[PPT] - 2 Conservation in Proof Theory Michael Rathjen und Andreas Weiermann PowerPoint Presentation

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2 Conservation in Proof Theory

Michael Rathjen und Andreas Weiermann

Department of Pure Mathematics, University of Leeds Vakgroep Zuivere Wiskunde en Computeralgebra, Universiteit Gent

Festkolloquium anläßlich des 60. Geburtstages von Wilfried Buchholz München, 5. April, 2008

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How is it that infinitary proof theory gives rise to finitistic reductions?

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How is it that infinitary proof theory gives rise to finitistic reductions?

It suffices to use recursive proof trees.

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How is it that infinitary proof theory gives rise to finitistic reductions?

It suffices to use recursive proof trees.
Recursive proof trees can be encoded by natural numbers.

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How is it that infinitary proof theory gives rise to finitistic reductions?

It suffices to use recursive proof trees.
Recursive proof trees can be encoded by natural numbers.
E.g. Pohlers: Proof-theoretical analysis of IDν by the

method of local predicativity. In: W. Buchholz, S. Feferman,

W. Pohlers, W. Sieg: Iterated inductive definitions and

subsystems of analysis: Recent proof-theoretical studies (1981)

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Recursive proof trees

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Recursive proof trees

Provides a reduction of a theory T (e.g. T = ∆1

2-CA + BI)

to PA +

α<λ TI(α) where λ is a sufficiently large ordinal

(representation system).

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Recursive proof trees

Provides a reduction of a theory T (e.g. T = ∆1

2-CA + BI)

to PA +

α<λ TI(α) where λ is a sufficiently large ordinal

(representation system).

Drawback: Has probably never been sufficiently
formalized. Glossing over detail. A lot of handwaving.

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Primitive recursive proof trees

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Primitive recursive proof trees

Schwichtenberg: Some applications of cut-elimination. In:
J. Barwise (ed.): Handbook of Mathematical Logic (1977).

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Primitive recursive proof trees

Schwichtenberg: Some applications of cut-elimination. In:
J. Barwise (ed.): Handbook of Mathematical Logic (1977).

1

Primitive recursive proof trees suffice for PA +

α<λ TI(α).

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Primitive recursive proof trees

Schwichtenberg: Some applications of cut-elimination. In:
J. Barwise (ed.): Handbook of Mathematical Logic (1977).

1

Primitive recursive proof trees suffice for PA +

α<λ TI(α).

2

Uses the Primitive Recursion Theorem of Kleene 1958.

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Primitive recursive proof trees

Schwichtenberg: Some applications of cut-elimination. In:
J. Barwise (ed.): Handbook of Mathematical Logic (1977).

1

Primitive recursive proof trees suffice for PA +

α<λ TI(α).

2

Uses the Primitive Recursion Theorem of Kleene 1958.

3

Continuous cut-elimination: The repetition rule appears in the guise of improper instances of the ω-rule.

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Primitive recursive proof trees

Schwichtenberg: Some applications of cut-elimination. In:
J. Barwise (ed.): Handbook of Mathematical Logic (1977).

1

Primitive recursive proof trees suffice for PA +

α<λ TI(α).

2

Uses the Primitive Recursion Theorem of Kleene 1958.

3

Continuous cut-elimination: The repetition rule appears in the guise of improper instances of the ω-rule.

Drawback: Proofs inchoate. Hand waving exacerbated.

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Primitive recursive proof trees

Schwichtenberg: Some applications of cut-elimination. In:
J. Barwise (ed.): Handbook of Mathematical Logic (1977).

1

Primitive recursive proof trees suffice for PA +

α<λ TI(α).

2

Uses the Primitive Recursion Theorem of Kleene 1958.

3

Continuous cut-elimination: The repetition rule appears in the guise of improper instances of the ω-rule.

Drawback: Proofs inchoate. Hand waving exacerbated.
It works for PA +

α<λ TI(α). Would it work for

KPi = ∆1

2-CA + BI?

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Primitive recursive proof trees

Schwichtenberg: Some applications of cut-elimination. In:
J. Barwise (ed.): Handbook of Mathematical Logic (1977).

