David Fernandez-Duque´ Joint work with Andres Cord´ on ...gmferreira... · David Fernandez-Duque´ Joint work with Andres Cord´ on-Franco, Joost J. Joosten and´ Francisco Felix

From weak to strong arithmetics via reflection

David Fernández-DuqueJoint work with Andrés Cordón-Franco, Joost J. Joosten and

Francisco Félix Lara

International Centre for Mathematics and Computer Science in Toulouse

Journées sur les Arithmétiques Faibles 2016Lisbon, Portugal

Plan of the presentation

I Review uniform reflection principles in first-orderarithmetic, and discuss how PA can be represented asuniform reflection over EA.

I Present reflection principles in second-order arithmetic andshow how ACA0 can also be represented as reflection overRCA0.

I Show how infinitary reflection principles may also be usedto represent more of the ‘Big Five’ systems of reversemathematics.

I Discuss to what extent RCA0 could be replaced by weakertheories.

First-order arithmeticL1 = Π0ω denotes the language of first-order arithmetic over thesignature

〈0,1,+,×〉

I ∆00 formulas: all quantifiers are of the form ∃x < t or∀x < t .

I Σ0n : ∃xn∀xn−1 . . . δ(x1, . . . , xn)I Π0n : ∀xn∃xn−1 . . . δ(x1, . . . , xn)

We will fix a Gödel numbering ϕ 7→ pϕq and define numerals

n̄ = 1 + . . .+ 1︸︷︷︸n

.

We assume all theories are elementarily presented:

T ` ϕ ⇔ ∃x Proof T (x , pϕq)︸︷︷︸∈∆00(exp)

.

Notation

I �Tϕ := ∃x Proof T (x , pϕq)

I ♦Tϕ := ¬�T¬ϕ

I > := 0 = 0

I ⊥ := ¬>

I Cons[T ] := ♦T> ∈ Π01

Some first-order theories of arithmeticInduction schema:

Iϕ = ϕ(0) ∧ ∀x(ϕ(x)→ ϕ(x + 1)

)→ ∀xϕ(x).

IΓ = {Iϕ : ϕ ∈ Γ}.

I Robinson’s Q: Includes axioms for +,× but no induction.

I Kalmár elementary arithmetic:

EA := Q + I∆00 + exp

I Peano arithmetic:PA := Q + IΠ0ω

Arithmetical reflection principlesStatements of the form

“If ϕ is provable in T then ϕ is true.”

Formally,�Tϕ→ ϕ.

I If ϕ is a sentence, this is an instance of local reflection.I Uniform reflection generalizes this to formulas ϕ = ϕ(x):

RFNϕ[T ] = ∀x(�Tϕ(x̄)→ ϕ(x)

).

If Γ is a set of formulas,

RFNΓ[T ] := {RFNϕ[T ] : ϕ ∈ Γ}.

Extending theories by reflection

Löb’s rule: T only proves its reflection instances when wealready have that T ` ϕ:

T ` �Tϕ→ ϕT ` ϕ

.

This generalizes Gödel’s second incompleteness theorem ifϕ = ⊥ :

T ` ♦T>T ` ⊥

.

Extending theories by reflection

Löb’s rule: T only proves its reflection instances when wealready have that T ` ϕ:

T ` �Tϕ→ ϕT ` ϕ

.

This generalizes Gödel’s second incompleteness theorem ifϕ = ⊥ :

T ` ♦T>T ` ⊥

.

Arithmetic through reflection

Theorem (Kreisel and Levy)PA ≡ EA + RFN[EA].

More specifically:

TheoremFor all n ≥ 1, IΣn ≡ EA + RFNΣn+1 [EA].

This was proven for n ≥ 2 by Leviant using primitive recursivearithmetic instead of EA, and extended to n = 1 byBeklemishev.

Arithmetic through reflection

Theorem (Kreisel and Levy)PA ≡ EA + RFN[EA].

More specifically:

TheoremFor all n ≥ 1, IΣn ≡ EA + RFNΣn+1 [EA].

This was proven for n ≥ 2 by Leviant using primitive recursivearithmetic instead of EA, and extended to n = 1 byBeklemishev.

Strong extensions of T

We may obtain stronger reflection principles by passing topossibly non-computable extensions of T .

For n ∈ N, define [n]Tϕ if and only if ϕ is provable from T usingan oracle for Π0n sentences.

