[PPT] - Negative Churchs Thesis and Russian Constructivism SATO Kentaro PowerPoint Presentation

SLIDE 1

Negative Church’s Thesis and Russian Constructivism

SATO Kentaro

sato@inf.unibe.ch

Institution for Computer Science, University of Bern * partially supported by John Templeton Foundation

Negative Church’s Thesis and Russian Constructivism – p. 1

SLIDE 2

Plan of This Talk

Russian Recursive Constructive Mathematics

(vs. Classical Mathematics and

vs. Brouwer’s Intuitionism);
New Principle: Negative Church’s Thesis NCT

— representing RRCM-spirit better than CT?

Realizability model of NCT + BCP + MP

and “any f : R → R is continuous”;

Consequences of NCT + MP.

Negative Church’s Thesis and Russian Constructivism – p. 2

SLIDE 3

Plan of This Talk

Russian Recursive Constructive Mathematics

(vs. Classical Mathematics and

vs. Brouwer’s Intuitionism);
New Principle: Negative Church’s Thesis NCT

— representing RRCM-spirit better than CT?

Realizability model of NCT + BCP + MP

and “any f : R → R is continuous”;

Consequences of NCT + MP.

All the technical results are proved in:

T. Nemoto and K. Sato

“A marriage of Brouwer’s Intuitionism and Hilbert’s Finitism I: Arithmetic”, to appear in The Journal of Symbolic Logic

Negative Church’s Thesis and Russian Constructivism – p. 2

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Varieties of Mathematics

Classical Mathematics Brouwer’s Intuitionistic Mathematics

Negative Church’s Thesis and Russian Constructivism – p. 3

SLIDE 5

Varieties of Mathematics

Classical Mathematics Brouwer’s Intuitionistic Mathematics

the precursor of (serious) constructivism;

Negative Church’s Thesis and Russian Constructivism – p. 3

SLIDE 6

Varieties of Mathematics

Classical Mathematics Brouwer’s Intuitionistic Mathematics

the precursor of (serious) constructivism;
rejection of LEM (intuitionistic logic);

Negative Church’s Thesis and Russian Constructivism – p. 3

SLIDE 7

Varieties of Mathematics

Classical Mathematics Brouwer’s Intuitionistic Mathematics

the precursor of (serious) constructivism;
rejection of LEM (intuitionistic logic);
infinite sequences can be captured only via

finite fragments → Brouwer’s continuous principle;

Bar Induction.

Negative Church’s Thesis and Russian Constructivism – p. 3

SLIDE 8

Varieties of Mathematics

Classical Mathematics Brouwer’s Intuitionistic Mathematics

the precursor of (serious) constructivism;
rejection of LEM (intuitionistic logic);
infinite sequences can be captured only via

finite fragments → Brouwer’s continuous principle;

Bar Induction.

Russian Recursive Constructive Mathematics

rejection of LEM (intuitionistic logic);

Negative Church’s Thesis and Russian Constructivism – p. 3

SLIDE 9

Varieties of Mathematics

Classical Mathematics Brouwer’s Intuitionistic Mathematics

the precursor of (serious) constructivism;
rejection of LEM (intuitionistic logic);
infinite sequences can be captured only via

finite fragments → Brouwer’s continuous principle;

Bar Induction.

Russian Recursive Constructive Mathematics

rejection of LEM (intuitionistic logic);
everything is computable/recursive;

Negative Church’s Thesis and Russian Constructivism – p. 3

SLIDE 10

Varieties of Mathematics

Classical Mathematics Brouwer’s Intuitionistic Mathematics

the precursor of (serious) constructivism;
rejection of LEM (intuitionistic logic);
infinite sequences can be captured only via

finite fragments → Brouwer’s continuous principle;

Bar Induction.

Russian Recursive Constructive Mathematics

rejection of LEM (intuitionistic logic);
everything is computable/recursive;
Markov’s Principle: ¬¬∃xϕ[x] → ∃xϕ[x],

for decidable ϕ.

