Post on 30-Mar-2021
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
JAF29 On Kreisel-Levy Theorem 2/26
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).
JAF29 On Kreisel-Levy Theorem 3/26
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.
JAF29 On Kreisel-Levy Theorem 4/26
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
�
�
JAF29 On Kreisel-Levy Theorem 5/26
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 ).
JAF29 On Kreisel-Levy Theorem 6/26
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 )?
JAF29 On Kreisel-Levy Theorem 7/26
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+).
JAF29 On Kreisel-Levy Theorem 8/26
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 .
JAF29 On Kreisel-Levy Theorem 9/26
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——————–)
JAF29 On Kreisel-Levy Theorem 10/26
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.
JAF29 On Kreisel-Levy Theorem 11/26
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.
JAF29 On Kreisel-Levy Theorem 12/26
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 .
JAF29 On Kreisel-Levy Theorem 13/26
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 .
JAF29 On Kreisel-Levy Theorem 14/26
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 .
JAF29 On Kreisel-Levy Theorem 15/26
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 .
JAF29 On Kreisel-Levy Theorem 16/26
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).
JAF29 On Kreisel-Levy Theorem 17/26
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)
JAF29 On Kreisel-Levy Theorem 18/26
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.
JAF29 On Kreisel-Levy Theorem 19/26
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.
JAF29 On Kreisel-Levy Theorem 20/26
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.
JAF29 On Kreisel-Levy Theorem 21/26
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)
JAF29 On Kreisel-Levy Theorem 22/26
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.
JAF29 On Kreisel-Levy Theorem 23/26
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.
JAF29 On Kreisel-Levy Theorem 24/26
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.
JAF29 On Kreisel-Levy Theorem 25/26
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.
JAF29 On Kreisel-Levy Theorem 26/26