Some principles weaker than Markovs principle Makoto Fujiwara - - PowerPoint PPT Presentation

Background Results

Some principles weaker than Markov’s principle

Makoto Fujiwara (joint work with Hajime Ishihara and Takako Nemoto)

School of Information Science, Japan Advanced Institute of Science and Technology (JAIST)

CTFM 2015 9 September, 2015

This work is supported by Grant-in-Aid for JSPS Fellows, JSPS Core-to-Core Program (A. Advanced Research Networks) and JSPS Bilateral Programs Joint Research Projects/Seminars.

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Background Results

Constructive Mathematics (Early 20th Century –)

Constructive mathematics is distinguished from its traditional counterpart, classical mathematics, by the strict interpretation of the phrase “there exists” as “we can construct”.∗

∗This exposition is taken from Douglas Bridges and Erik Palmgren,

Constructive Mathematics, The Stanford Encyclopedia of Philosophy (Winter 2013 Edition).

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Background Results

Constructive Mathematics (Early 20th Century –)

Constructive mathematics is distinguished from its traditional counterpart, classical mathematics, by the strict interpretation of the phrase “there exists” as “we can construct”.∗ In order to work constructively, we need to re-interpret not only the existential quantifier but all the logical connectives and quantifiers as instructions on how to construct a proof of the statement involving these logical expressions (BHK-interpretation).

∗This exposition is taken from Douglas Bridges and Erik Palmgren,

Constructive Mathematics, The Stanford Encyclopedia of Philosophy (Winter 2013 Edition).

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Background Results

Constructive Mathematics (Early 20th Century –)

Constructive mathematics is distinguished from its traditional counterpart, classical mathematics, by the strict interpretation of the phrase “there exists” as “we can construct”.∗ In order to work constructively, we need to re-interpret not only the existential quantifier but all the logical connectives and quantifiers as instructions on how to construct a proof of the statement involving these logical expressions (BHK-interpretation). Heyting (1930’s -) and Kolmogorov (1920’s -) tried to formalize constructive mathematics and introduced intuitionistic logic.

∗This exposition is taken from Douglas Bridges and Erik Palmgren,

Constructive Mathematics, The Stanford Encyclopedia of Philosophy (Winter 2013 Edition).

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Background Results

Heyting Arithmetic HA

As language, HA has variables (for natural numbers), 0, successor S, function constants for all primitive recursive functions and a binary predicate constant =. HA is based on intuitionistic first order predicate logic and in addition contains

the defining axioms for the primitive recursive function constants, the equality axioms, IND: A(0) ∧ ∀x (A(x) → A(Sx)) → ∀xA(x).

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Background Results

Hierarchy of Logical Principles over HA (Akama, Berardi, Hayashi and Kohlenbach, 2004)

Γ-LEM: A ∨ ¬A, where A ∈ Γ (Γ ∈ {Σ0

0, Σ0 1, Π0 1}).

Σ0

1-LLPO: ¬(A ∧ B) → (¬A ∨ ¬B), where A, B ∈ Σ0 1.

Σ0

1-DNE: ¬¬A → A, where A ∈ Σ0 1.

∆0

1-LEM: (A ↔ B) → (A ∨ ¬A), where A ∈ Σ0 1, B ∈ Π0 1.

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Background Results

Hierarchy of Logical Principles over HA (Akama, Berardi, Hayashi and Kohlenbach, 2004)

Γ-LEM: A ∨ ¬A, where A ∈ Γ (Γ ∈ {Σ0

0, Σ0 1, Π0 1}).

Σ0

1-LLPO ≡ Σ0 1-DML: ¬(A ∧ B) → (¬A ∨ ¬B), where

A, B ∈ Σ0

1.

Σ0

1-DNE ≡ MP: ¬¬A → A, where A ∈ Σ0 1.

∆0

1-LEM: (A ↔ B) → (A ∨ ¬A), where A ∈ Σ0 1, B ∈ Π0 1.

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Background Results

Elementary Analysis EL

Elementary analysis EL is a conservative extension of HA, which is served as base theory formalizing (Bishop-style) constructive mathematics.

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Background Results

Elementary Analysis EL

Elementary analysis EL is a conservative extension of HA, which is served as base theory formalizing (Bishop-style) constructive mathematics. As language, EL has two-sorted variables (for numbers and functions), abstraction operators λx.(only for numbers), a recursor R in addition to that for HA. Axioms and rules of EL contain

λ-CON: (λx.t)t′ = t[t′/x] REC: Rtϕ0 = 0 and Rtϕ(St′) = ϕ(Rtϕt′, t′) QF-AC0,0: ∀x∃yAqf (x, y) → ∃f ∀xAqf (x, fx) IND: A(0) ∧ ∀x (A(x) → A(Sx)) → ∀xA(x)

EL0 is a fragment of EL where IND is replaced by QF-IND.