Negative Church’s Thesis and Russian Constructivism – p. 3

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Standard Formalization of RRCM

0. Heyting Arithmetic HA plus

Negative Church’s Thesis and Russian Constructivism – p. 4

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Standard Formalization of RRCM

0. Heyting Arithmetic HA plus
1. Markov’s Principle MP:
¬¬∃xϕ[x] → ∃xϕ[x] for a ∆0 formula ϕ[x]
r
∀x(ϕ(x) ∨ ¬ϕ[x]) → (¬¬∃xϕ[x] → ∃xϕ[x])

for any formula ϕ[x];

Negative Church’s Thesis and Russian Constructivism – p. 4

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Standard Formalization of RRCM

0. Heyting Arithmetic HA plus
1. Markov’s Principle MP:
¬¬∃xϕ[x] → ∃xϕ[x] for a ∆0 formula ϕ[x]
r
∀x(ϕ(x) ∨ ¬ϕ[x]) → (¬¬∃xϕ[x] → ∃xϕ[x])

for any formula ϕ[x];

2. and Church’s Thesis:
∀x∃yϕ[x, y] → ∃e∀x({e}(x)↓ ∧ ϕ[x, {e}(x)])
r
∀α∃e∀x({e}(x)↓ ∧ {e}(x) = α(x))

+ choice ∀x∃yϕ[x, y] → ∃α∀xϕ[x, α(x)] (in the function-based 2nd order setting).

Negative Church’s Thesis and Russian Constructivism – p. 4

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Kleene’s Number Realizability

For a formula ϕ, define another formula e nr ϕ by e nr ϕ ≡ ϕ for atomic ϕ; e nr ϕ ∧ ψ ≡ ((e)0 nr ϕ) ∧ ((e)1 nr ψ); e nr ϕ ∨ ψ ≡ ((e)0 = 0 ∧ (e)1 nr ϕ) ∨ ((e)0 = 0 ∧ (e)1 nr ψ); e nr ϕ → ψ ≡ ∀x((x nr ϕ) → ({e}(x)↓ ∧ {e}(x) nr ψ)); e nr ∃xϕ[x] ≡ (e)1 nr ϕ[(e)0]; e nr ∀xϕ[x] ≡ ∀x({e}(x)↓ ∧ {e}(x) nr ϕ[x]).

Negative Church’s Thesis and Russian Constructivism – p. 5

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Kleene’s Number Realizability

For a formula ϕ, define another formula e nr ϕ by e nr ϕ ≡ ϕ for atomic ϕ; e nr ϕ ∧ ψ ≡ ((e)0 nr ϕ) ∧ ((e)1 nr ψ); e nr ϕ ∨ ψ ≡ ((e)0 = 0 ∧ (e)1 nr ϕ) ∨ ((e)0 = 0 ∧ (e)1 nr ψ); e nr ϕ → ψ ≡ ∀x((x nr ϕ) → ({e}(x)↓ ∧ {e}(x) nr ψ)); e nr ∃xϕ[x] ≡ (e)1 nr ϕ[(e)0]; e nr ∀xϕ[x] ≡ ∀x({e}(x)↓ ∧ {e}(x) nr ϕ[x]). A straightforward formalization of BHK interpretation by “algorithms” = “partial recursive functions”.

Negative Church’s Thesis and Russian Constructivism – p. 5

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Kleene’s Number Realizability

For a formula ϕ, define another formula e nr ϕ by e nr ϕ ≡ ϕ for atomic ϕ; e nr ϕ ∧ ψ ≡ ((e)0 nr ϕ) ∧ ((e)1 nr ψ); e nr ϕ ∨ ψ ≡ ((e)0 = 0 ∧ (e)1 nr ϕ) ∨ ((e)0 = 0 ∧ (e)1 nr ψ); e nr ϕ → ψ ≡ ∀x((x nr ϕ) → ({e}(x)↓ ∧ {e}(x) nr ψ)); e nr ∃xϕ[x] ≡ (e)1 nr ϕ[(e)0]; e nr ∀xϕ[x] ≡ ∀x({e}(x)↓ ∧ {e}(x) nr ϕ[x]). A straightforward formalization of BHK interpretation by “algorithms” = “partial recursive functions”. But “BHK interpretation”=“Brouwer’s world”?