Formally,

[n]Tϕ ≡ ∃ψ(TrueΠ0n (ψ) ∧�T (ψ → ϕ)

)∈ Σ0n+1.

Reflection and n-consistency

We may then consider principles of the form [n]Tϕ→ ϕ, orsimply 〈n〉Tϕ := ¬[n]T¬ϕ:

LemmaFor all n ∈ N,

EA ` 〈n〉T> ↔ RFNΣ0n [T ].

CorollaryPA ≡ EA + {〈n〉EA> : n < ω}.

Second-order arithmetic

Language: Add to the first-order arithmetical language:

I set-variables X ,Y ,Z , . . .

I new atomic formulas t ∈ X

I second-order quantifiers ∀X , ∃X

Π1n, Σ1n formulas have n alternating second-order quantifiers.

Basic second-order axioms

Induction axiom (Ind):

∀X(

0 ∈ X ∧ ∀n (n ∈ X → (n + 1) ∈ X )→ ∀n (n ∈ X ))

Comprehension axioms: State the existence of sets of the form

{n ∈ N : ϕ(n)}.

CA(Γ): comprehension for ϕ ∈ Γ.

CA(∆01) : for π ∈ Π01, σ ∈ Σ

01,

∀n(π(n)↔ σ(n)

)→ ∃X∀n

(n ∈ X ↔ σ(n)

).

The Big Five

I RCA0: Q + CA(∆01) + IΣ01.

I WKL0: Weak König’s Lemma.

I ACA0: Q + CA(Σ01) + Ind .

I ATR0: Arithmetical transfinite recursion.

I Π11-CA0: Q + CA(Π11) + Ind .

The Big Five

I RCA0: Q + CA(∆01) + IΣ01. Base theory

I WKL0: Weak König’s Lemma.

I Π11-CA0: Q + CA(Π11) + Ind .

The Big Five

I RCA0: Q + CA(∆01) + IΣ01. Base theory

I WKL0: Weak König’s Lemma.⊗

I Π11-CA0: Q + CA(Π11) + Ind .

Recursive comprehension

I RCA0 is the standard ‘weak base theory’ of reversemathematics.

I Conservative over IΣ1.

I Sufficient for proving the intermediate value theorem fromcalculus.

I Proof-theoretic ordinal ωω.

RCA0 will be our base theory through most of our presentation,but later we will discuss weaker alternatives.

Arithmetical comprehension

I ACA0 is equivalent to either:

I Q + CA(Σ01) + Ind . I Q + CA(Π0ω) + Ind .

Important: Free set-variables are allowed incomprehension instances.

I Conservative over PA.I Equivalent to the Bolzano-Weierstrass theorem over RCA0.I Proof-theoretic ordinal

ε0 = supn

First approximation:

Let us first consider the theory

RCA0 +{∀n(�RCA0ϕ(n̄)→ ϕ(n)

): ϕ ∈ Π0ω

}.

This will indeed give us the first-order part of ACA0 (Peanoarithmetic).

However, we do not get any new comprehension.

): ϕ ∈ Π0ω

}.

): ϕ ∈ Π0ω

}.

Second-order formalization of provability

Instead, we will formalize provability using a least fixed point.

For a theory T , let ThmT (P) be a formula stating:

P is the least set containing all axioms of T andclosed under the rules of T .

Then define

[0]Tϕ = ∀P(ThmT (P)→ ϕ ∈ P

).

Note: 〈0〉T> implies ∃P ThmT (P).

Our second approximation

For a set of formulas Γ, we define the schema

0-RFNΓ[T ] = ∀X ∀n([0]Tϕ(n̄)→ ϕ(n)

).

(for ϕ ∈ Γ).

Our second approximation is

RCA0 + 0-RFNΣ01 [RCA0].

Does 0-RFNΣ01[RCA0] prove ACA0?

Idea: Let ϕ(n) be a Σ01 formula. We wish to form the set

{n : ϕ(n)}.

Reasoning in RCA0 + 0-RFNΣ01 [RCA0], there exists P such thatThmRCA0(P) holds. We instead form the set

E = {n : ϕ(n̄) ∈ P}.

I If n ∈ E then ϕ(n) holds by reflection.I If ϕ(n) then n ∈ E should hold by Σ01-completeness.I But we lose completeness when ϕ has free set variables!

{n : ϕ(n)}.

E = {n : ϕ(n̄) ∈ P}.

