Constructivism and weak logical principles over arithmetic Makoto - - PowerPoint PPT Presentation

Introduction Structure of Logical Principles in Arithmetic Discussions

Constructivism and weak logical principles

• ver arithmetic

Makoto Fujiwara

JSPS Research Fellow (PD), School of Science and Technology, Meiji University; Visiting researcher, Zukunftskolleg, University of Konstanz

Mathematical Logic and Constructivity, Stockholm 23 August 2019

This work is supported by JSPS KAKENHI Grant Numbers JP18K13450 and JP19J01239, as well as JSPS Core-to-Core Program (A. Advanced Research Networks).

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Introduction Structure of Logical Principles in Arithmetic Discussions

We study the interrelation between syntactical restrictions of the following logical principles over (many-sorted) intuitionistic arithmetic. LEM (Law of Excluded Middle): ϕ ∨ ¬ϕ DML (De Morgan’s Law): ¬(ϕ ∧ ψ) → ¬ϕ ∨ ¬ψ DNE (Double Negation Elimination): ¬¬ϕ → ϕ DNS (Double Negation Shift): ∀x¬¬ϕ(x) → ¬¬∀xϕ(x) Motivations

1 One Conceptual Motivation:

Foundation (Axiomatization) of Constructivism

2 One Practical Motivation:

Framework for Constructive Reverse Mathematics

3 Applications to other fields (Proof mining, Limit

computable math., Computable math. etc.)

4 . . .

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Introduction Structure of Logical Principles in Arithmetic Discussions

Motivation 1: Foundation of Constructivism

In early 20 centuries, L. E. J. Brouwer tried to reconstruct mathematics in favor of his intuitionism. In Brouwer’s intuitionistic mathematics (which is a first school of constructive mathematics), everything has to be built from the ground up, including the meaning of the logical symbols.

• A. Heyting, a pupil of Brouwer, formalized Brouwer’s

“proofs as constructions”-concept via his theory Heyting arithmetic (HA) with an informal semantics (BHK interpretation, below). HA is a variant of Peano arithmetic (PA) based on intuitionistic logic, and one can obtain PA just by adding LEM (ϕ ∨ ¬ϕ) into the axioms of HA.

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Introduction Structure of Logical Principles in Arithmetic Discussions

BHK(Brouwer/Heyting/Kolmogorov)-interpretation

There is no proof for ⊥. A proof of A ∧ B is a pair (q, r) of proofs, where q is a proof of A and r is a proof of B. A proof of A ∨ B is a pair (n, q) consisting of an integer n and a proof q, where q is a proof of A if n = 0 and q is a proof of B if n ̸= 0. A proof p of ∃xA(x) is a pair (cd, q), where cd is the construction of an element d of the domain and q is a proof of A(d). A proof p of A → B is a construction which transforms any proof of A into a proof of B. A proof p of ∀xA(x) is a construction which transforms any construction of d of the domain into a proof of A(d).

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Introduction Structure of Logical Principles in Arithmetic Discussions

The reasoning in constructive/intuitionistic mathematics is somewhat restricted than that in classical (usual) mathematics.

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Introduction Structure of Logical Principles in Arithmetic Discussions

The reasoning in constructive/intuitionistic mathematics is somewhat restricted than that in classical (usual) mathematics. Then classical logic is too strong as a formalization of constructive reasoning: Typical Example. A classical rule RAA(reductio ad absurdum)

✟✟ ✟

¬S . . . ⊥ S RAA

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Introduction Structure of Logical Principles in Arithmetic Discussions

• Question. (Heyting?)

What is the logic which captures constructive reasoning?

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Introduction Structure of Logical Principles in Arithmetic Discussions

• Question. (Heyting?)

What is the logic which captures constructive reasoning?

A simple answer is intuitionistic logic.

Constructive Mathematics ≈ Formalized Const. Math. If we assume Γ as axioms, then we can constructively prove a statement S. ≈ Γ ⊢i S

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• Question. (Heyting?)

What is the logic which captures constructive reasoning?

A simple answer is intuitionistic logic.

Constructive Mathematics ≈ Formalized Const. Math. If we assume Γ as axioms, then we can constructively prove a statement S. ≈ Γ ⊢i S The large amount of Bishop’s constructive mathematics can be formalized in a second-order theory EL, or even EL0 which has Σ0

1 induction only and is often served as a

base theory for constructive reverse mathematics.