Negative Church’s Thesis and Russian Constructivism – p. 5

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Standard Formalization of BIM

Brouwer’s Intuitionistic Mathematics is formalized, within the function-based 2nd order language, by

0. Heyting Arithmetic HA plus
1. Induction: for any formula ϕ

ϕ[0] ∧ ∀x(ϕ[x] → ϕ[x+1) → ∀xϕ[x];

Negative Church’s Thesis and Russian Constructivism – p. 6

SLIDE 18

Standard Formalization of BIM

Brouwer’s Intuitionistic Mathematics is formalized, within the function-based 2nd order language, by

0. Heyting Arithmetic HA plus
1. Induction: for any formula ϕ

ϕ[0] ∧ ∀x(ϕ[x] → ϕ[x+1) → ∀xϕ[x];

2. Axiom of Choice AC: for any formula ϕ

∀x∃yϕ[x, y] → ∃α∀xϕ[x, α(x)];

Negative Church’s Thesis and Russian Constructivism – p. 6

SLIDE 19

Standard Formalization of BIM

Brouwer’s Intuitionistic Mathematics is formalized, within the function-based 2nd order language, by

0. Heyting Arithmetic HA plus
1. Induction: for any formula ϕ

ϕ[0] ∧ ∀x(ϕ[x] → ϕ[x+1) → ∀xϕ[x];

2. Axiom of Choice AC: for any formula ϕ

∀x∃yϕ[x, y] → ∃α∀xϕ[x, α(x)];

3. Brouwer Continuous Principle: for any formula ϕ

∀α∃xϕ[α, x] → ∀α∃n, x∀β(β↾n = α↾n → ϕ[β, x]);

Negative Church’s Thesis and Russian Constructivism – p. 6

SLIDE 20

Standard Formalization of BIM

Brouwer’s Intuitionistic Mathematics is formalized, within the function-based 2nd order language, by

0. Heyting Arithmetic HA plus
1. Induction: for any formula ϕ

ϕ[0] ∧ ∀x(ϕ[x] → ϕ[x+1) → ∀xϕ[x];

2. Axiom of Choice AC: for any formula ϕ

∀x∃yϕ[x, y] → ∃α∀xϕ[x, α(x)];

3. Brouwer Continuous Principle: for any formula ϕ

∀α∃xϕ[α, x] → ∀α∃n, x∀β(β↾n = α↾n → ϕ[β, x]);

4. Brouwer’s Bar Induction: for a ∆0

0 formula ϕ

∀α∃nϕ[α↾n] ∧ ∀u(∀xϕ[u∗x] → ϕ[u]) → ϕ[ ]

Negative Church’s Thesis and Russian Constructivism – p. 6

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Church’s Thesis Contradicts BCP

Church’s Thesis:

∀α∃e(α = {e}).

Brouwer’s Continuity Principle implies:

∀α∃n∀β(β↾n = α↾n → β = {e}).

Negative Church’s Thesis and Russian Constructivism – p. 7

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Church’s Thesis Contradicts BCP

Church’s Thesis:

∀α∃e(α = {e}).

Brouwer’s Continuity Principle implies:

∀α∃n∀β(β↾n = α↾n → β = {e}).

Particularly, ∃n∀β(β↾n = 0↾n → β = {e}).

Negative Church’s Thesis and Russian Constructivism – p. 7

SLIDE 23

Different Treatments of Reals

The contradiction seems to be on the difference:

In Brouwer’s idea:

infinite sequences can be captured only through finite fragments;

In Kleene’s number realizability:

infinite sequences are given (wholly) by recursive indices.

Negative Church’s Thesis and Russian Constructivism – p. 8

SLIDE 24

Different Treatments of Reals

The contradiction seems to be on the difference:

In Brouwer’s idea:

infinite sequences can be captured only through finite fragments;

In Kleene’s number realizability:

infinite sequences are given (wholly) by recursive indices. This contrast becomes clearer in

∀α-case of BHK interpretation:

given via finite fragments vs. given by an index;

Negative Church’s Thesis and Russian Constructivism – p. 8

SLIDE 25

Different Treatments of Reals

The contradiction seems to be on the difference:

In Brouwer’s idea:

infinite sequences can be captured only through finite fragments;

In Kleene’s number realizability:

infinite sequences are given (wholly) by recursive indices. This contrast becomes clearer in

∀α-case of BHK interpretation:

given via finite fragments vs. given by an index;

game semantics: Opponent’s challenges can be

seen via finite fragments vs. given by an index.