{n : ϕ(n)}.

E = {n : ϕ(n̄) ∈ P}.

I If n ∈ E then ϕ(n) holds by reflection.

I If ϕ(n) then n ∈ E should hold by Σ01-completeness.I But we lose completeness when ϕ has free set variables!

{n : ϕ(n)}.

E = {n : ϕ(n̄) ∈ P}.

I If n ∈ E then ϕ(n) holds by reflection.I If ϕ(n) then n ∈ E should hold by Σ01-completeness.

I But we lose completeness when ϕ has free set variables!

{n : ϕ(n)}.

E = {n : ϕ(n̄) ∈ P}.

Theories with oracles

For a theory T and a set X , let T |X be the extension of T with anew set-constant O and all axioms of the form

I n̄ ∈ O for n ∈ X

I n̄ 6∈ O for n 6∈ X

Let [0|X ]Tϕ = ∀P(ThmT |X (P)→ ϕ ∈ P

).

Oracle reflection

Let 0-OrRFNΓ[T ] = ∀X ∀n([0|X ]Tϕ(n̄,O)→ ϕ(n,X )

).

TheoremACA0 ≡ RCA0 + 0-OrRFNΠ12 [RCA0].

Oracle reflection

Let 0-OrRFNΓ[T ] = ∀X ∀n([0|X ]Tϕ(n̄,O)→ ϕ(n,X )

).

TheoremACA0 ≡ RCA0 + 0-OrRFNΠ12 [RCA0].

The ‘standard’ proof of reflection

All axioms of RCA0 are true, and all rules preserve truth. Thusby induction on the length of a derivation, all theorems of RCA0are true.

Formally, we are proving by induction on n that

∀ϕ(Proof RCA0(n, ϕ)→ True(ϕ)

).

Question: Can we formalize this proof in ACA0?

Partial truth classes and cut-elimination

I ACA0 cannot do this induction since it cannot form the setof true arithmetical statements.

I It can, however, form the set of all true Πn-sentences forfixed n.

I Thus we would need to bound the complexity of allformulas appearing in a derivation.

I Solution: Cut elimination!

The Tait calculus

Sequent-based calculus, where all negations are pushed downto atomic formulas.

(LEM)Γ, α,¬α

(∧) Γ, ϕ Γ, ψΓ, ϕ ∧ ψ

(∨) Γ, ϕ, ψΓ, ϕ ∨ ψ

(∀) Γ, ϕ(v)Γ,∀xϕ(x)

(∃) Γ, ϕ(t)Γ, ∃xϕ(x)

(CUT)Γ, ϕ Γ,¬ϕ

Γ,

where α is atomic and v does not appear free in Γ.

Cut elimination

TheoremIt is provable in ACA0 that any sequent derivable in the Taitcalculus can be derived without the cut rule.

In fact, we do not need full ACA0.

Let EA+ be the theory EA+“the superexponential is total”.

Then, EA+ suffices to prove cut-elimination.

Cut elimination

TheoremIt is provable in ACA0 that any sequent derivable in the Taitcalculus can be derived without the cut rule.

In fact, we do not need full ACA0.

Let EA+ be the theory EA+“the superexponential is total”.

Then, EA+ suffices to prove cut-elimination.

Proving reflection in ACA0

I A set satisfying ThmRCA0|X (P) can be constructed withinACA0 by

P = {ϕ : ∃xProof RCA0|X (x , ϕ)}.

I Then we may use ω-models of RCA0 to prove reflection.

ω-models

DefinitionAn ω-model is a set M = (Mn)n∈N of subsets of N. We writeM |= ϕ if ϕ holds when all first-order quantifiers range over Nand all second-order quantifiers over {Mn}n∈N

A satisfaction class on M is a set S such that for all ϕ, ϕ ∈ S ifand only if M |= ϕ.

A partial satisfaction class on M for Γ is a set S such that for allϕ ∈ Γ, ϕ ∈ S if and only if M |= ϕ.

ω-models of RCA0

TheoremACA0 proves that, given any set X , there is an ω-model M ofRCA0 such that M |= RCA0 and M0 = X.

TheoremGiven a finite set of formulas Γ, ACA0 proves that, given anω-model M, there is a partial satisfaction class S for the set ofsubstitution instances of formulas of Γ and their subformulas.