1

Primitive recursive proof trees suffice for PA +

α<λ TI(α).

2

Uses the Primitive Recursion Theorem of Kleene 1958.

3

Continuous cut-elimination: The repetition rule appears in the guise of improper instances of the ω-rule.

Drawback: Proofs inchoate. Hand waving exacerbated.
It works for PA +

α<λ TI(α). Would it work for

KPi = ∆1

2-CA + BI?

Does this machinery work for elementary in place of

primitive recursive functions?

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Complexity of Ordinal Representation Systems

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Complexity of Ordinal Representation Systems

Rick Sommer has investigated the question of complexity
f ordinal representation systems at great length. His case

studies revealed that with regard to complexity measures considered in complexity theory the complexity of ordinal representation systems involved in ordinal analyses is rather low. It appears that computations on ordinals in actual proof-theoretic ordinal analyses can be handled in the theory I∆0 + Ω1 where Ω1 is the assertion that the function x → xlog2(x) is total.

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Elementary descent recursion and proof theory

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Elementary descent recursion and proof theory

H. Friedman, M. Sheard: Elementary descent recursion

and proof theory (1995)

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Elementary descent recursion and proof theory

H. Friedman, M. Sheard: Elementary descent recursion

and proof theory (1995)

Proof theory of infinitary proof figures

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Elementary descent recursion and proof theory

H. Friedman, M. Sheard: Elementary descent recursion

and proof theory (1995)

Proof theory of infinitary proof figures
Provides a lot of details about the provably computable

functions of theories of the form PA +

α<λ TI(α).

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Elementary descent recursion and proof theory

H. Friedman, M. Sheard: Elementary descent recursion

and proof theory (1995)

Proof theory of infinitary proof figures
Provides a lot of details about the provably computable

functions of theories of the form PA +

α<λ TI(α).

Characterizes them as the λ-descent recursive

functions.

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Elementary descent recursion and proof theory

H. Friedman, M. Sheard: Elementary descent recursion

and proof theory (1995)

Proof theory of infinitary proof figures
Provides a lot of details about the provably computable

functions of theories of the form PA +

α<λ TI(α).

Characterizes them as the λ-descent recursive

functions.

Treatment is semi-rigorous.

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The Descent Computable Functions

Theorem (Friedman, Sheard) The provably computable functions of PA +

α<λ

TI(α) 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 and, for some α ∈ A, EA ⊢ ∀ xy h( x, y) ∈ Aα. The above class of functions is called the descent computable functions over Aλ.

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Main Step

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Main Step

Ordinal analysis reduces a theory T which is a subsystem
f second order arithmetic or set theory to

PA +

α<λ

TI(α), where λ is the proof-theoretic ordinal of T.

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Friedman’s question:

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 transfinite induction on a notation system, for Σ0

1

sentences, or even Π0

2 sentences.

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Friedman’s question:

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 transfinite induction on a notation system, for Σ0

1

sentences, or even Π0

2 sentences.

2

The conservative extension statement itself is a Π0

2

sentence. All I really need is that this Π0

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, 21000 symbols.

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Friedman’s question:

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 transfinite induction on a notation system, for Σ0

1

sentences, or even Π0

2 sentences.

2

The conservative extension statement itself is a Π0

2

sentence. All I really need is that this Π0

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, 21000 symbols.

3

From this, I can apply Friedman/Sheard with the old classical theory of infinitary derivations, linking to elementary recursive descent recursion, and no longer need your expertise.

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Friedman’s question:

1

21000 ≃ 10300 > ♯ atoms in the visible universe

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An intuitionistic fixed point theory

Wilfried Buchholz: An intuitionistic fixed point theory

Archive Math Logic (1997)

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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 ∧, ∨, ∀, ∃.

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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

i 1 is obtained from HA by adding for each strongly

positive operator form Φ(P, x) a new predicate symbol IΦ and the axiom (FPΦ) ∀x [Φ(IΦ, x) ↔ IΦ(x)]. Moreover, the induction schema is extended to the new language.

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Let CT0 be Church’s thesis, i.e. the schema ∀x∃y B(x, y) → ∃e∀x B(x, {e}(x)).

