Local versions of the Kreisel–Levy theorem

A. Cordon–Franco,A. Fernandez–Margarit

F. F. Lara–Martın

University of Seville (Spain)

29eme Journees sur les Arithmetiques FaiblesWarsaw, 2010

JAF29 On Kreisel-Levy Theorem 1/26

Outline

Reflection PrinciplesUniform and Local ReflectionKreisel–Levy Theorem

Local InductionInduction up to Σn–definable elementsThe main result

Final remarksRelativizationLocal versions of the Kreisel–Levy Theorem

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Reflection principles

I Our basic theory is EA (Elementary Arithmetic) with languageLexp = {0, S ,+, · , exp, <}

I For each theory T , elementary presented, we considerformulas

I PrfT (y , x) expresing “y is (codes) a proof of x in T”I ProvT (x) ≡ ∃y PrfT (y , x)

I Local Reflection for T is the following scheme, Rfn(T ),

ProvT (pϕq)→ ϕ

for each sentence ϕ.

I Uniform Reflection for T is the following scheme, RFN(T ),

∀x1 . . . ∀xn (ProvT (pϕ(x1, . . . , xn)q→ ϕ(x1, . . . , xn))

for each formula ϕ(x1, . . . , xn).

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Partial Reflection

Partial Reflection: Reflection scheme restricted to a class offormulas Σ.

I Partial Local Reflection, RfnΣ(T ) is given by

ProvT (pϕq)→ ϕ

for every ϕ ∈ Σ ∩ Sent

I Partial Uniform Reflection, RFNΣ(T ) is given by

∀x1 . . . ∀xn (ProvT (pϕ(x1, . . . , xn)q)→ ϕ(x1, . . . , xn))

for all ϕ(~x) ∈ Σ.

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Uniform and Local Reflection

Let T be an elementary presented extension of EA. Here RFNΓ

denotes T + RFNΓ(T ) and RfnΓ denotes T + RfnΓ(T ).

...

RFNΠ3 ≡ RFNΣ2

RFNΠ2 ≡ RFNΣ1

?RfnΣ2

RfnΣ1 ≡ RfnB(Σ1)

�

Σ2-

RfnB(Σ2)

�

· · ·

RFNΠ1 ≡ RfnΠ1

?�RfnΠ2

�

�

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Basic properties of Partial Reflection

Unboundedness (Kreisel–Levy)

I RfnΠn(T ) is not contained in any consistent extension of T bya finite set of Σn formulas.

I RFNΠn(T ) is not contained in any consistent extension of Tby a set of Σn formulas.

I The duals of these results for Σn also hold.

Conservation (Beklemishev)

Let Γ = Σn or Πn with n ≥ 2 or Γ = B(Σk), with k ≥ 1, then

I T + Rfn(T ) is Γ–conservative over T + RfnΓ(T ).

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Conservation for Local Reflection

RfnΣ2 RfnΣ3

RfnΣ1 ≡ RfnB(Σ1)

�B(Σ

1)

RfnB(Σ2)

�B(Σ

2)

�

Σ2

RfnB(Σ3)

�

Σ3

· · ·

RfnΠ1

?RfnΠ2

�Π 2

�

Σ2

RfnΠ3

�Π 3

�B

(Σ2 )

Problem (Beklemishev): Is T + RfnΣ2(T ) a Π2–conservativeextension of T + RfnΣ1(T )?

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Kreisel–Levy Theorem and refinements

Theorem (Kreisel–Levy)

PA ≡ EA + RFN(EA)

Theorem (Leivant-Ono)

(n ≥ 1)IΣn ≡ EA + RFNΣn+1(EA)

Theorem (Beklemishev)

1. Over EA, RfnΣ2(EA) implies IΠ−1 .

2. Over EA+, IΠ−1 ≡ RfnΣ2(EA).

3. EA+ + IΠ−1 ≡ EA+ + RfnΣ2(EA) ≡ EA+ + RfnΣ2(EA+).

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Main result

TheoremEA + RfnΣ2(EA) is not a Π2–conservative extension ofEA + RfnΣ1(EA)

I Since EA + RfnΣ2(EA) extends EA + IΠ−1 , it is enough toshow that there exists a Π2–sentence ψ such that

I EA + IΠ−1 ` ψ, butI EA + RfnΣ1 (EA) 6` ψ

I Proof strategy: characterize the class of Π2–consequences ofEA + IΠ−1 .

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Definable and minimal elements

I a is Γ–definable in A (with parameters b) if there isϕ(x , v) ∈ Γ such that A |= ϕ(a, b) ∧ ∃!x ϕ(x , b).

I a is Γ–minimal in A (with parameters b) if there isϕ(x , v) ∈ Γ such that A |= a = (µx) (ϕ(x , b)).

I K1(A) = Σ1–definable elements of A.