Negative Church’s Thesis and Russian Constructivism – p. 8

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Our Motivating Questions

The actual content of Church’s Thesis is “infinite sequences of numbers (or functions) are given by recursive indices” rather than merely “any infinite sequence of numbers (or function) is recursive”

Negative Church’s Thesis and Russian Constructivism – p. 9

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Our Motivating Questions

The actual content of Church’s Thesis is “infinite sequences of numbers (or functions) are given by recursive indices” rather than merely “any infinite sequence of numbers (or function) is recursive” Questions

How to formalize the latter statement?

Negative Church’s Thesis and Russian Constructivism – p. 9

SLIDE 28

Our Motivating Questions

The actual content of Church’s Thesis is “infinite sequences of numbers (or functions) are given by recursive indices” rather than merely “any infinite sequence of numbers (or function) is recursive” Questions

How to formalize the latter statement?
Does it contradict BCP?

Negative Church’s Thesis and Russian Constructivism – p. 9

SLIDE 29

Our Motivating Questions

The actual content of Church’s Thesis is “infinite sequences of numbers (or functions) are given by recursive indices” rather than merely “any infinite sequence of numbers (or function) is recursive” Questions

How to formalize the latter statement?
Does it contradict BCP?
Can it play the role of Church’s Thesis for

Russian Recursive Constructive Mathematic?

Negative Church’s Thesis and Russian Constructivism – p. 9

SLIDE 30

Our Motivating Questions

The actual content of Church’s Thesis is “infinite sequences of numbers (or functions) are given by recursive indices” rather than merely “any infinite sequence of numbers (or function) is recursive” Questions

How to formalize the latter statement?
Does it contradict BCP?
Can it play the role of Church’s Thesis for

Russian Recursive Constructive Mathematic?

Does it fit with Markov’s original idea?

Negative Church’s Thesis and Russian Constructivism – p. 9

SLIDE 31

Negative Church’s Thesis

Negative Church’s Thesis NCT: ∀α¬∀e¬∀x({e}(x)↓ ∧ α(x) = {e}(x))

Negative Church’s Thesis and Russian Constructivism – p. 10

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Negative Church’s Thesis

Negative Church’s Thesis NCT: ∀α¬∀e¬∀x({e}(x)↓ ∧ α(x) = {e}(x)) Compared to the usual Church’s Thesis CT: ∀α∃e∀x({e}(x)↓ ∧ α(x) = {e}(x)).

Negative Church’s Thesis and Russian Constructivism – p. 10

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Negative Church’s Thesis

Negative Church’s Thesis NCT: ∀α¬∀e¬∀x({e}(x)↓ ∧ α(x) = {e}(x)) Compared to the usual Church’s Thesis CT: ∀α∃e∀x({e}(x)↓ ∧ α(x) = {e}(x)). NCT is equivalent to CTN over HA + MP where ϕN :≡ ¬¬ϕ (ϕ is atomic); (ϕ ∧ ψ)N :≡ ϕN ∧ ψN; (ϕ ∨ ψ)N :≡ ¬(¬ϕN ∧ ¬ψN); (ϕ → ψ)N :≡ ϕN→ ψN; (∃xϕ[x])N :≡ ¬∀x¬(ϕ[x]N); (∀xϕ[x])N :≡ ∀x(ϕ[x]N); (∃xϕ[α])N :≡ ¬∀x¬(ϕ[α]N); (∀xϕ[α])N :≡ ∀x(ϕ[α]N).

Negative Church’s Thesis and Russian Constructivism – p. 10

SLIDE 34

Functional Algebra

Functionals can be coded by functions (by Brouwer?) (α|β)(n) = α(n∗β) where α(β) = α(β↾m)−1 for m = min{k | α(β↾k) > 0} undefined if there is no such k.

Negative Church’s Thesis and Russian Constructivism – p. 11

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Functional Algebra

Functionals can be coded by functions (by Brouwer?) (α|β)(n) = α(n∗β) where α(β) = α(β↾m)−1 for m = min{k | α(β↾k) > 0} undefined if there is no such k. Note: α(β) ∈ ω and α|β ∈ ωω (if defined) for α, β ∈ ωω.

Negative Church’s Thesis and Russian Constructivism – p. 11

SLIDE 36

Functional Algebra

Functionals can be coded by functions (by Brouwer?) (α|β)(n) = α(n∗β) where α(β) = α(β↾m)−1 for m = min{k | α(β↾k) > 0} undefined if there is no such k. Note: α(β) ∈ ω and α|β ∈ ωω (if defined) for α, β ∈ ωω. (ωω, | ) forms a Partial Combinator Algebra (PCA): ∃κ(κ|α|β ≃ α); ∃σ(σ|α|β|γ ≃ (α|γ)|(β|γ)).