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Background Results

Intuitionistic Logic Classical Logic Non-sorted HA PA Two-sorted EL RCA EL0 RCA0

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Background Results

Intuitionistic Logic Classical Logic Non-sorted HA PA Two-sorted EL RCA EL0 RCA0

RCA0 is the most popular base system of reverse mathematics, which consists of

basic axioms BA of arithmetic based on classical logic, Σ0

1 induction scheme Σ0 1-IND,

∆0

1 comprehension scheme ∆0 1-CA:

∀α, β ( ∀y ( ∃x(α(y, x) = 0) ↔ ¬∃x(β(y, x) = 0) ) → ∃γ∀y ( γ(y) = 0 ↔ ∃x(α(y, x) = 0) ) ) .

RCA consists of BA, IND and ∆0

1-CA.

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Background Results

Proposition EL0 (containing only QF-IND) ⊢ Σ0

1-IND.

EL0 + LEM(A ∨ ¬A) ⊢ ∆0

1-CA.

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Background Results

Proposition EL0 (containing only QF-IND) ⊢ Σ0

1-IND.

EL0 + LEM(A ∨ ¬A) ⊢ ∆0

1-CA.

In fact, ∆0

1-CA is intuitionistically derived from QF-AC0,0

and Markov’s principle MP: ∀α ( ¬¬∃x(α(x) = 0) → ∃x(α(x) = 0) ) .

Note that ∆0

1-CA is equivalent to ∆0 1-LEM over

EL0 + AC.

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Background Results

Proposition EL0 (containing only QF-IND) ⊢ Σ0

1-IND.

EL0 + LEM(A ∨ ¬A) ⊢ ∆0

1-CA.

In fact, ∆0

1-CA is intuitionistically derived from QF-AC0,0

and Markov’s principle MP: ∀α ( ¬¬∃x(α(x) = 0) → ∃x(α(x) = 0) ) .

Note that ∆0

1-CA is equivalent to ∆0 1-LEM over

EL0 + AC. Inspecting the proofs in [Akama et al. 2004] reveals that there is also a corresponding hierarchy over EL or EL0.

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Background Results

Proposition EL0 (containing only QF-IND) ⊢ Σ0

1-IND.

EL0 + LEM(A ∨ ¬A) ⊢ ∆0

1-CA.

In fact, ∆0

1-CA is intuitionistically derived from QF-AC0,0

and Markov’s principle MP: ∀α ( ¬¬∃x(α(x) = 0) → ∃x(α(x) = 0) ) .

Note that ∆0

1-CA is equivalent to ∆0 1-LEM over

EL0 + AC. Inspecting the proofs in [Akama et al. 2004] reveals that there is also a corresponding hierarchy over EL or EL0. In particular, ∆0

1-LEM is derived from either MP or

Σ0

1-DML.

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Background Results

• Proposition. (Ishihara 1993)

1 EL0 ⊢ MP → Π0 1-DML. 2 EL0 ⊢ Σ0 1-DML → Π0 1-DML.

Note that Π0

1-DML is denoted as MP∨ in the literature.

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Background Results

Situation

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Background Results

Situation

Question. How is the relationship between Π0

1-DML and ∆0 1-LEM?

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Background Results

Warning.

Constructively, there is a couple of (classically equivalent) ways to define a formula being ∆0

1:

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Background Results

Warning.

Constructively, there is a couple of (classically equivalent) ways to define a formula being ∆0

1:

(a) α ∈ ∆a :≡ ∃β ( ∃xα(x) = 0 ↔ ¬∃xβ(x) = 0 ) . (b) α ∈ ∆b :≡ ∃β ( ¬∃xα(x) = 0 ↔ ∃xβ(x) = 0 ) . (c) α ∈ ∆c :≡ ∃β ( ¬∃xα(x) = 0 ↔ ¬¬∃xβ(x) = 0 ) . (ab) α ∈ ∆ab :≡ ∃β   ∃xα(x) = 0 ↔ ¬∃xβ(x) = 0 & ¬∃xα(x) = 0 ↔ ∃xβ(x) = 0  .

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Background Results

Warning.

Constructively, there is a couple of (classically equivalent) ways to define a formula being ∆0

1:

(a) α ∈ ∆a :≡ ∃β ( ∃xα(x) = 0 ↔ ¬∃xβ(x) = 0 ) . (b) α ∈ ∆b :≡ ∃β ( ¬∃xα(x) = 0 ↔ ∃xβ(x) = 0 ) . (c) α ∈ ∆c :≡ ∃β ( ¬∃xα(x) = 0 ↔ ¬¬∃xβ(x) = 0 ) . (ab) α ∈ ∆ab :≡ ∃β   ∃xα(x) = 0 ↔ ¬∃xβ(x) = 0 & ¬∃xα(x) = 0 ↔ ∃xβ(x) = 0  . Note that ∆0

1-LEM in [Akama et al. 2004] has been defined in

the sense of (a).