Reflection via ω-models

Reason in ACA0:

I Fix a formula ϕ(n,Z ) = ∀X∃Yψ(n,X ,Y ,Z ) ∈ Π12 andassume that RCA0|Z proves ϕ(n̄,O).

I Then there is a cut-free derivation ofΓ = ¬Ax1, . . . ,¬Axm, ϕ(n̄,O).

I Let X be an arbitrary set and build an ω-model Mcontaining X with a satisfaction class S for Γ.

I Prove by induction on the length of a derivation that Γ ∈ S.

I By upwards-perseverance of Σ11 formulas, ∃Yψ(n,X ,Y ,Z )holds in N.

I But X was arbitrary, so ∀X∃Yψ(n,X ,Y ,Z ) holds in N.

Reason in ACA0:

Strong provability operators

Notation: [n]PΓ := 〈n, Γ〉 ∈ P

DefinitionA iterated provability class of depth n > 0 for a theory T is a setP closed under

1.[m]P∆1 . . . , [m]P∆j

[m]PΓ(ρ)

2.[m]P Γ, ϕ(0̄), [m]P Γ, ϕ(1̄), [m]P Γ, ϕ(2̄), [m]P Γ, ϕ(3̄), . . .

[k ]P Γ,∀xϕ(x)

(m < k ≤ n and ρ is any rule of T ).

Reasoning about strong provability operators in ACA0

TheoremGiven n > 0 and a theory T , it is provable in ACA0 that aniterated provability class of depth n exists for T .

DefinitionFix an axiomatization AxT for T and define:

I IPCnT (P) to be a formula expressing that P is an iteratedprovability class of depth n for T .

I [n]T Γ := ∀P(IPCnT (n,P)→ [n]P¬AxT , Γ

).

I [n|X ]T Γ is defined similarly but with an oracle for X .

I If cuts are not allowed we will write [n|X ]cfT Γ.

Strong reflection and consistency

DefinitionGiven a theory T , we defineI n-OrConscf [T ] = ∀X 〈n̄|X 〉cfT >I n-OrRFNcfΓ [T ] = ∀X ∀x

([n|X ]cfT ϕ(x̄)→ ϕ(x)

), where ϕ ∈ Γ

TheoremThe following theories are equivalent:I ACA0I RCA0 + 0-OrRFNΣ01 [RCA0]

I RCA0 + 1-OrConscf [RCA0]I RCA0 + {n̄-OrConscf [RCA0] : n ∈ N}I RCA0 + {n̄-OrRFNcfΠ12

[RCA0] : n ∈ N}

), where ϕ ∈ Γ

TheoremThe following theories are equivalent:

I ACA0I RCA0 + 0-OrRFNΣ01 [RCA0]

[RCA0] : n ∈ N}

), where ϕ ∈ Γ

TheoremThe following theories are equivalent:I ACA0

I RCA0 + 0-OrRFNΣ01 [RCA0]

[RCA0] : n ∈ N}

), where ϕ ∈ Γ

[RCA0] : n ∈ N}

), where ϕ ∈ Γ

I RCA0 + 1-OrConscf [RCA0]

I RCA0 + {n̄-OrConscf [RCA0] : n ∈ N}I RCA0 + {n̄-OrRFNcfΠ12

[RCA0] : n ∈ N}

), where ϕ ∈ Γ

I RCA0 + 1-OrConscf [RCA0]I RCA0 + {n̄-OrConscf [RCA0] : n ∈ N}

I RCA0 + {n̄-OrRFNcfΠ12[RCA0] : n ∈ N}

), where ϕ ∈ Γ

[RCA0] : n ∈ N}

Beyond ACA0

We may obtain stronger reflection principles if we iterateω-rules transfinitely. For this, we must represent well-orders insecond-order arithmetic.

They can be represented by pairs Λ = (|Λ|,

Transfinite reflection principles

We can iterate ω-rules along any well-order Λ using the samedefinition as in the finite case.

Given a definable well-order Λ and a theory T , we can consider

T Λ = T + ∀X 〈Λ| X 〉cfT >.

We have seen that ACA0 =⋃

n

T Λ = T + ∀X 〈Λ| X 〉cfT >.

n

T Λ = T + ∀X 〈Λ| X 〉cfT >.

n

Arithmetical Transfinite Recursion

I ATR0 is associated with predicative reductionism.

I Equivalent to the perfect set theorem (every uncountableclosed subset of a complete separable metric spacecontains a non-empty, perfect closed set).