Negative Church’s Thesis and Russian Constructivism – p. 11

SLIDE 37

Functional Algebra

Functionals can be coded by functions (by Brouwer?) (α|β)(n) = α(n∗β) where α(β) = α(β↾m)−1 for m = min{k | α(β↾k) > 0} undefined if there is no such k. Note: α(β) ∈ ω and α|β ∈ ωω (if defined) for α, β ∈ ωω. (ωω, | ) forms a Partial Combinator Algebra (PCA): ∃κ(κ|α|β ≃ α); ∃σ(σ|α|β|γ ≃ (α|γ)|(β|γ)). This is called Kleene second model.

Negative Church’s Thesis and Russian Constructivism – p. 11

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Functional Realizability

The realizability can be defined by this PCA: α fr ϕ ≡ ϕ for atomic ϕ; α fr ϕ ∧ ψ ≡ ((α)0 fr ϕ) ∧ ((α)1 fr ψ); α fr ϕ ∨ ψ ≡ (α(0) = 0 ∧ (α)1 nr ϕ) ∨ (α(0) = 0 ∧ (α)1 fr ψ); α fr ϕ → ψ ≡ ∀β((β fr ϕ) → ((α|β)↓ ∧ (α|β) fr ψ)); α fr ∃ξϕ[ξ] ≡ (α)1 fr ϕ[(α)0]; α fr ∀ξϕ[ξ] ≡ ∀ξ((α|ξ)↓ ∧ (α|ξ) fr ϕ[ξ]).

Negative Church’s Thesis and Russian Constructivism – p. 12

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Functional Realizability

The realizability can be defined by this PCA: α fr ϕ ≡ ϕ for atomic ϕ; α fr ϕ ∧ ψ ≡ ((α)0 fr ϕ) ∧ ((α)1 fr ψ); α fr ϕ ∨ ψ ≡ (α(0) = 0 ∧ (α)1 nr ϕ) ∨ (α(0) = 0 ∧ (α)1 fr ψ); α fr ϕ → ψ ≡ ∀β((β fr ϕ) → ((α|β)↓ ∧ (α|β) fr ψ)); α fr ∃ξϕ[ξ] ≡ (α)1 fr ϕ[(α)0]; α fr ∀ξϕ[ξ] ≡ ∀ξ((α|ξ)↓ ∧ (α|ξ) fr ϕ[ξ]). If ϕ is ess. (∨, ∃)-free (i.e., built by ∧, →, ∀ from Σ0

1),

ϕ ↔ ∃α(α fr ϕ)

Negative Church’s Thesis and Russian Constructivism – p. 12

SLIDE 40

Functional Realizability

The realizability can be defined by this PCA: α fr ϕ ≡ ϕ for atomic ϕ; α fr ϕ ∧ ψ ≡ ((α)0 fr ϕ) ∧ ((α)1 fr ψ); α fr ϕ ∨ ψ ≡ (α(0) = 0 ∧ (α)1 nr ϕ) ∨ (α(0) = 0 ∧ (α)1 fr ψ); α fr ϕ → ψ ≡ ∀β((β fr ϕ) → ((α|β)↓ ∧ (α|β) fr ψ)); α fr ∃ξϕ[ξ] ≡ (α)1 fr ϕ[(α)0]; α fr ∀ξϕ[ξ] ≡ ∀ξ((α|ξ)↓ ∧ (α|ξ) fr ϕ[ξ]). If ϕ is ess. (∨, ∃)-free (i.e., built by ∧, →, ∀ from Σ0

1),

EL0 ⊢ ϕ ↔ ∃α(α fr ϕ)

Negative Church’s Thesis and Russian Constructivism – p. 12

SLIDE 41

Elementary Arithmetic EL

iQex consists of: x+0 = x; x+(y+1) = (x+y)+1; x·0 = 0; x·(y+1) = (x·y)+x; exp(x, 0) = 1; exp(x, y+1) = exp(x, y)·x; ¬(x < x); x < y ∧ y < z → x < z; x < x+1; x < y → (x+1 < y) ∨ (x+1 = y). EL0 extends iQex by

α↾0 = ; and α↾(x+1) = (α↾x)∗α(x);
Induction for Σ0

1 formulae;

∆0

0 bounded search: for ∆0 0 formula ϕ

∃α∀x((∃y < t[x])ϕ[x, y] → α(x) < t[x] ∧ ϕ[x, α(x)]).