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Introduction Structure of Logical Principles in Arithmetic Discussions

• Question. (Heyting?)

What is the logic which captures constructive reasoning?

A simple answer is intuitionistic logic.

Constructive Mathematics ≈ Formalized Const. Math. If we assume Γ as axioms, then we can constructively prove a statement S. ≈ Γ ⊢i S The large amount of Bishop’s constructive mathematics can be formalized in a second-order theory EL, or even EL0 which has Σ0

1 induction only and is often served as a

base theory for constructive reverse mathematics. From a foundational point of view, however, a determinative answer on suitable underlying logic for constructive mathematics is still missing (in my thought).

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• Remark. If Γ ⊢i S, then one can usually obtain a constructive

(in the sense of BHK) proof of S from Γ.

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Introduction Structure of Logical Principles in Arithmetic Discussions

• Remark. If Γ ⊢i S, then one can usually obtain a constructive

(in the sense of BHK) proof of S from Γ. The converse direction (Constructive ⇒ Intuitionistic+α), however, is somewhat serious.

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• Remark. If Γ ⊢i S, then one can usually obtain a constructive

(in the sense of BHK) proof of S from Γ. The converse direction (Constructive ⇒ Intuitionistic+α), however, is somewhat serious. In the case of arithmetic: Some weak logical principles (like ∆a-LEM below) are modified realizable (and also Dialectica interpretable) in G¨

• del’s T but NOT provable in HAω.

DNS is modified realizable in T (while DNS itself is used for the verification) but NOT provable HAω.

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• Remark. If Γ ⊢i S, then one can usually obtain a constructive

(in the sense of BHK) proof of S from Γ. The converse direction (Constructive ⇒ Intuitionistic+α), however, is somewhat serious. In the case of arithmetic: Some weak logical principles (like ∆a-LEM below) are modified realizable (and also Dialectica interpretable) in G¨

• del’s T but NOT provable in HAω.

DNS is modified realizable in T (while DNS itself is used for the verification) but NOT provable HAω.

• Question. How much α capture constructive provability?

Constructive Mathematics ≈ Formalized Const. Math. If we assume Γ as axioms, then we can constructively prove a statement S. ≈? Γ ⊢i+α S

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Motivation 2: Framework for Constructive Reverse Mathematics

The aim of reverse mathematics is classifying mathematical theorems from a perspective of logical complexity.

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Introduction Structure of Logical Principles in Arithmetic Discussions

Motivation 2: Framework for Constructive Reverse Mathematics

The aim of reverse mathematics is classifying mathematical theorems from a perspective of logical complexity. In reverse mathematics, one formalizes mathematical statements in second-order arithmetic, and investigates the relation between such statements and logical axioms.

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Classical Reverse Mathematics (1970’s–) Many of ordinary (non-set theoretic) mathematical theorems are provable in RCA0, or equivalent to WKL or ACA over RCA0. (Friedman, Simpson etc.) A well-known exception is Ramsey’s theorem for pairs RT2

• 2. (Cholak/Jockusch/Slaman 2001, Liu 2012)

ACA

• WKL
• ⧹

RT2

2

RCA0 (or RCA) ≈ Classical Computable Math.

WKL: Weak K¨

• nig’s Lemma, ACA: Arithmetical Comprehension.

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Constructive Reverse Mathematics (2000’s–)

Constructive reverse mathematics (Ishihara, Nemoto, Berger etc.), which is reverse mathematics over intuitionistic fragment EL0 (resp. EL) of RCA0 (resp. RCA) In contrast to other attempts in reverse mathematics, its underlying logic is NOT classical logic.

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Constructive Reverse Mathematics (2000’s–)

Constructive reverse mathematics (Ishihara, Nemoto, Berger etc.), which is reverse mathematics over intuitionistic fragment EL0 (resp. EL) of RCA0 (resp. RCA) In contrast to other attempts in reverse mathematics, its underlying logic is NOT classical logic. One can obtain a much sharper classification rather than classical reverse mathematics: Intuitionistic Logic Classical Logic First-order HA PA Second-order EL0 EL RCA0 RCA

EL0, EL ≈ (Bishop’s) Constructive Math.