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Background Results

Warning.

Constructively, there is a couple of (classically equivalent) ways to define a formula being ∆0

1:

(a) α ∈ ∆a :≡ ∃β ( ∃xα(x) = 0 ↔ ¬∃xβ(x) = 0 ) . (b) α ∈ ∆b :≡ ∃β ( ¬∃xα(x) = 0 ↔ ∃xβ(x) = 0 ) . (c) α ∈ ∆c :≡ ∃β ( ¬∃xα(x) = 0 ↔ ¬¬∃xβ(x) = 0 ) . (ab) α ∈ ∆ab :≡ ∃β   ∃xα(x) = 0 ↔ ¬∃xβ(x) = 0 & ¬∃xα(x) = 0 ↔ ∃xβ(x) = 0  . Note that ∆0

1-LEM in [Akama et al. 2004] has been defined in

the sense of (a).

⇒ We consider the fragments of LEM with respect to ∆i (i ∈ {a, b, c, ab}).

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Background Results

Definition. ∆i-LEM :≡ ∀α ( α ∈ ∆i → ∃xα(x) = 0 ∨ ¬∃xα(x) = 0 ) .

(a) α ∈ ∆a :≡ ∃β ( ∃xα(x) = 0 ↔ ¬∃xβ(x) = 0 ) . (b) α ∈ ∆b :≡ ∃β ( ¬∃xα(x) = 0 ↔ ∃xβ(x) = 0 ) . (c) α ∈ ∆c :≡ ∃β ( ¬∃xα(x) = 0 ↔ ¬¬∃xβ(x) = 0 ) . (ab) α ∈ ∆ab :≡ ∃β   ∃xα(x) = 0 ↔ ¬∃xβ(x) = 0 & ¬∃xα(x) = 0 ↔ ∃xβ(x) = 0  .

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Background Results

Definition. ∆i-LEM :≡ ∀α ( α ∈ ∆i → ∃xα(x) = 0 ∨ ¬∃xα(x) = 0 ) .

(a) α ∈ ∆a :≡ ∃β ( ∃xα(x) = 0 ↔ ¬∃xβ(x) = 0 ) . (b) α ∈ ∆b :≡ ∃β ( ¬∃xα(x) = 0 ↔ ∃xβ(x) = 0 ) . (c) α ∈ ∆c :≡ ∃β ( ¬∃xα(x) = 0 ↔ ¬¬∃xβ(x) = 0 ) . (ab) α ∈ ∆ab :≡ ∃β   ∃xα(x) = 0 ↔ ¬∃xβ(x) = 0 & ¬∃xα(x) = 0 ↔ ∃xβ(x) = 0  .

Remark. ∆ab ↙ ↘ ∆a ∆b ↘ ↙ ∆c

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Background Results

Proposition. The following are pairwise equivalent over EL (even over EL0). MP. ∆c-LEM. ∆b-LEM.

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Background Results

Proposition. The following are pairwise equivalent over EL (even over EL0). MP. ∆c-LEM. ∆b-LEM. Proposition. Π0

1-DML implies ∆a-LEM over EL (even over EL0).

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Background Results

Proposition. The following are pairwise equivalent over EL (even over EL0). MP. ∆c-LEM. ∆b-LEM. Proposition. Π0

1-DML implies ∆a-LEM over EL (even over EL0).

Proposition.(Kohlenbach) EL + AC + ∆a-LEM does not prove Π0

1-DML.

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Background Results

Proposition. The following are pairwise equivalent over EL (even over EL0). MP. ∆c-LEM. ∆b-LEM. Proposition. Π0

1-DML implies ∆a-LEM over EL (even over EL0).

Proposition.(Kohlenbach) EL + AC + ∆a-LEM does not prove Π0

1-DML.

• Fact. ∆a-LEM implies ∆ab-LEM over EL (even over EL0).

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Background Results

Proposition. The following are pairwise equivalent over EL (even over EL0). MP. ∆c-LEM. ∆b-LEM. Proposition. Π0

1-DML implies ∆a-LEM over EL (even over EL0).

Proposition.(Kohlenbach) EL + AC + ∆a-LEM does not prove Π0

1-DML.

• Fact. ∆a-LEM implies ∆ab-LEM over EL (even over EL0).

⇒ How is the converse direction?

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Background Results

Definition. ∆[a → b] :≡ ∀α(α ∈ ∆a → α ∈ ∆b).

• Fact. ∆ab-LEM + ∆[a → b] implies ∆a-LEM.

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Background Results

Definition. ∆[a → b] :≡ ∀α(α ∈ ∆a → α ∈ ∆b).