I I1(A) = initial segment determined by K1(A)

[——–)ω——–)I1——————–)

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Iterating Σ1–definability: I∞1

I01 (A) = I1(A)

Ik+11 (A) = initial segment determined by K1(A, Ik

1 (A))

I∞1 (A) =⋃k≥0

Ik1 (A)

[———–)I01——)I1

1—— )I2

1————–)I∞1

—————-)

I I∞1 (A) is the least initial segment of A containing all the

Σ1–definable elements and closed under Σ1–definability.

I If A |= IΣ1 with nonstandard Σ1–definable elements,

{Ik1 (A) : k ≥ 0} form a proper hierarchy.

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Induction up to Σ1–definable elements

I We denote by I(Σ1,K1) the theory given by the inductionscheme

ϕ(0) ∧ ∀x (ϕ(x)→ ϕ(x + 1))→∀x1, x2 (δ(x1) ∧ δ(x2)→ x1 = x2)→ ∀x (δ(x)→ ϕ(x))

where ϕ(x) ∈ Σ1 and δ(x) ∈ Σ−1 .I We consider an inference rule associated with this scheme:

I (Σ1,K1)–IR denotes the following inference rule:

ϕ(0) ∧ ∀x (ϕ(x)→ ϕ(x + 1))

∀x1, x2 (δ(x1) ∧ δ(x2)→ x1 = x2)→ ∀x (δ(x)→ ϕ(x))

where δ(x) ∈ Σ−1 and ϕ(x) ∈ Σ1.

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Some basic facts on I(Σ1,K1)

LemmaLet T be an Π2-axiomatizable extension of EA. ThenT + I(Σ1,K1) is Π2–conservative over T + (Σ1,K1)–IR.

I So, a version of Parsons’s theorem holds for Σ1–induction upto Σ1–definable elements.

I The parameter free version of I(Σ1,K1) is equivalent to IΠ−1 .

I EA + (Σ1,K1)–IR provides an upper bound for the class ofΠ2–consequences of EA + IΠ−1 .

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Characterizing ThΠ2(EA + I(Σ1,K1))

Proposition

The following theories are equivalent:

1. EA + (Σ1,K1)–IR.

2. [EA, (Σ1,K1)–IR].

3. EA + ∀x ∀u ∈ K1 ∃y (2xu = y)

Here 2xu denotes the iteration of the exponential function

exp(x) = 2x .

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IΠ−1 and Restricted Σ1–induction

Proposition

Over I∆0 the following are equivalent:

1. A |= IΠ−1 .

2. For each ϕ(x , v) ∈ Σ1 and a, b ∈ I1(A):

A |= ϕ(0, b) ∧ ∀x (ϕ(x , b)→ ϕ(x + 1, b)) → ϕ(a, b)

Proof: Pick c ∈ K1(A) with a, b ≤ c .

Step 1: IΠ−1 ` minimization for each Π1 formula satisfiable inK1(A).

Step 2: Show that ϕ(x , b) ≡ x ∈ d , for all x ≤ c.

Step 3: Apply ∆0–induction to x ∈ d .

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IΠ−1 and Restricted Exponentiation

[———)I1——–)I1

1——– )I2

1—————-)I∞1

—————-)

TheoremEA + IΠ−1 ` ∀a ∈ I1(A) ∀b ∈ I1

1 (A) ∃y (y = 2ba)

Proof:Pick c ∈ K1(A, I1(A)) with b ≤ c . If δ(u, d) ∈ Σ1 defines c with

d ∈ I1(A), apply Σ1–induction up to a to:

∃y (y = 2cx) ≡ ∃y , u (δ(u, d) ∧ y = 2u

x )

Then, 2ca exists and so does 2b

a since b ≤ c .

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Expressing ”∀x ∈ I1(A)” in the language ofArithmetic

I For each a ∈ K1(A) there is b Π0–minimal such that a = (b)0.

“∀x ∈ I1(A) Φ(x , v)”

m

{∀z , x (

{z = (µt) (δ(t))

∧ x ≤ (z)0

}→ Φ(x , v) ) : δ(t) ∈ Π−0 }

I “∀a ∈ I1(A) ∀b ∈ I11 (A) ∃y (y = 2b

a)” can be reexpressed as aset of Π2 sentences in the usual language of Arithmetic.

I So, ThΠ2(EA + IΠ−1 ) ` ∀a ∈ I1(A) ∀b ∈ I11 (A) ∃y (y = 2b

a).

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Separating ThΠ2(EA + IΠ−1 ) and EA + RfnΣ1

(EA)

A refinement of Bigorajska’s Theorem holds for every extension ofEA by a set of B(Σ1) sentences.