Negative Church’s Thesis and Russian Constructivism – p. 13

SLIDE 42

Elementary Arithmetic EL

iQex consists of: x+0 = x; x+(y+1) = (x+y)+1; x·0 = 0; x·(y+1) = (x·y)+x; exp(x, 0) = 1; exp(x, y+1) = exp(x, y)·x; ¬(x < x); x < y ∧ y < z → x < z; x < x+1; x < y → (x+1 < y) ∨ (x+1 = y). EL0 extends iQex by

α↾0 = ; and α↾(x+1) = (α↾x)∗α(x);
Induction for Σ0

1 formulae;

∆0

0 bounded search: for ∆0 0 formula ϕ

∃α∀x((∃y < t[x])ϕ[x, y] → α(x) < t[x] ∧ ϕ[x, α(x)]). Note: EL0 is consistent with CT.

Negative Church’s Thesis and Russian Constructivism – p. 13

SLIDE 43

Realizability Model

It is known that EL0 + BCP ⊢ ϕ = ⇒ EL0 ⊢ ∃α(α fr ϕ). Actually, usually this is used in the form: EL0 + BCP + WFT ⊢ ϕ = ⇒ EL0 + WKL ⊢ ∃α(α fr ϕ); EL0 + BCP + FT ⊢ ϕ = ⇒ EL0 + KL ⊢ ∃α(α fr ϕ);

Negative Church’s Thesis and Russian Constructivism – p. 14

SLIDE 44

Realizability Model

It is known that EL0 + BCP ⊢ ϕ = ⇒ EL0 ⊢ ∃α(α fr ϕ). Actually, usually this is used in the form: EL0 + BCP + WFT ⊢ ϕ = ⇒ EL0 + WKL ⊢ ∃α(α fr ϕ); EL0 + BCP + FT ⊢ ϕ = ⇒ EL0 + KL ⊢ ∃α(α fr ϕ); EL0 + BCP + BIM ⊢ ϕ = ⇒ EL0 + BI ⊢ ∃α(α fr ϕ).

Negative Church’s Thesis and Russian Constructivism – p. 14

SLIDE 45

Realizability Model

It is known that EL0 + BCP ⊢ ϕ = ⇒ EL0 ⊢ ∃α(α fr ϕ). Actually, usually this is used in the form: EL0 + BCP + WFT ⊢ ϕ = ⇒ EL0 + WKL ⊢ ∃α(α fr ϕ); EL0 + BCP + FT ⊢ ϕ = ⇒ EL0 + KL ⊢ ∃α(α fr ϕ); EL0 + BCP + BIM ⊢ ϕ = ⇒ EL0 + BI ⊢ ∃α(α fr ϕ). We can use the same technique to show EL0+BCP+NCT ⊢ ϕ = ⇒ EL0+CT ⊢ ∃α(α fr ϕ).

Negative Church’s Thesis and Russian Constructivism – p. 14

SLIDE 46

Realizability Model

It is known that EL0 + BCP ⊢ ϕ = ⇒ EL0 ⊢ ∃α(α fr ϕ). Actually, usually this is used in the form: EL0 + BCP + WFT ⊢ ϕ = ⇒ EL0 + WKL ⊢ ∃α(α fr ϕ); EL0 + BCP + FT ⊢ ϕ = ⇒ EL0 + KL ⊢ ∃α(α fr ϕ); EL0 + BCP + BIM ⊢ ϕ = ⇒ EL0 + BI ⊢ ∃α(α fr ϕ). We can use the same technique to show EL0+BCP+NCT ⊢ ϕ = ⇒ EL0+CT ⊢ ∃α(α fr ϕ). Conclusion: NCT is consistent with BCP.

Negative Church’s Thesis and Russian Constructivism – p. 14

SLIDE 47

Classical Interpreting Theories

In classical theories, we can realize MP: EL0 + MP + BCP + NCT ⊢ ϕ = ⇒ EL0 + LEM + CT ⊢ ∃α(α fr ϕ).