I Proof-theoretical ordinal the Feferman-Schütte ordinal Γ0.

Goal: Use strong reflection principles to represent ATR0.

Define

TRΛϕ(X ,Y ) ≡ ∀n∀λ(〈n, λ〉 ∈ Y ↔ ϕ(n, λ,X ,Y

X

X Y0

Y0 = {n ∈ N : ϕ(n,1,X ,∅)}

X Y0︸︷︷︸Y

X Y0 Y1︸︷︷︸Y

X Y0 Y1 Y2 . . .︸︷︷︸Y

X Y0 Y1 Y2 . . . Yω︸︷︷︸Y

Predicative reflection principlesPredicative consistency:

PredCons[T ] := ∀Λ∀X (wo(Λ)→ 〈Λ|X 〉T>)

Predicative reflection:

PredRFNϕ[T ] := ∀Λ∀X(

wo(Λ)→([Λ|X ]Tϕ→ ϕ

))PredRFNΓ[T ] := {PredRFNϕ[T ] : ϕ ∈ Γ}

Theorem (Cordón-Franco, DFD, Joosten, Lara-Martı́n)The following theories are equivalent:I ATR0

I RCA0 + PredCons[RCA0]

I ACA0 + PredRFNΠ12 [ACA0]

Predicative reflection principlesPredicative consistency:

PredCons[T ] := ∀Λ∀X (wo(Λ)→ 〈Λ|X 〉T>)

Predicative reflection:

PredRFNϕ[T ] := ∀Λ∀X(

wo(Λ)→([Λ|X ]Tϕ→ ϕ

))PredRFNΓ[T ] := {PredRFNϕ[T ] : ϕ ∈ Γ}

Theorem (Cordón-Franco, DFD, Joosten, Lara-Martı́n)The following theories are equivalent:I ATR0

I RCA0 + PredCons[RCA0]

I ACA0 + PredRFNΠ12 [ACA0]

Predicative reflection proves transfinite recursion

Reason in ACA0 + PredRFNΠ0ω [RCA0]

Given a well-order Λ and arithmetical ϕ(n,X ) ∈ Π02m, we wish toconstruct a set R satisfying

∀n ∀λ(

n ∈ Rλ ↔ ϕ(n,R

∀n ∀λ(

ω-models in ATR0

TheoremIt is provable in ATR0 that any ω-model M admits a fullsatisfaction class.

TheoremIt is provable in ATR0 that any set X can be included in anω-model M[X ] for ACA0.

Transfinite recursion proves predicative reflection

Reasoning in ATR0:

I Pick a well-order Λ, a Π12-formula ϕ = ∀X∃Yψ(X ,Y ), a setX and assume that [Λ|Z ]ACA0ϕ for some Z .

I Construct a Λ-IPC P and consider the ω-model M[X ,Z ] ofACA0 with full satisfaction class S.

I By a straightforward transfinite induction show that [λ]Pθimplies θ ∈ S.

I In particular M[X ,Z ] |= ∃Yψ(X ,Y ), so ∃Yψ(X ,Y ) holds.Since X was arbitrary, so does ϕ.

Reasoning in ATR0:

Π11-comprehension

I Π11-CA0 is equivalent to Σ11-CA0.

I Proper extension of ATR0.

I Equivalent to the Cantor-Bendixson theorem over RCA0(every closed set of reals is the union of a perfect set and acountable set).

I Proof-theoretic ordinal ψ0(Ωω).

Π11-comprehension

Π11-CA0: Add to RCA0 all axioms of the form

∀X∃Y∀n(n ∈ Y ↔ ∀Z ϕ(n,X ,Z )

)where ϕ is arithmetical.

Impredicativity: The set Y is defined using a collection whichincludes Y !

Unbounded ω-logic

Define [∞]Tϕ as“ϕ is provable using an arbitrary number of ω-rules.”

Formally:I Say that P is a saturated provability class for T if

I P contains all axioms of T and is closed under all rules ofthe Tait calculus as well as the ω-rule;

I P is the least set with this property.

We can write this in a formula SPCT (P).I Define

[∞]T Γ := ∀P(

SPCT (P)→ Γ ∈ P).

I As before, [∞|X ]T Γ means that we also have an oracle forX .

Existence of an SPCTheoremIt is provable in Π11-CA that given a theory T and a set X thereexists an SPC for T with an oracle for X .