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Examples of Constructive Reverse Mathematics.

1 UCc: Every continuous mapping from 2N to N with a

continuous modulus is uniformly continuous.

RCA0 ⊢ WKL ↔ UCc. EL0 ⊢ FAND ↔ UCc and EL0 + FAND ⊬ WKL.

2 IVT: Intermediate Value Theorem

RCA0 ⊢ IVT. EL0 ⊢ IVT ↔ WKLc and EL0 ⊬ IVT. WKL: Every infinite binary tree has a infinite path. FAND: Contrapositive of WKL. WKLc: Every infinite binary convex tree has a infinite path.

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• Proposition. (Ishihara 2005)

EL0 ⊢ ACA ↔ Σ0

1-LEM + Π0 1-AC0,0.

EL0 ⊢ WKL ↔ Σ0

1-DML + Π0 1-AC∨ 0 .

Σ0

1-LEM: ∀α∀x(∃y α(x, y) = 0 ∨ ¬∃y α(x, y) = 0).

Σ0

1-DML:

∀α, β∀x ( ¬(∃y α(x, y) = 0 ∧ ∃z β(x, z) = 0) → ¬∃y α(x, y) = 0 ∨ ¬∃z β(x, z) = 0 ) . Notation: We use x, y, z . . . for number variables and α, β, γ . . . for function variables. Remark. RCA0 = EL0 + LEM = EL0 + DNE ⊃ EL0 + DML.

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Structure of the weak logical principles over EL0, EL or HA.

Σ0

1-LEM

• Σ0

1-DNE ⧹

• Π0

1-LEM ⧹

• ∆b-LEM
• Σ0

1-DML

• Π0

1-DML

• ∆a-LEM

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Σ0

1-LEM: ∀α∀x(∃y α(x, y) = 0 ∨ ¬∃y α(x, y) = 0).

Π0

1-LEM: ∀α∀x(∀y α(x, y) = 0 ∨ ¬∀y α(x, y) = 0).

Σ0

1-DML: ∀α, β∀x(¬(∃y α(x, y) = 0 ∧ ∃z β(x, z) = 0) →

¬∃y α(x, y) = 0 ∨ ¬∃z β(x, z) = 0).

Π0

1-DML: ∀α, β∀x(¬(∀y α(x, y) = 0 ∧ ∀z β(x, z) = 0) →

¬∀y α(x, y) = 0 ∨ ¬∀z β(x, z) = 0).

Σ0

1-DNE: ∀α∀x(¬¬∃y α(x, y) = 0 → ∃y α(x, y) = 0).

∆a-LEM: ∀α, β∀x((∃y α(x, y) = 0 ↔ ¬∃z β(x, z) = 0) →

∃y α(x, y) = 0 ∨ ¬∃y α(x, y) = 0).

∆b-LEM: ∀α, β∀x((¬∃y α(x, y) = 0 ↔ ∃z β(x, z) = 0) →

∃y α(x, y) = 0 ∨ ¬∃y α(x, y) = 0).

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Realizability and arithmetic in all finite types

Kreisel’s modified realizability interpretation (1959–) is a kind of formal treatment of the BHK interpretation in the context of arithmetic in all finite types.

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• Definition. (Modified realizability interpretation)

For a formulas of HAω (containing only =0 in the language), its realizability interpretation Amr is defined as follows: Amr :≡ ∃x (x mr A) :≡ A for atomic A; Let Amr :≡ ∃x (x mr A), Bmr :≡ ∃u (u mr B). Then, (A ∧ B)mr :≡ ∃x, u (x, u mr (A ∧ B)) :≡ ∃x, u (x mr A ∧ u mr B); (A ∨ B)mr :≡ ∃z0, x, y(z, x, y mr (A ∨ B)) :≡ ∃z, x, y (( z =0 0 → x mr A ) ∧ ( z ̸=0 0 → y mr B )) ; (A → B)mr :≡ ∃y (y mr (A → B)) :≡ ∃y∀x ( x mr A → yx mr B ) ; (∀zρA(z))mr :≡ ∃x (x mr ∀zA(z)) :≡ ∃x∀z(x z mr A(z)); (∃zρA(z))mr :≡ ∃z, x (z, x mr ∃zA(z)) :≡ ∃z, x (x mr A(z)).

