ) of Z 2 such that S(τ) proves τ. Moreover, it has turned out that S(τ) often belongs to a small list of specific subsystems of Z 2. Reverse Mathematics has singled out five subsystems of Z 2 : RCA 0 WKL 0 ACA 0 Recursive Comprehension Weak König s Lemma Arithmetic Comprehension

124 Five Systems For many mathematical theorems τ, there is a weakest natural subsystem S(τ) of Z 2 such that S(τ) proves τ. Moreover, it has turned out that S(τ) often belongs to a small list of specific subsystems of Z 2. Reverse Mathematics has singled out five subsystems of Z 2 : RCA 0 WKL 0 ACA 0 ATR 0 Recursive Comprehension Weak König s Lemma Arithmetic Comprehension Arithmetic Transfinite Recursion

125 Five Systems For many mathematical theorems τ, there is a weakest natural subsystem S(τ) of Z 2 such that S(τ) proves τ. Moreover, it has turned out that S(τ) often belongs to a small list of specific subsystems of Z 2. Reverse Mathematics has singled out five subsystems of Z 2 : RCA 0 WKL 0 ACA 0 ATR 0 (Π 1 1 CA) 0 Recursive Comprehension Weak König s Lemma Arithmetic Comprehension Arithmetic Transfinite Recursion Π 1 1 -Comprehension

126 Mathematical Equivalences: Examples RCA 0 Every countable field has an algebraic closure"; Every countable ordered field has a real closure"

127 Mathematical Equivalences: Examples RCA 0 Every countable field has an algebraic closure"; Every countable ordered field has a real closure" WKL 0 Cauchy-Peano existence theorem for solutions of ordinary differential equations"; Hahn-Banch theorem for separable Banach spaces"

128 Mathematical Equivalences: Examples RCA 0 Every countable field has an algebraic closure"; Every countable ordered field has a real closure" WKL 0 Cauchy-Peano existence theorem for solutions of ordinary differential equations"; Hahn-Banch theorem for separable Banach spaces" ACA 0 Bolzano-Weierstrass theorem"; Every countable commutative ring with a unit has a maximal ideal"

129 Mathematical Equivalences: Examples RCA 0 Every countable field has an algebraic closure"; Every countable ordered field has a real closure" WKL 0 Cauchy-Peano existence theorem for solutions of ordinary differential equations"; Hahn-Banch theorem for separable Banach spaces" ACA 0 Bolzano-Weierstrass theorem"; Every countable commutative ring with a unit has a maximal ideal" ATR 0 Every countable reduced abelian p-group has an Ulm resolution"

130 Mathematical Equivalences: Examples RCA 0 Every countable field has an algebraic closure"; Every countable ordered field has a real closure" WKL 0 Cauchy-Peano existence theorem for solutions of ordinary differential equations"; Hahn-Banch theorem for separable Banach spaces" ACA 0 Bolzano-Weierstrass theorem"; Every countable commutative ring with a unit has a maximal ideal" ATR 0 Every countable reduced abelian p-group has an Ulm resolution" (Π 1 1 CA) 0 Every uncountable closed set of real numbers is the union of a perfect set and a countable set"; Every countable abelian group is a direct sum of a divisible group and a reduced group"

131 ATR 0 = Γ 0 ACA 0 = ε 0 RCA 0 = ω ω = WKL 0 0

132 (Σ 1 2 -AC) + BI = ψ Ω 1 I ( 1 2 -CA) = ψ Ω 1 Ω ε0 (Π 1 1 CA) 0 = ψ Ω1 Ω ω ATR 0 = Γ 0

133 (Σ 1 2 -AC) + BI = ψ Ω 1 I

134 (Π 1 2 CA) 0 = ψ Ω1 R ω

135 A Brief History of Ordinal Analysis Gentzen 1936 theory PA ordinal ε 0

136 A Brief History of Ordinal Analysis Gentzen 1936 theory PA ordinal ε 0 Feferman, Schütte 1963 Predicative Second Order Arithmetic ordinal Γ 0

137 A Brief History of Ordinal Analysis Gentzen 1936 theory PA ordinal ε 0 Feferman, Schütte 1963 Predicative Second Order Arithmetic ordinal Γ 0 Takeuti 1967 (Π 1 1 -CA) 0, (Π 1 1-CA) + BI ordinals ψ Ω1 Ω ω, ψ Ω1 ε Ωω+1 cardinal analogue: ω-many regular cardinals