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Let CT0 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 AΦ(x) such that HA + CT0 ⊢ ∀x[Φ(AΦ, x) ↔ AΦ(x)].

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Theorem 2 ID

i 1 is conservative over HA w.r.t. almost

negative formulas.

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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 IΦ(t) by AΦ(t).

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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 IΦ(t) by AΦ(t).

1

ID

i 1 ⊢ B ⇒ HA + CT0 ⊢ B∗

(Theorem 1)

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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 IΦ(t) by AΦ(t).

1

ID

i 1 ⊢ B ⇒ HA + CT0 ⊢ B∗

(Theorem 1)

2

HA + CT0 ⊢ C ⇒ HA ⊢ ∃e (erC).

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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 IΦ(t) by AΦ(t).

1

ID

i 1 ⊢ B ⇒ HA + CT0 ⊢ B∗

(Theorem 1)

2

HA + CT0 ⊢ C ⇒ HA ⊢ ∃e (erC).

3

HA ⊢ ∃e (erC) → C for almost negative C.

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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:

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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 (Γi)i∈I such that for every i ∈ I there exists βi < α with D(βi, ρ, Γi), and if the inference is a cut it has rank < ρ.

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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 (Γi)i∈I such that for every i ∈ I there exists βi < α with D(βi, ρ, Γi), 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.

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The strength of Kruskal’s theorem

M. Rathjen, A. Weiermann: Proof–theoretic investigations
n Kruskal’s theorem (1993)

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The strength of Kruskal’s theorem

M. Rathjen, A. Weiermann: Proof–theoretic investigations
n Kruskal’s theorem (1993)

Theorem 1. The proof-theoretic ordinal ordinal of Π1

2 − BI0

and Π1

2 − BI− 0 is the Ackermann ordinal θΩω0.

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The strength of Kruskal’s theorem

M. Rathjen, A. Weiermann: Proof–theoretic investigations
n Kruskal’s theorem (1993)

Theorem 1. The proof-theoretic ordinal ordinal of Π1

2 − BI0

and Π1

2 − BI− 0 is the Ackermann ordinal θΩω0.

Theorem 2. For every n, Π1

2 − BI− 0 proves KTn, i.e.

Kruskal’s theorem for finite at most n branching trees. Π1

2 − BI0 ⊢ ∀x KTx.

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The strength of Kruskal’s theorem

M. Rathjen, A. Weiermann: Proof–theoretic investigations
n Kruskal’s theorem (1993)

Theorem 1. The proof-theoretic ordinal ordinal of Π1

2 − BI0

and Π1

2 − BI− 0 is the Ackermann ordinal θΩω0.

Theorem 2. For every n, Π1

2 − BI− 0 proves KTn, i.e.

Kruskal’s theorem for finite at most n branching trees. Π1

2 − BI0 ⊢ ∀x KTx.

Theorem 3. ACA0 proves that Kruskal’s theorem is equivalent to the uniform Π1

1-reflection for Π1 2 − BI0.

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Buchholz’s Ω-rule

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

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Buchholz’s Ω-rule

Ω-rule particularly suited to deal with Bar induction, BI.
BI is equivalent (over RCA0 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).

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Buchholz’s Ω-rule

Ω-rule particularly suited to deal with Bar induction, BI.
BI is equivalent (over RCA0 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.

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Buchholz’s Ω-rule

Ω-rule particularly suited to deal with Bar induction, BI.
BI is equivalent (over RCA0 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)?

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Buchholz’s Ω-rule

Ω-rule particularly suited to deal with Bar induction, BI.
BI is equivalent (over RCA0 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.

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Buchholz’s Ω-rule cont’d

Weak formulas are formulas that are arithmetic or Π1

1.

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Buchholz’s Ω-rule cont’d

Weak formulas are formulas that are arithmetic or Π1

1.

Inductive definition of T ∗

α ̺ Γ for α ∈ OT(ψ) and

̺ < ω + ω.

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Buchholz’s Ω-rule cont’d

Weak formulas are formulas that are arithmetic or Π1

1.

Inductive definition of T ∗

α ̺ Γ for α ∈ OT(ψ) and

̺ < ω + ω.