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• Proposition. (Soundness of the modified realizability)

If HAω + ACω + IPω

ef ⊢ A, then there exist terms t in G¨

• del’s

T s.t. HAω ⊢ t mr A.

• Proposition. (Characterization of the modified realizability)

HAω + ACω + IPω

ef ⊢ A ⇐

⇒ HAω ⊢ Amr. ACρ,τ : ∀xρ∃y τA(x, y) → ∃Y τ(ρ)∀xρA(x, Y (x)) (axiom scheme of choice) IPρ

ef : (Aef → ∃xρB(x)) → ∃xρ(Aef → B(xρ))

(independence-of-premise schema for ∃-free formulas)

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Introduction Structure of Logical Principles in Arithmetic Discussions

• Proposition. (Soundness of the modified realizability)

If HAω + ACω + IPω

ef ⊢ A, then there exist terms t in G¨

• del’s

T s.t. HAω ⊢ t mr A.

• Proposition. (Characterization of the modified realizability)

HAω + ACω + IPω

ef ⊢ A ⇐

⇒ HAω ⊢ Amr. ACρ,τ : ∀xρ∃y τA(x, y) → ∃Y τ(ρ)∀xρA(x, Y (x)) (axiom scheme of choice) IPρ

ef : (Aef → ∃xρB(x)) → ∃xρ(Aef → B(xρ))

(independence-of-premise schema for ∃-free formulas) Remark. ∆a-LEM is provable in HAω + IPω

ef, but not provable in HAω.

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• Definition. (Dialectica interpretation)

For a formulas of HAω (containing only =0 in the language), its Dialectica interpretation AD is defined as follows: AD :≡ AD :≡ A (x, y) for atomic A; Let AD ≡ ∃x∀yAD(x, y), BD :≡ ∃u∀vBD(u, v). Then, (A ∧ B)D :≡ ∃x, u∀y, v (A ∧ B)D :≡ ∃x, u∀y, v(AD(x, y) ∧ BD(u, v)); (A ∨ B)D :≡ ∃z0, x, u∀y, v (A ∨ B)D :≡ ∃z0, x, u∀y, v (( z = 0 → AD(x, y) ) ∧ ( z ̸= 0 → BD(u, v) )) ; (A → B)D :≡ ∃U, Y ∀x, v (A → B)D :≡ ∃U, Y ∀x, v ( AD(x, Y x v) → BD(U x, v) ) ; (∀zρA(z))D :≡ ∃X∀z, y (∀zA(z))D :≡ ∃X∀z, yAD(X z, y, z); (∃zρA(z))D :≡ ∃z, x∀y (∃zA(z))D :≡ ∃z, x∀yAD(x, y, z).

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• Proposition. (Soundness of the Dialectica interpretation)

If HAω + ACω + IPω

∀ + Mω ⊢ A, then there exist terms t in

G¨

• del’s T s.t. HAω ⊢ ∀yAD(t, y).
• Proposition. (Characterization of the Dialectica interpretation)

HAω + ACω + IPω

∀ + Mω ⊢ A ⇐

⇒ HAω ⊢ AD.

IPρ,τ

∀

: (∀uτAqf (u) → ∃xρB(x)) → ∃xρ (∀uτAqf (u) → B(x)) (independence-of-premise schema for purely univ. formulas)

Mρ : ¬¬∃xρAqf (x) → ∃xρAqf (x)

(Markov’s principle in all finite types)

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• Proposition. (Soundness of the Dialectica interpretation)

If HAω + ACω + IPω

∀ + Mω ⊢ A, then there exist terms t in

G¨

• del’s T s.t. HAω ⊢ ∀yAD(t, y).
• Proposition. (Characterization of the Dialectica interpretation)

HAω + ACω + IPω

∀ + Mω ⊢ A ⇐

⇒ HAω ⊢ AD.

IPρ,τ

∀

: (∀uτAqf (u) → ∃xρB(x)) → ∃xρ (∀uτAqf (u) → B(x)) (independence-of-premise schema for purely univ. formulas)

Mρ : ¬¬∃xρAqf (x) → ∃xρAqf (x)

(Markov’s principle in all finite types)

Remark. Mω is a finite type extension of Σ0

1-DNE, and hence the above

system contains the weak logical principles below Σ0

1-DNE.