• Fact. ∆ab-LEM + ∆[a → b] implies ∆a-LEM.

Lemma. ∆a-LEM implies ∆[a → b] over EL0.

• Proof. We reason in EL0. Let α ∈ ∆a.

Then ∃xα(x) = 0 ∨ ¬∃xα(x) = 0 holds by ∆a-LEM. In the case of ∃xα(x) = 0, take β as β ≡ 1. In the case of ¬∃xα(x) = 0, take β as β ≡ 0. Then we have ¬∃xα(x) = 0 ↔ ∃xβ(x) = 0 in both cases.

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Background Results

Definition. ∆[a → b] :≡ ∀α(α ∈ ∆a → α ∈ ∆b).

• Fact. ∆ab-LEM + ∆[a → b] implies ∆a-LEM.

Lemma. ∆a-LEM implies ∆[a → b] over EL0.

• Proof. We reason in EL0. Let α ∈ ∆a.

Then ∃xα(x) = 0 ∨ ¬∃xα(x) = 0 holds by ∆a-LEM. In the case of ∃xα(x) = 0, take β as β ≡ 1. In the case of ¬∃xα(x) = 0, take β as β ≡ 0. Then we have ¬∃xα(x) = 0 ↔ ∃xβ(x) = 0 in both cases. Proposition. ∆a-LEM is equivalent to ∆ab-LEM + ∆[a → b] over EL0.

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Background Results

Definition. ∆[a → b] :≡ ∀α(α ∈ ∆a → α ∈ ∆b).

• Fact. ∆ab-LEM + ∆[a → b] implies ∆a-LEM.

Lemma. ∆a-LEM implies ∆[a → b] over EL0.

• Proof. We reason in EL0. Let α ∈ ∆a.

Then ∃xα(x) = 0 ∨ ¬∃xα(x) = 0 holds by ∆a-LEM. In the case of ∃xα(x) = 0, take β as β ≡ 1. In the case of ¬∃xα(x) = 0, take β as β ≡ 0. Then we have ¬∃xα(x) = 0 ↔ ∃xβ(x) = 0 in both cases. Proposition. ∆a-LEM is equivalent to ∆ab-LEM + ∆[a → b] over EL0. Open Problem. Does ∆ab-LEM imply ∆[a → b] over EL?

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Background Results

• Remark. Reverse Mathathematical Hierarchy vs

Hierarchy of Logical Principles

Reverse Mathematics Phenomenon:

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Background Results

• Remark. Reverse Mathathematical Hierarchy vs

Hierarchy of Logical Principles

Reverse Mathematics Phenomenon:

Some relationship between the uniform provability in classical reverse mathematics and the hierarchy of logical principles has been recently established by [Hirst/Mummert

2011], [Dorais 2014], [Kohlenbach/F. 2015] etc.

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Background Results

• Remark. Reverse Mathathematical Hierarchy vs

Hierarchy of Logical Principles

Reverse Mathematics Phenomenon:

Some relationship between the uniform provability in classical reverse mathematics and the hierarchy of logical principles has been recently established by [Hirst/Mummert

2011], [Dorais 2014], [Kohlenbach/F. 2015] etc.

EL0 ⊢ ACA ↔ Σ0

1-LEM + Π0 1-AC0,0. (Ishihara, 2005)

EL0 ⊢ WKL ↔ Σ0

1-DML + Π0 1-AC∨ 0,0. (Ishihara, 2005)

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Background Results

• Remark. Reverse Mathathematical Hierarchy vs

Hierarchy of Logical Principles

Reverse Mathematics Phenomenon:

Some relationship between the uniform provability in classical reverse mathematics and the hierarchy of logical principles has been recently established by [Hirst/Mummert

2011], [Dorais 2014], [Kohlenbach/F. 2015] etc.

EL0 ⊢ ACA ↔ Σ0

1-LEM + Π0 1-AC0,0. (Ishihara, 2005)

EL0 ⊢ WKL ↔ Σ0

1-DML + Π0 1-AC∨ 0,0. (Ishihara, 2005)

However, the corresponding system to Π0

1-DML or

∆i-LEM is still missing.

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Background Results

References

Makoto Fujiwara, Hajime Ishihara and Takako Nemoto, Some principles weaker than Markov’s principle, Arch.

• Math. Logic, to appear.

Urlich Kohlenbach, On the disjunctive Markov principle, Studia Logica, to appear.

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

arithmetical hierarchy of the law of excluded middle and related principles. Proc. of the 19th Annual IEEE Symposium on Logic in Computer Science (LICS’04), pp. 192–201, IEEE Press, 2004. Hajime Ishihara, Markov’s principle, Church’s thesis and Lindel¨

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

1993.

Thank you for your attention!

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