(Ω-rule). Let f be a fundamental function satisfying

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Buchholz’s Ω-rule cont’d

Weak formulas are formulas that are arithmetic or Π1

1.

Inductive definition of T ∗

α ̺ Γ for α ∈ OT(ψ) and

̺ < ω + ω.

(Ω-rule). Let f be a fundamental function satisfying

1

Ω ∈ dom(f) and f(Ω) ✂ α,

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Buchholz’s Ω-rule cont’d

Weak formulas are formulas that are arithmetic or Π1

1.

Inductive definition of T ∗

α ̺ Γ for α ∈ OT(ψ) and

̺ < ω + ω.

(Ω-rule). Let f be a fundamental function satisfying

1

Ω ∈ dom(f) and f(Ω) ✂ α,

2

T ∗

f(0) ̺

Γ, ∀XF(X) , where ∀XF(X) ∈ Π1

1, and

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Buchholz’s Ω-rule cont’d

Weak formulas are formulas that are arithmetic or Π1

1.

Inductive definition of T ∗

α ̺ Γ for α ∈ OT(ψ) and

̺ < ω + ω.

(Ω-rule). Let f be a fundamental function satisfying

1

Ω ∈ dom(f) and f(Ω) ✂ α,

2

T ∗

f(0) ̺

Γ, ∀XF(X) , where ∀XF(X) ∈ Π1

1, and

3

T ∗

β 0 Ξ, ∀XF(X) implies T ∗ f(β) ̺

Ξ, Γ for every set of weak formulas Ξ and β < Ω.

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Buchholz’s Ω-rule cont’d

Weak formulas are formulas that are arithmetic or Π1

1.

Inductive definition of T ∗

α ̺ Γ for α ∈ OT(ψ) and

̺ < ω + ω.

(Ω-rule). Let f be a fundamental function satisfying

1

Ω ∈ dom(f) and f(Ω) ✂ α,

2

T ∗

f(0) ̺

Γ, ∀XF(X) , where ∀XF(X) ∈ Π1

1, and

3

T ∗

β 0 Ξ, ∀XF(X) implies T ∗ f(β) ̺

Ξ, Γ for every set of weak formulas Ξ and β < Ω.

Then T ∗

α ̺ Γ holds.

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The other rules

1

If A is a true constant prime formula or negated prime formula and A ∈ Γ, then T ∗

α ̺ Γ.

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The other rules

1

If A is a true constant prime formula or negated prime formula and A ∈ Γ, then T ∗

α ̺ Γ.

2

If Γ contains formulas A(s1, . . . , sn) and ¬A(t1, . . . , tn) of grade 0 or ω, where si and ti (1 ≤ i ≤ n) are equivalent terms, then T ∗

α ̺ Γ.

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The other rules

1

If A is a true constant prime formula or negated prime formula and A ∈ Γ, then T ∗

α ̺ Γ.

2

If Γ contains formulas A(s1, . . . , sn) and ¬A(t1, . . . , tn) of grade 0 or ω, where si and ti (1 ≤ i ≤ n) are equivalent terms, then T ∗

α ̺ Γ.

3

If T ∗

β ̺ Γi and β ✁ α hold for every premiss Γi of an

inference (∧), (∨), (∃1), (∀2) or (Cut) with a cut formula having grade < ̺, and conclusion Γ, then T ∗

α ̺ Γ.

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The other rules

1

If A is a true constant prime formula or negated prime formula and A ∈ Γ, then T ∗

α ̺ Γ.

2

If Γ contains formulas A(s1, . . . , sn) and ¬A(t1, . . . , tn) of grade 0 or ω, where si and ti (1 ≤ i ≤ n) are equivalent terms, then T ∗

α ̺ Γ.

3

If T ∗

β ̺ Γi and β ✁ α hold for every premiss Γi of an

inference (∧), (∨), (∃1), (∀2) or (Cut) with a cut formula having grade < ̺, and conclusion Γ, then T ∗

α ̺ Γ.

4

If T ∗

α0 ̺ Γ, F(U) holds for some α0 ✁ α and a

non-arithmetic formula F(U), then T ∗

α ̺ Γ, ∃XF(X) .

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SLIDE 65

Metatheory for the Ω-rule?