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Unification (Oliva 2006)

• Definition. (Modified realizability in the Dialectica style)

AR :≡ AR :≡ A for atomic A; Let AR ≡ ∃x∀yAR(x, y), BR :≡ ∃u∀vBR(u, v). Then, (A ∧ B)R :≡ ∃x, u∀y, v (A ∧ B)R :≡ ∃x, u∀y, v(AR(x, y) ∧ BR(u, v)); (A ∨ B)R :≡ ∃z0, x, u∀y, v (A ∨ B)R

:≡ ∃z0, x, u∀y, v (( z = 0 → AR(x, y) ) ∧ ( z ̸= 0 → BR(u, v) )) ;

(A → B)R :≡ ∃U∀x (A → B)R :≡ ∃U∀x ( ∀y AR(x, y) → ∀v BR(U x, v) ) ;

(∀zρA(z))R :≡ ∃X∀z, y (∀zA(z))R :≡ ∃X∀z, yAR(X z, y, z);

(∃zρA(z))R :≡ ∃z, x∀y (∃zA(z))R :≡ ∃z, x∀yAR(x, y, z).

• Remark. One can show that AR is ∃-free.

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Proposition. HAω ⊢ ∀x ( x mr C ↔ ∀y CR(x, y) ) . Thus, one can see the modified realizability as a kind of Dialectica (Oliva 2006), and vice versa!

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Modified realizability and logical principles

Σ0

1-LEM

• Σ0

1-DNE

• Π0

1-LEM

• ∆b-LEM
• Σ0

1-DML

• Π0

1-DML

• ∆a-LEM
• Tnonotone

Dialectic a

/

modified realizability

/

• monotone

/

,

Dialectics

=

modified realizeability

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On the status of 2 important restrictions of DNS

DNS : ∀x¬¬ϕ(x) → ¬¬∀xϕ(x) is a principle which is modified realizable with an empty realizer. For the verification, however, one need DNS itself (as well as ACω).

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G¨

• del-Spector’s Reduction of Consistency

1 In his 1958 paper, G¨

• del gave a consistency proof of PA

by using the Dialectica interpretation.

2 In his 1962 paper, Spector extended G¨

• del’s method to

second-order arithmetic (classical analysis). Their method consists of the following steps: Firstly, by negative translation, one reduces a classical theory to the corresponding (semi-)intuitionistic theory, Secondly, by the Dialectica interpretation, one reduces the (semi-)intuitionistic theory to the quantifier-free functional theory T (+ bar recursion). PA(ω) ⊢⊥= = = = = = = = ⇒

Negative translation

HA(ω) ⊢ ⊥N(≡⊥) = = = = = = = = = ⇒

Dialectica interpretation

T | =⊥ .

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The difficulty of the consistency reduction for (subsystems of) second-order arithmetic is in the interpretation of AC0,0.

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The difficulty of the consistency reduction for (subsystems of) second-order arithmetic is in the interpretation of AC0,0. Note that RCA0 contains QF-AC0,0 : ∀α(∀x∃y α(x, y) = 0 → ∃γ∀x α(x, γ(x)) = 0)

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The difficulty of the consistency reduction for (subsystems of) second-order arithmetic is in the interpretation of AC0,0. Note that RCA0 contains QF-AC0,0 : ∀α(∀x∃y α(x, y) = 0 → ∃γ∀x α(x, γ(x)) = 0) and RCA0 + ACA is equivalent to RCA0 augmented with Π0

1-AC 0,0 : ∀α(∀x∃y∀z α(x, y, z) = 0 → ∃γ∀x, z α(x, γ(x), z) = 0).

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Negative Translations of RCA0 and ACA0. RCA0 ⊢ ϕ = ⇒ EL0 + Σ0

1-DNS0 ⊢ ϕN.

RCA0+Π0

1-AC0,0 ⊢ ϕ ⇒ EL0+Π0 1-AC0,0+Σ0 2-DNS0 ⊢ ϕN.

