Nonstandard Analysis: a new way to compute Sam Sanders 1 Model - - PowerPoint PPT Presentation

Nonstandard Analysis: a new way to compute

Sam Sanders1

Model Theory and Proof Theory of Arithmetic

A Memorial Conference in Honour of H. Kotlarski and Z. Ratajczyk

July 25, 2012

1This research is generously supported by the John Templeton Foundation.

Take-home message

Take-home message

In Nonstandard Analysis, an algorithm is any object whose definition is independent of the choice of infinitesimal (Ω-invariance).

Take-home message

In Nonstandard Analysis, an algorithm is any object whose definition is independent of the choice of infinitesimal (Ω-invariance).

More technically, we define a translation between Constructive Analysis (BISH) and Nonstandard Analysis (NSA):

Take-home message

In Nonstandard Analysis, an algorithm is any object whose definition is independent of the choice of infinitesimal (Ω-invariance).

More technically, we define a translation between Constructive Analysis (BISH) and Nonstandard Analysis (NSA):

(Proof and Algorithm) in BISH = (Transfer and Ω-invariance) in NSA

Take-home message

In Nonstandard Analysis, an algorithm is any object whose definition is independent of the choice of infinitesimal (Ω-invariance).

More technically, we define a translation between Constructive Analysis (BISH) and Nonstandard Analysis (NSA):

(Proof and Algorithm) in BISH = (Transfer and Ω-invariance) in NSA Most results from CRM (= RM based on BISH) translate to NSA under a natural translation B.

Son of a. . .

INT CLASS RUSS

BISH

Son of a. . .

INT CLASS RUSS

BISH ≈Math. programmable on TM

Son of a. . .

INT CLASS RUSS

BISH ≈Math. programmable on TM LEM (P ∨ ¬P) Classical Math. ≈

Son of a. . .

INT CLASS RUSS

BISH ≈Math. programmable on TM LEM (P ∨ ¬P) Classical Math. ≈

(proof by contradiction)

Son of a. . .

INT CLASS RUSS

BISH ≈Math. programmable on TM LEM (P ∨ ¬P) Classical Math. ≈

(proof by contradiction)

≈Brouwer’s Intuitionistic Math.

Son of a. . .

INT CLASS RUSS

BISH ≈Math. programmable on TM LEM (P ∨ ¬P) Classical Math. ≈

(proof by contradiction)

≈Brouwer’s Intuitionistic Math. MP

(¬¬P → P)

Son of a. . .

INT CLASS RUSS

BISH ≈Math. programmable on TM LEM (P ∨ ¬P) Classical Math. ≈

(proof by contradiction)

≈Brouwer’s Intuitionistic Math. MP

(¬¬P → P)

CPF

Son of a. . .

INT CLASS RUSS

BISH ≈Math. programmable on TM LEM (P ∨ ¬P) Classical Math. ≈

(proof by contradiction)

≈Brouwer’s Intuitionistic Math. MP

(¬¬P → P)

CPF CONT

Son of a. . .

INT CLASS RUSS

BISH ≈Math. programmable on TM LEM (P ∨ ¬P) Classical Math. ≈

(proof by contradiction)

≈Brouwer’s Intuitionistic Math. MP

(¬¬P → P)

CPF CONT FAN

Son of a. . .

INT CLASS RUSS

BISH ≈Math. programmable on TM LEM (P ∨ ¬P) Classical Math. ≈

(proof by contradiction)

≈Brouwer’s Intuitionistic Math. MP

(¬¬P → P)

CPF CONT FAN CONT′

Son of a. . .

Errett Bishop’s Constructive Analysis (also ‘BISH’) is a constructive redevelopment of Mathematics, consistent with CLASS, RUSS and INT.

INT CLASS RUSS

BISH ≈Math. programmable on TM LEM (P ∨ ¬P) Classical Math. ≈ (proof by contradiction) ≈Brouwer’s Intuitionistic Math. MP

(¬¬P → P)

CPF CONT FAN CONT′

Algorithm and Proof in Constructive Analysis

Errett Bishop’s Constructive Analysis (BISH) is a constructive redevelopment of Mathematics, where algorithm and proof are central.