The derivability notion T ∗

α ̺ Γ seems to require an iterated

inductive definition.

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SLIDE 66

Metatheory for the Ω-rule?

The derivability notion T ∗

α ̺ Γ seems to require an iterated

inductive definition.

1

First inductively defined set, T∞: Infinitary (cut-free) proofs without Ω-rule.

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SLIDE 67

Metatheory for the Ω-rule?

The derivability notion T ∗

α ̺ Γ seems to require an iterated

inductive definition.

1

First inductively defined set, T∞: Infinitary (cut-free) proofs without Ω-rule.

2

Inductive definition of T ∗

α ̺ Γ involves T∞ negatively.

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SLIDE 68

Metatheory for the Ω-rule?

The derivability notion T ∗

α ̺ Γ seems to require an iterated

inductive definition.

1

First inductively defined set, T∞: Infinitary (cut-free) proofs without Ω-rule.

2

Inductive definition of T ∗

α ̺ Γ involves T∞ negatively.

Buchholz’s result can be extended to finitely iterated

inductive definitions.

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SLIDE 69

Metatheory for the Ω-rule?

The derivability notion T ∗

α ̺ Γ seems to require an iterated

inductive definition.

1

First inductively defined set, T∞: Infinitary (cut-free) proofs without Ω-rule.

2

Inductive definition of T ∗

α ̺ 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).

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SLIDE 70

Metatheory for the Ω-rule?

The derivability notion T ∗

α ̺ Γ seems to require an iterated

inductive definition.

1

First inductively defined set, T∞: Infinitary (cut-free) proofs without Ω-rule.

2

Inductive definition of T ∗

α ̺ 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

i n(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

ver HA by Goodman’s theorem.

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Metatheory for the Ω-rule cont’d

Two Drawbacks: (1) The employment of Goodman’s theorem makes the proof less explicit. Hard to say how long a fully formalized proof would be.

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SLIDE 72

Metatheory for the Ω-rule cont’d

Two Drawbacks: (1) The employment of Goodman’s theorem makes the proof less explicit. Hard to say how long a fully formalized proof would be. (2) ID

i n is formulated for strongly positive operator forms.

But the iterated inductive definition of T ∗

α ̺ Γ seems to

require a strictly positive iterated inductive definition.

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Metatheory for the Ω-rule cont’d

Two Drawbacks: (1) The employment of Goodman’s theorem makes the proof less explicit. Hard to say how long a fully formalized proof would be. (2) ID

i n is formulated for strongly positive operator forms.

But the iterated inductive definition of T ∗

α ̺ Γ seems to

require a strictly positive iterated inductive definition.

The strictly positive (with respect to P) formulas of

L1(P, Q) formulas are closed under the following clause: If A is an L1(Q) formula and B is strictly positive, then A → B is strictly positive.

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Metatheory for the Ω-rule cont’d

C. Rüede, T. Strahm: Intuitionistic fixed point theories for

strictly positive operators. (MLQ 2002)

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SLIDE 75

Metatheory for the Ω-rule cont’d

C. Rüede, T. Strahm: Intuitionistic fixed point theories for

strictly positive operators. (MLQ 2002)

Theorem.

ID

i n(strict) is conservative over HA w.r.t. Π0 2

sentences.

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SLIDE 76

Metatheory for the Ω-rule cont’d

C. Rüede, T. Strahm: Intuitionistic fixed point theories for

strictly positive operators. (MLQ 2002)

Theorem.

ID

i n(strict) is conservative over HA w.r.t. Π0 2

sentences. Proof uses a realizability interpretation of ID

i n(strict) into

ID

i n(acc) (preserves almost negative formulas).

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SLIDE 77

Metatheory for the Ω-rule cont’d

C. Rüede, T. Strahm: Intuitionistic fixed point theories for

strictly positive operators. (MLQ 2002)

Theorem.

ID

i n(strict) is conservative over HA w.r.t. Π0 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.

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SLIDE 78

Metatheory for the Ω-rule cont’d

C. Rüede, T. Strahm: Intuitionistic fixed point theories for

strictly positive operators. (MLQ 2002)

Theorem.