Σ0

1-DNS0 : ∀α(∀x¬¬∃y α(x, y) = 0 → ¬¬∀x∃y α(x, y) = 0).

Σ0

2-DNS0 :

∀α(∀x¬¬∃y∀z α(x, y, z) = 0 → ¬¬∀x∃y∀z α(x, y, z) = 0).

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Negative Translations of RCA0 and ACA0. RCA0 ⊢ ϕ = ⇒ EL0 + Σ0

1-DNS0 ⊢ ϕN.

RCA0+Π0

1-AC0,0 ⊢ ϕ ⇒ EL0+Π0 1-AC0,0+Σ0 2-DNS0 ⊢ ϕN.

Σ0

1-DNS0 : ∀α(∀x¬¬∃y α(x, y) = 0 → ¬¬∀x∃y α(x, y) = 0).

Σ0

2-DNS0 :

∀α(∀x¬¬∃y∀z α(x, y, z) = 0 → ¬¬∀x∃y∀z α(x, y, z) = 0). Observation.

1 EL0 + AC0,0 + Σ0 1-DNE (which derives Σ0 1-DNS0) is

Dialectica interpretable in T.

2 Σ0 2-DNS0 is NOT Dialectica interpretable in T (while it is

Dialectica interpretable in T + BR0,1).

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Observation from Pure (Predicate) Logic

Fact 1. CQC = IQC + LEM = IQC + DML + DNE. Let us consider about ¬¬LEM. Fact 2. IQC ⊢ ∀x¬¬(ϕ(x) ∨ ¬ϕ(x)). IQC ⊬ ¬¬∀x(ϕ(x) ∨ ¬ϕ(x)). Fact 3. IQC ⊢ DNS ↔ ¬¬LEM. DNS : ∀x¬¬ϕ(x) → ¬¬∀xϕ(x). ¬¬LEM : ¬¬∀x(ϕ(x) ∨ ¬ϕ(x)). The proof shows Σ0

2-DNS0 → ¬¬Σ0 1-LEM → Σ0 1-DNS0.

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Results (Kohlenbach/F. 2018) Whenever an implication does not follow by transitivity, then it actually does not hold!

Σ0

2-DNS0

¬¬Σ0

1-LEM

• ¬¬Σ0

1-DNE

• ¬¬Π0

1-LEM

• ¬¬∆b-LEM
• ¬¬Σ0

1-DML

• ¬¬Π0

1-DML

• Σ0

1-DNS0

¬¬∆a-LEM EL0, EL or HA

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Warning

The hierarchy above ¬¬Σ0

1-LEM collapses in the

presence of the axiom of choice!

Fact. For all n, EL0 + Π0

1-AC0,0 + ¬¬Σ0 1-LEM ⊢ ¬¬Σ0 n-LEM.

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• Proposition. (Kohlenbach 2015)

Π0

1-DML is NOT modified realizable in T.

∆a-LEM is modified realizable in T.

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• Proposition. (Kohlenbach 2015)

Π0

1-DML is NOT modified realizable in T.

∆a-LEM is modified realizable in T.

• Lemma. (Ishihara/Nemoto/F. 2015)

EL + ECT0 + ∆a-LEM ⊢ Π0

1-DML, where ECT0 is the

following (classically false but intuitionistically consistent) statement: ∀x(A → ∃yB(x, y)) → ∃z∀x(A → ∃u(T(z, x, u)∧B(x, U(u))), where A is almost negative.

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• Proposition. (Kohlenbach 2015)

Π0

1-DML is NOT modified realizable in T.

∆a-LEM is modified realizable in T.

• Lemma. (Ishihara/Nemoto/F. 2015)

EL + ECT0 + ∆a-LEM ⊢ Π0

1-DML, where ECT0 is the

following (classically false but intuitionistically consistent) statement: ∀x(A → ∃yB(x, y)) → ∃z∀x(A → ∃u(T(z, x, u)∧B(x, U(u))), where A is almost negative. In the same manner, we have EL + ECT0 + ¬¬∆a-LEM ⊢ ¬¬Π0

1-DML.

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Introduction Structure of Logical Principles in Arithmetic Discussions

• Proposition. (Kohlenbach 2015)

Π0

1-DML is NOT modified realizable in T.

∆a-LEM is modified realizable in T.