Proposition

Let Γ a set of B(Σ1) sentences and let A |= EA + Γ. Letϕ(x , y) ∈ Σ1 such that

EA + Γ ` ∀x ∃y ϕ(x , y)

Then, there exist a ∈ I1(A) and k ∈ ω such that

A |= ∀x > a ∃y ≤ 2xk ϕ(x , y)

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Separating ThΠ2(EA + IΠ−1 ) and EA + RfnΣ1

(EA)

TheoremEA + RfnΣ1(EA) 6` ThΠ2(EA + IΠ−1 )

Proof:

I Given a, c Π0–minimal and δ(z , v) ∈ Π0:

g(x) =

2

(x)3

(x)0if

(x)0 = a ∧ (x)1 = c ∧ (x)2 ≤ (x)1

(x)3 = (µz) (δ(z , (x)2))

0 otherwise

I ThΠ2(EA + IΠ−1 ) ` ∀x ∃y (y = g(x)).Since (x)3 ∈ I1

1 (A) and (x)0 ∈ I1(A).

I EA + RfnΣ1(EA) 6` ∀x ∃y (y = g(x)).Otherwise, it would follow from the strong version of Bigorajska’s

Theorem that 2ba ≤ 2b

k+1 for some b ∈ I11 (A)− I1(A) and k ∈ ω.

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Extensions of the main result

Some interesting directions for further work:

I Generalization of the main result to every finiteΠ2–axiomatizable extension of EA.

I Extensions for IΠ−n+1, n > 1. This involves relativization.

I Extensions for RfnΣn with n > 2.

I Versions of Kreisel–Levy theorem for Local Induction. Anappropiate reflection principle must be isolated.

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Relativized reflection principles

I The natural generalization of Beklemishev result for IΠ−1 andRfnΣ2(EA) to IΠ−n and RfnΣn+1(EA) does not hold.

I A stronger reflection scheme is needed. For each n ≥ 1 andeach theory T , elementary presented, we consider the formulas

I PrfnT (y , x) expressing

“y is (codes) a proof of x in T + ThΠn(N )”

I ProvnT (x) ≡ ∃y PrfnT (y , x)

I Relativized Local Reflection for T is the scheme, Rfnn(T ),

ProvnT (pϕq)→ ϕ

for each sentence ϕ.

I Relativized Uniform Reflection for T , RFNn(T ) is defined in asimilar way.

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Relativized reflection principles

RFNΠ3 ≡ RFNΣ2 Rfn1Σ3

Rfn1Σ2≡ Rfn1

B(Σ2)

� B(Σ

2)Σ

3 -

RfnB(Σ3)

�Σ

3

· · ·

Rfn1Π2

?

Rfn1Π3

�Π 3

�Σ

3

Theorem (Beklemishev)

1. EA + IΣ−n ≡ EA + RfnnΣn+1

(EA)

2. EA + IΠ−n+1 ≡ EA + RfnnΣn+2

(EA)

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Proof of K–L Theorem

(⇐) Given σ(x , y) ∈ Σn, let ψ(y) be the formula

σ(0, y) ∧ ∀x (σ(x , y)→ σ(x + 1, y))

Then ψ(y)→ σ(x , y) is a Σn+1 formula and

EA ` ∀x , y ProvEA(ψ(y)→ σ(x , y))

By Uniform reflection,

EA + RFNΣn+1(EA) ` ∀x , y (ψ(y)→ σ(x , y)),

as required.

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Proof of K–L Theorem (continued)

(⇒) First we observe that

EA ` RFNΣn+1(EA)↔ RFNΣn+1(PC)

where PC denotes the pure predicate calculus in the language offirst order arithmetic (without exponentiation).

I Let σ(z) ∈ Σn+1 such that ProvPC(σ(z)) and assume that¬σ(a). Then considering a Tait sequent calculus, there existsa cut–free proof of the sequent ¬σ(a).

I Using a partial truth predicate, TrΠn(ψ), we show by inductionover the height of a cut–free derivation, that

¬σ(a)→ TrΠn(¬σ(a))

A contradiction.

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Restricted Reflection

A restricted form of reflection can be informally defined as follows.

I Consider a model A and two subsets of A, I and J.

I For a theory T , elementary presented, and each formula ϕ(x)we consider a reflection principle RFNI ,J

ϕ (T ) expressing that:

I “For every element a ∈ J if there exists a proof of ϕ(a) in Twith length bounded by an element in I then ϕ(a) holds.”

I If I and J are of the form Ik1 then RFNI ,J

ϕ (T ) can beexpressed in the language of first order arithmetic.

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References

Beklemishev, L. D., “Reflection principles and provabilityalgebras in formal arthimetic”, Logic Group Preprint Series236, University of Utrecht, 2005.http://www.phil.uu.nl/preprints/lgps/

Kreisel, G., Levy, A., “Reflection Principles and Their Use forEstablishing the Complexity of Axiomatic Systems”, Zeitschriftfur Mathematische Logik und Grundlagen der Mathematik,14:97–142, 1968.

Leivant, D., “The optimality of induction as an axiomatizationof arithmetic”, The Journal of Symbolic Logic, 48:182–184,1983.

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