ID

i n(strict) is conservative over HA w.r.t. Π0 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 L1(Q).

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Friedman’s impractical matters

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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)

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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

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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.

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SLIDE 83

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 T0, . . . , Tk of trees in Q with |Ti| ≤ c · (i + 1) for all i ≤ k, there exist indices i < j ≤ k such that Ti is homoemorphically embeddable into Tj.

Let ΨQ(c) be the smallest such k.

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Impractical matters cont’d

Theorem. SWQ(Q) is provable in Π1

2-BI but not in Π1 2-BI0.

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Impractical matters cont’d

Theorem. SWQ(Q) is provable in Π1

2-BI but not in Π1 2-BI0.

When c is specialized we obtain a Σ0

1 statement SWQc(Q).

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Impractical matters cont’d

Theorem. SWQ(Q) is provable in Π1

2-BI but not in Π1 2-BI0.

When c is specialized we obtain a Σ0

1 statement SWQc(Q).

Theorem. SWQ1(Q) is provable in Π1

2-BI0 (of course!) but

any proof requires at least 2[900] symbols. 2[0] := 1, 2[n+1] := 22[n].

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Impractical matters cont’d

Theorem. SWQ(Q) is provable in Π1

2-BI but not in Π1 2-BI0.

When c is specialized we obtain a Σ0

1 statement SWQc(Q).

Theorem. SWQ1(Q) is provable in Π1

2-BI0 (of course!) but

any proof requires at least 2[900] symbols. 2[0] := 1, 2[n+1] := 22[n]. Why?

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Impractical matters cont’d

For a theory T define

χT (n) to be the least integer k such that if ∃x A(x) is any Σ0

1

statement provable in T using ≤ n symbols, then A(m) is true for some m ≤ k.

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Impractical matters cont’d

For a theory T define

χT (n) to be the least integer k such that if ∃x A(x) is any Σ0

1

statement provable in T using ≤ n symbols, then A(m) is true for some m ≤ k.

Let

χλ(n) be the least integer k such that if ∃x A(x) is any Σ0

1

statement provable in PA +

α<λ

TI(α) using ≤ n symbols, then A(m) is true for some m ≤ k.

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Impractical matters finale

1

Theorem. The proof in [RW] of the 1-consistency of

T := Π1

2-BI0 in PA + TI(θΩω0) can be carried out by a proof

shorter than 21000 symbols.

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Impractical matters finale

1

Theorem. The proof in [RW] of the 1-consistency of

T := Π1

2-BI0 in PA + TI(θΩω0) can be carried out by a proof

shorter than 21000 symbols.

2

Corollary. χT(n) ≤ χθΩω0(22(n+1)·1000)

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Impractical matters finale

1

Theorem. The proof in [RW] of the 1-consistency of

T := Π1

2-BI0 in PA + TI(θΩω0) can be carried out by a proof

shorter than 21000 symbols.

2

Corollary. χT(n) ≤ χθΩω0(22(n+1)·1000)

3

(Friedman: in [Smith] Lemma 22) χθΩω0(2[1000]) ≤ ΨQ(1).

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Impractical matters finale

1

Theorem. The proof in [RW] of the 1-consistency of

T := Π1

2-BI0 in PA + TI(θΩω0) can be carried out by a proof

shorter than 21000 symbols.

2

Corollary. χT(n) ≤ χθΩω0(22(n+1)·1000)

3

(Friedman: in [Smith] Lemma 22) χθΩω0(2[1000]) ≤ ΨQ(1).

4

χT(n) ≤ χθΩω0(22(n+1)·1000)

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Impractical matters finale

1

Theorem. The proof in [RW] of the 1-consistency of

T := Π1

2-BI0 in PA + TI(θΩω0) can be carried out by a proof

shorter than 21000 symbols.

2

Corollary. χT(n) ≤ χθΩω0(22(n+1)·1000)

3

(Friedman: in [Smith] Lemma 22) χθΩω0(2[1000]) ≤ ΨQ(1).

4

χT(n) ≤ χθΩω0(22(n+1)·1000)

5

χT(2[900]) ≤ χθΩω0(2[1000]) ≤ ΨQ(1)

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The End

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The End

Thank you!

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