• Lemma. (Ishihara/Nemoto/F. 2015)

EL + ECT0 + ∆a-LEM ⊢ Π0

1-DML, where ECT0 is the

following (classically false but intuitionistically consistent) statement: ∀x(A → ∃yB(x, y)) → ∃z∀x(A → ∃u(T(z, x, u)∧B(x, U(u))), where A is almost negative. In the same manner, we have EL + ECT0 + ¬¬∆a-LEM ⊢ ¬¬Π0

1-DML.

Theorem. EL + ECT0 + Σ0

1-DNS0 ⊬ ¬¬∆a-LEM.

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Proof Sketch. Suppose that EL + ECT0 + Σ0

1-DNS0 ⊢ ¬¬∆a-LEM.

By the previous lemma, we have that EL + ECT0 + Σ0

1-DNS0

(equivalently Σ0

1-DNS0−) proves ¬¬Π0 1-DML, in particular,

¬¬∀x (

¬(∀y s(x, y) = 0 ∧ ∀z t(x, z) = 0) → ¬∀y s(x, y) = 0 ∨ ¬∀z t(x, z) = 0

) (1) where s, t are from Kohlenbach 2015. Then HA + ECT0 + Σ0

1-DNS0− ⊢ (1).

By Kleene realizability, one can drop ECT0 and obtain HAω + AC + Σ0

1-DNS0 proves

¬¬∃f ∀x   ¬(∀y s(x, y) = 0 ∧ ∀z t(x, z) = 0) → ( (f (x) = 0 → ¬∀y s(x, y) = 0) ∧(f (x) ̸= 0 → ¬∀z t(x, z) = 0 )   . (2)

By modified realizability, we have HAω + QF-AC0,0 + Σ0

1-DNS0 ⊢ (2).

HRO is a model of the former but (2) is false in HRO.

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Corollary EL + AC0,0 + Σ0

1-DNS0 ⊬ ¬¬∆a-LEM

and HA + Σ0

1-DNS0 ⊬ ¬¬∆a-LEM in the language of HA.

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Introduction Structure of Logical Principles in Arithmetic Discussions

References

1 H. Ishihara, Markov’s principle, Church’s thesis and Lindel¨

• f

theorem, Indag. Math. (N.S.) 4, pp. 321–325, 1993.

2 Y. Akama, S. Berardi, S. Hayashi and U. Kohlenbach, An

arithmetical hierarchy of the law of excluded middle and related principles, LICS’04, pp. 192–201, IEEE Press, 2004.

3 M. Fujiwara, H. Ishihara and T. Nemoto, Some principles

weaker than Markov’s principle, Arch. Math. Logic, vol. 54, issue 7-8, pp. 861–870, 2015.

4 U. Kohlenbach, On the disjunctive Markov principle, Studia

Logica, vol. 103, pp. 1313–1317, 2015.

5 M. Fujiwara and U. Kohlenbach, Interrelation between weak

fragments of double negation shift and related principles, Journal of Symb. Logic, vol. 83, issue 3, pp. 991–1012, 2018.

6 P. Oliva. Unifying functional interpretations. Notre Dame J.

Formal Logic, 47(2): pp. 263–290, 2006.

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A Possible Direction

An interesting direction concerning foundations of constructivism is to investigate the weak logical principles in

• ther theories with respect to constructivism (as Martin-L¨
• f

type theory).

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Introduction Structure of Logical Principles in Arithmetic Discussions

“Neutral” constructivism and the logical principles

The logical principles weaker than Markov’s principle (Σ0

1-DNE) are accepted both in classical mathematics

and Russian constructive recursive mathematics. While Markov’s principle is not accepted in intuitionistic mathematics, it is consistent with e.g. HAω + ACω + C-N + BIM (Ishihara/Nemoto). Then it is interesting to see whether some weak logical principles are provable in (the system corresponding to) intuitionistic mathematics (without Kripke scheme). Proposition. Σ0

2-DNS0 is equivalent to (¬¬Σ0 1)-BIM over

EL0 + Π0

1-AC0,0 + Σ0 1-DNS1, where

Σ0

1-DNS1 : ∀ᬬ∃xAqf(α, x) → ¬¬∀α∃xAqf(α, x).

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Thank you for your attention!

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