138 A Brief History of Ordinal Analysis Gentzen 1936 theory PA ordinal ε 0 Feferman, Schütte 1963 Predicative Second Order Arithmetic ordinal Γ 0 Takeuti 1967 (Π 1 1 -CA) 0, (Π 1 1-CA) + BI ordinals ψ Ω1 Ω ω, ψ Ω1 ε Ωω+1 cardinal analogue: ω-many regular cardinals Takeuti, Yasugi 1983 ( 1 2 -CA) ordinal ψ Ω1 Ω ε0 cardinal analogue: ε 0 -many regular cardinals

139 A Brief History of Ordinal Analysis cont d Buchholz, Pohlers, Sieg 1977 Theories of Iterated Inductive Definitions ordinals ψ Ω1 Ω ν cardinal analogue: ν-many regular cardinals

140 A Brief History of Ordinal Analysis cont d Buchholz, Pohlers, Sieg 1977 Theories of Iterated Inductive Definitions ordinals ψ Ω1 Ω ν cardinal analogue: ν-many regular cardinals Buchholz 1977 Ω ν+1 -rules

141 A Brief History of Ordinal Analysis cont d Buchholz, Pohlers, Sieg 1977 Theories of Iterated Inductive Definitions ordinals ψ Ω1 Ω ν cardinal analogue: ν-many regular cardinals Buchholz 1977 Ω ν+1 -rules Pohlers Method of Local Predicativity

142 A Brief History of Ordinal Analysis cont d Buchholz, Pohlers, Sieg 1977 Theories of Iterated Inductive Definitions ordinals ψ Ω1 Ω ν cardinal analogue: ν-many regular cardinals Buchholz 1977 Ω ν+1 -rules Pohlers Method of Local Predicativity Girard 1979 Π 1 2 -Logic

143 A Brief History of Ordinal Analysis cont d Buchholz, Pohlers, Sieg 1977 Theories of Iterated Inductive Definitions ordinals ψ Ω1 Ω ν cardinal analogue: ν-many regular cardinals Buchholz 1977 Ω ν+1 -rules Pohlers Method of Local Predicativity Girard 1979 Π 1 2 -Logic Jäger 1979 Constructible Hierarchy in Proof Theory

144 A Brief History of Ordinal Analysis cont d Jäger, Pohlers 1982 (Σ 1 2-AC) + BI, KPi ordinal ψ Ω1 I cardinal analogue: I inaccessible cardinal

145 A Brief History of Ordinal Analysis cont d Jäger, Pohlers 1982 (Σ 1 2-AC) + BI, KPi ordinal ψ Ω1 I cardinal analogue: I inaccessible cardinal R 1989 KPM ordinal ψ Ω1 M cardinal analogue: M Mahlo cardinal

146 A Brief History of Ordinal Analysis cont d Jäger, Pohlers 1982 (Σ 1 2-AC) + BI, KPi ordinal ψ Ω1 I cardinal analogue: I inaccessible cardinal R 1989 KPM ordinal ψ Ω1 M cardinal analogue: M Mahlo cardinal Buchholz 1990 Operator Controlled Derivations

147 A Brief History of Ordinal Analysis cont d R 1992 Π 3 -reflection ordinal ψ Ω1 K cardinal analogue: K weakly compact cardinal

148 A Brief History of Ordinal Analysis cont d R 1992 Π 3 -reflection ordinal ψ Ω1 K cardinal analogue: K weakly compact cardinal R 1992 First-order reflection cardinal analogue: totally indescribable cardinal

149 A Brief History of Ordinal Analysis cont d R 1992 Π 3 -reflection ordinal ψ Ω1 K cardinal analogue: K weakly compact cardinal R 1992 First-order reflection cardinal analogue: totally indescribable cardinal R 1995 Π 1 2 -Comprehension cardinal analogue: ω-many reducible cardinals

150 A Brief History of Ordinal Analysis cont d R 1992 Π 3 -reflection ordinal ψ Ω1 K cardinal analogue: K weakly compact cardinal R 1992 First-order reflection cardinal analogue: totally indescribable cardinal R 1995 Π 1 2 -Comprehension cardinal analogue: ω-many reducible cardinals Arai Ordinal Analysis of Theories up to Π 1 2 -Comprehension using Reductions on Finite Proof Figures and Ordinal Diagrams.