Proof.I Let C(Y ) be a formula stating

Y contains T |X and is closed under all the rulesof ω-logic.

I DefineP =

{Γ : ∀Y

(C(Y )→ Γ ∈ Y

)}.

I P exists by Π11 comprehension.I It is not hard to check that P itself satisfies C(P).I By definition, P is contained in any set Y satisfying C(Y ),

hence it is the least such set.

I DefineP =

{Γ : ∀Y

(C(Y )→ Γ ∈ Y

)}.

I DefineP =

{Γ : ∀Y

(C(Y )→ Γ ∈ Y

)}.

I DefineP =

{Γ : ∀Y

(C(Y )→ Γ ∈ Y

)}.

I P exists by Π11 comprehension.

I It is not hard to check that P itself satisfies C(P).I By definition, P is contained in any set Y satisfying C(Y ),

I DefineP =

{Γ : ∀Y

(C(Y )→ Γ ∈ Y

)}.

I P exists by Π11 comprehension.I It is not hard to check that P itself satisfies C(P).

I By definition, P is contained in any set Y satisfying C(Y ),hence it is the least such set.

I DefineP =

{Γ : ∀Y

(C(Y )→ Γ ∈ Y

)}.

Completeness

TheoremGiven any theory T extending Q it is provable in ACA0 thatϕ→ [∞|X ]Tϕ holds.

Proof: We proceed by contrapositive.

I If Γ(Y ) is not provable, use a standard proof-search to build

Γ = Γ0 ⊆ Γ1 ⊆ Γ2 . . .

which decides any subformula of Γ and such that no Γi isderivable.

I Define Y ∗ :={

n : ∃i(n̄ 6∈ Y ∈ Γi

)}.

I Y ∗ is a witness for ¬∨

Γ, hence ∀Y∨

Γ(Y ) is not true.

CorollaryFor ϕ ∈ Σ12 it is provble in ACA0 that ϕ→ ∃Z [∞|Z ]Tϕ.

Completeness

TheoremGiven any theory T extending Q it is provable in ACA0 thatϕ→ [∞|X ]Tϕ holds.Proof: We proceed by contrapositive.

Γ = Γ0 ⊆ Γ1 ⊆ Γ2 . . .

I Define Y ∗ :={

n : ∃i(n̄ 6∈ Y ∈ Γi

)}.

Γ, hence ∀Y∨

Γ(Y ) is not true.

Completeness

Γ = Γ0 ⊆ Γ1 ⊆ Γ2 . . .

I Define Y ∗ :={

n : ∃i(n̄ 6∈ Y ∈ Γi

)}.

Γ, hence ∀Y∨

Γ(Y ) is not true.

Completeness

Γ = Γ0 ⊆ Γ1 ⊆ Γ2 . . .

I Define Y ∗ :={

n : ∃i(n̄ 6∈ Y ∈ Γi

)}.

Γ, hence ∀Y∨

Γ(Y ) is not true.

Completeness

Γ = Γ0 ⊆ Γ1 ⊆ Γ2 . . .

I Define Y ∗ :={

n : ∃i(n̄ 6∈ Y ∈ Γi

)}.

Γ, hence ∀Y∨

Γ(Y ) is not true.

Completeness

Γ = Γ0 ⊆ Γ1 ⊆ Γ2 . . .

I Define Y ∗ :={

n : ∃i(n̄ 6∈ Y ∈ Γi

)}.

Γ, hence ∀Y∨

Γ(Y ) is not true.

Impredicative consistency and reflection

DefinitionWe define:I ∞-OrCons[T ] := ∀X 〈∞|X 〉T>;I ∞-OrRFNΓ[T ] := ∀X

([∞|X ]Tϕ→ ϕ

)for ϕ ∈ Γ.

Observation: Over ACA0,∞-Cons[T ] implies∞-RFNΠ12 [T ] byΠ11 completeness.

I Π11-CA0I RCA0 +∞-OrCons[RCA0];I RCA0 +∞-OrRFNΠ13 [ATR0].

Impredicative consistency and reflection

DefinitionWe define:I ∞-OrCons[T ] := ∀X 〈∞|X 〉T>;I ∞-OrRFNΓ[T ] := ∀X

([∞|X ]Tϕ→ ϕ

)for ϕ ∈ Γ.

Observation: Over ACA0,∞-Cons[T ] implies∞-RFNΠ12 [T ] byΠ11 completeness.