Negative Church’s Thesis and Russian Constructivism – p. 15

SLIDE 48

Classical Interpreting Theories

In classical theories, we can realize MP: EL0 + MP + BCP + NCT ⊢ ϕ = ⇒ EL0 + LEM + CT ⊢ ∃α(α fr ϕ). Question for Russian Constructivists: Can the functional realizability model fr (defined inside EL0 + LEM + CT) be considered as a model of Russian Recursive Constructive Mathematics?

Negative Church’s Thesis and Russian Constructivism – p. 15

SLIDE 49

Classical Interpreting Theories

In classical theories, we can realize MP: EL0 + MP + BCP + NCT ⊢ ϕ = ⇒ EL0 + LEM + CT ⊢ ∃α(α fr ϕ). Question for Russian Constructivists: Can the functional realizability model fr (defined inside EL0 + LEM + CT) be considered as a model of Russian Recursive Constructive Mathematics?

It satisfies Markov’s Principle;
It satisfies the Axiom of Choice as well as
ur formulation of “every function is recursive”;
It satisfies also Brouwer’s Continuity Principle.

Negative Church’s Thesis and Russian Constructivism – p. 15

SLIDE 50

KLS Theorem

One of the most important theorems in RRCM is Any function f : R → R is continuous where R is the real numbers (defined constructively).

Negative Church’s Thesis and Russian Constructivism – p. 16

SLIDE 51

KLS Theorem

One of the most important theorems in RRCM is Any function f : R → R is continuous where R is the real numbers (defined constructively).

The proof in EL0 + MP + ECT is involved

(due to Kreisel-Lacombe-Shoenfield);

That in EL0 + BCP is straightforward.

Negative Church’s Thesis and Russian Constructivism – p. 16

SLIDE 52

KLS Theorem

One of the most important theorems in RRCM is Any function f : R → R is continuous where R is the real numbers (defined constructively).

The proof in EL0 + MP + ECT is involved

(due to Kreisel-Lacombe-Shoenfield);

That in EL0 + BCP is straightforward.

Question for Russian Constructivists 2: Are there other important theorems in RRCM?

Negative Church’s Thesis and Russian Constructivism – p. 16

SLIDE 53

Early Intuitionism?

In early stages, Brouwer rejected lawless sequences:

lawful sequence “=” recursive function?
lawless sequence “=” non-recursive one?

Negative Church’s Thesis and Russian Constructivism – p. 17

SLIDE 54

Early Intuitionism?

In early stages, Brouwer rejected lawless sequences:

lawful sequence “=” recursive function?
lawless sequence “=” non-recursive one?

He changed his mind; indeed Kleene found a recursive finitary branching tree with no recursive infinite path nor finite height; particularly Brouwer’s FT contradicts CT.

Negative Church’s Thesis and Russian Constructivism – p. 17

SLIDE 55

Early Intuitionism?

In early stages, Brouwer rejected lawless sequences:

lawful sequence “=” recursive function?
lawless sequence “=” non-recursive one?

He changed his mind; indeed Kleene found a recursive finitary branching tree with no recursive infinite path nor finite height; particularly Brouwer’s FT contradicts CT. Question for Historians of Brouwer Can our model fr (defined in EL0 + LEM + CT) be seen as a model of this early stage of Intuitionism?

it satisfies “every sequence is recursive”;
it still satisfies BCP.

Negative Church’s Thesis and Russian Constructivism – p. 17

SLIDE 56

Kleene’s Alternative

Veldmann (2014) introduced Kleene’s Alternative KA: there is a finitary branching tree with no infinite path nor finite height.

Negative Church’s Thesis and Russian Constructivism – p. 18

SLIDE 57

Kleene’s Alternative

Veldmann (2014) introduced Kleene’s Alternative KA: there is a finitary branching tree with no infinite path nor finite height. This can be analyzed as

Bar[T] :≡ FinTr[T] ∧ ¬∃αPath[α, T];
KA is ∃T(Bar[T] ∧ ¬∃nHt[n, T]);
whereas FT is ∀T(Bar[T] → ∃nHt[n, T]).

Negative Church’s Thesis and Russian Constructivism – p. 18

SLIDE 58

Kleene’s Alternative

Veldmann (2014) introduced Kleene’s Alternative KA: there is a finitary branching tree with no infinite path nor finite height. This can be analyzed as

Bar[T] :≡ FinTr[T] ∧ ¬∃αPath[α, T];
KA is ∃T(Bar[T] ∧ ¬∃nHt[n, T]);
whereas FT is ∀T(Bar[T] → ∃nHt[n, T]).