I Π11-CA0I RCA0 +∞-OrCons[RCA0];I RCA0 +∞-OrRFNΠ13 [ATR0].

Impredicative reflection implies Π11 comprehension

Reason in ACA0 +∞-OrRFNΠ11 [ACA0]:

To construct the set {n : ∀Xϕ(n,X ,Y )

},

we instead consider{n : [∞|Y ]ACA0∀Xϕ(n,X ,Y )

}.

},

}.

},

}.

β-models

DefinitionAn ω-model M is a β-model if for every ϕ ∈ Π11 with parametersfrom M, ϕ holds if and only if M |= ϕ.

TheoremIt is provable in Π11-CA0 that, for every β-model M, M |= ATR0.

TheoremIt is provable in Π11-CA0 that for every set X there is a fullcountable coded β-model M[X ] such that M0 = X.

Π11 comprehension implies impredicative reflection

Reason in Π11-CA0.

I Let P be an SPC for ACA0 with oracle for Z .

I Given an arbitrary set X , consider the β-model M[X ,Z ] ofACA0 with full satisfaction class S.

I S is closed under all rules of ω-logic and contains allaxioms of ACA0, which by minimality of P means thatP ⊆ S.

I If ∀X∃Y∀Zϕ(X ,Y ,Z ) ∈ P, then M[X ] |= ∃Y∀Zϕ(X ,Y ,Z ),and since X can be taken arbitrary, ∀X∃Y∀Zϕ(X ,Y ,Z ) istrue.

Reason in Π11-CA0.

Weaker base theories

I RCA0: Q + CA(∆01) + IΣ01.

I RCA∗0: Q + CA(∆01) + Ind .

I ECA0: Q(exp) + CA(∆00(exp)) + Ind .(Second-order analogue of EA).

I RCA0: Q + CA(∆01) + IΣ01.

I RCA∗0: Q + CA(∆01) + Ind .

I RCA0: Q + CA(∆01) + IΣ01.

I RCA∗0: Q + CA(∆01) + Ind .

Elementary comprehension axiom

All of our proofs work verbatim if we replace RCA0 by ECA0:

I ACA0 ≡ ECA0 + 1-OrConscf [ECA0]

I ATR0 ≡ ECA0 + PredCons[ECA0];

I Π11-CA0 ≡ ECA0 +∞-OrCons[ECA0].

Question: Can we weaken our base theory further?

Weakening the provability unit

General Scheme:T ≡ B + C(U)

I B is the ‘base theory’;

I U is the ‘proof unit’.

I C(U) is some ‘consistency assertion’.

Weakening the consistency unit

Remark: All we need from the consistency unit U is that U besound and Σ1-complete.

But, Robinson’s Q already has this property.

Thus we may take U = Q.

Base theory breakdown

Suppose that B is such that:

1. B is able to form sets of the form {n : [m]Uϕ(n̄)};

2. given a ∆0(exp) formula δ, B can prove for some m that

∀n(δ(n)→ [m]cfU δ(n̄)

).

Then, ECA0 ⊆ B + {n-OrConscf [Q] : n < ω}.

Can we weaken the base theory?

If B is any theory between Q + Ind and ACA0 and satisfying1,2, we can show that:

I ACA0 ≡ B + {n-OrConscf [Q] : n < ω};

I ATR0 ≡ B + PredOrCons[Q];

I Π11-CA0 ≡ B +∞-OrCons[Q].

Question: Can we choose B to be weaker than ECA0?

Can we weaken the base theory?

If B is any theory between Q + Ind and ACA0 and satisfying1,2, we can show that:

I ACA0 ≡ B + {n-OrConscf [Q] : n < ω};

I ATR0 ≡ B + PredOrCons[Q];

I Π11-CA0 ≡ B +∞-OrCons[Q].

Question: Can we choose B to be weaker than ECA0?

Conclusions and future work

I We have shown how three of the Big Five theories ofreverse mathematics can be represented as strongconsistency or reflection principles over a weak basetheory.

I These principles naturally fall within a large spectrum oftheories between ACA0 and Π11-CA0.

I Can stronger theories such as Π12-CA0 be represented in asimilar fashion?

I How about natural theories in the language of set-theory?

I Can these principles be used for Π01 ordinal analysis in thespirit of Beklemishev’s analysis of PA?

FIN

Thank you!

FIN

Thank you!