Veldman realized that

Many equivalents of FT (e.g., int-value theorem)

are of the form ∀α(ϕ[α] → ∃βψ[α, β]) and

∃α(ϕ[α] ∧ ¬∃βψ[α, β])’s are equivalent to KA.

Negative Church’s Thesis and Russian Constructivism – p. 18

SLIDE 59

Kleene’s Alternative

Veldmann (2014) introduced Kleene’s Alternative KA: there is a finitary branching tree with no infinite path nor finite height. This can be analyzed as

Bar[T] :≡ FinTr[T] ∧ ¬∃αPath[α, T];
KA is ∃T(Bar[T] ∧ ¬∃nHt[n, T]);
whereas FT is ∀T(Bar[T] → ∃nHt[n, T]).

Veldman realized that

Many equivalents of FT (e.g., int-value theorem)

are of the form ∀α(ϕ[α] → ∃βψ[α, β]) and

∃α(ϕ[α] ∧ ¬∃βψ[α, β])’s are equivalent to KA.

Now EL0 + MP + NCT ⊢ KA.

Negative Church’s Thesis and Russian Constructivism – p. 18

SLIDE 60

Summary

1. A new axiom NCT is introduced which asserts

“every infinite sequence is recursive” without requiring “a infinite sequence is given by an index”.

Negative Church’s Thesis and Russian Constructivism – p. 19

SLIDE 61

Summary

1. A new axiom NCT is introduced which asserts

“every infinite sequence is recursive” without requiring “a infinite sequence is given by an index”.

2. NCT is consistent with BCP which requires

“infinite sequences can be accessed only via finite fragments”.

Negative Church’s Thesis and Russian Constructivism – p. 19

SLIDE 62

Summary

1. A new axiom NCT is introduced which asserts

“every infinite sequence is recursive” without requiring “a infinite sequence is given by an index”.

2. NCT is consistent with BCP which requires

“infinite sequences can be accessed only via finite fragments”.

3. There is a model of NCT + MP which satisfies

(one of) the most important theorem of RRCM: “any function f : R → R is continuous”.

Negative Church’s Thesis and Russian Constructivism – p. 19

SLIDE 63

Summary

1. A new axiom NCT is introduced which asserts

“every infinite sequence is recursive” without requiring “a infinite sequence is given by an index”.

2. NCT is consistent with BCP which requires

“infinite sequences can be accessed only via finite fragments”.

3. There is a model of NCT + MP which satisfies

(one of) the most important theorem of RRCM: “any function f : R → R is continuous”.

4. We already knew many mathematical

consequences of NCT + MP via Veldman’s KA.

Negative Church’s Thesis and Russian Constructivism – p. 19

SLIDE 64

Summary

1. A new axiom NCT is introduced which asserts

“every infinite sequence is recursive” without requiring “a infinite sequence is given by an index”.

2. NCT is consistent with BCP which requires

“infinite sequences can be accessed only via finite fragments”.

3. There is a model of NCT + MP which satisfies

(one of) the most important theorem of RRCM: “any function f : R → R is continuous”.

4. We already knew many mathematical

consequences of NCT + MP via Veldman’s KA.

5. This might model Brouwer’s idea before the

acceptance of lawless sequence.

Negative Church’s Thesis and Russian Constructivism – p. 19

SLIDE 65

References

The technical results in the talk will be published in

T. Nemoto and K. Sato

“A marriage of Brouwer’s Intuitionism and Hilbert’s Finitism I: Arithmetic”, to appear in The Journal of Symbolic Logic.

Negative Church’s Thesis and Russian Constructivism – p. 20

SLIDE 66

References

The technical results in the talk will be published in

T. Nemoto and K. Sato

“A marriage of Brouwer’s Intuitionism and Hilbert’s Finitism I: Arithmetic”, to appear in The Journal of Symbolic Logic. Other references:

A. Troelstra and D. van Dalen,

“ConstructivisminMathematics” Volumes I, Elsevier, 1988.

W. Veldman,

“Brouwer’s fan theorem as an axiom and as a contrast to Kleene’s alternative”, Archive for Mathematical Logic 53 pp.621-693, 2014.

Negative Church’s Thesis and Russian Constructivism – p. 20