Algorithm and Proof in Constructive Analysis

Errett Bishop’s Constructive Analysis (BISH) is a constructive redevelopment of Mathematics, where algorithm and proof are central.

Definition (Logical connectives in BISH: BHK)

Algorithm and Proof in Constructive Analysis

Errett Bishop’s Constructive Analysis (BISH) is a constructive redevelopment of Mathematics, where algorithm and proof are central.

Definition (Logical connectives in BISH: BHK)

1

P ∨ Q: we have an algorithm that outputs either P or Q, together with a proof of the chosen disjunct.

Algorithm and Proof in Constructive Analysis

Errett Bishop’s Constructive Analysis (BISH) is a constructive redevelopment of Mathematics, where algorithm and proof are central.

Definition (Logical connectives in BISH: BHK)

1

P ∨ Q: we have an algorithm that outputs either P or Q, together with a proof of the chosen disjunct.

2

P ∧ Q: we have both a proof of P and a proof of Q.

Algorithm and Proof in Constructive Analysis

Errett Bishop’s Constructive Analysis (BISH) is a constructive redevelopment of Mathematics, where algorithm and proof are central.

Definition (Logical connectives in BISH: BHK)

1

P ∨ Q: we have an algorithm that outputs either P or Q, together with a proof of the chosen disjunct.

2

P ∧ Q: we have both a proof of P and a proof of Q.

3

P → Q: by means of an algorithm we can convert any proof of P into a proof of Q.

Algorithm and Proof in Constructive Analysis

Errett Bishop’s Constructive Analysis (BISH) is a constructive redevelopment of Mathematics, where algorithm and proof are central.

Definition (Logical connectives in BISH: BHK)

1

P ∨ Q: we have an algorithm that outputs either P or Q, together with a proof of the chosen disjunct.

2

P ∧ Q: we have both a proof of P and a proof of Q.

3

P → Q: by means of an algorithm we can convert any proof of P into a proof of Q.

4

¬P ≡ P → (0 = 1).

Algorithm and Proof in Constructive Analysis

Errett Bishop’s Constructive Analysis (BISH) is a constructive redevelopment of Mathematics, where algorithm and proof are central.

Definition (Logical connectives in BISH: BHK)

1

P ∨ Q: we have an algorithm that outputs either P or Q, together with a proof of the chosen disjunct.

2

P ∧ Q: we have both a proof of P and a proof of Q.

3

P → Q: by means of an algorithm we can convert any proof of P into a proof of Q.

4

¬P ≡ P → (0 = 1).

5

(∃x)P(x): an algorithm computes an object x0 such that P(x0)

Algorithm and Proof in Constructive Analysis

Errett Bishop’s Constructive Analysis (BISH) is a constructive redevelopment of Mathematics, where algorithm and proof are central.

Definition (Logical connectives in BISH: BHK)

1

P ∨ Q: we have an algorithm that outputs either P or Q, together with a proof of the chosen disjunct.

2

P ∧ Q: we have both a proof of P and a proof of Q.

3

P → Q: by means of an algorithm we can convert any proof of P into a proof of Q.

4

¬P ≡ P → (0 = 1).

5

(∃x)P(x): an algorithm computes an object x0 such that P(x0)

6

(∀x ∈ A)P(x): for all x, x ∈ A → P(x).

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

From BISH to NSA

In BISH, proof and algorithm are central.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

From BISH to NSA

In BISH, proof and algorithm are central.

We define a system of Nonstandard Analyis NSA where Transfer (T) and Ω-invariant procedure play the same role.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

From BISH to NSA

In BISH, proof and algorithm are central.

We define a system of Nonstandard Analyis NSA where Transfer (T) and Ω-invariant procedure play the same role. NSA is similar to ∗RCA0, a nonstandard version of RCA0 (Keisler & Yokoyama).

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

From BISH to NSA

In BISH, proof and algorithm are central.

We define a system of Nonstandard Analyis NSA where Transfer (T) and Ω-invariant procedure play the same role. NSA is similar to ∗RCA0, a nonstandard version of RCA0 (Keisler & Yokoyama).

Three important features:

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

From BISH to NSA

In BISH, proof and algorithm are central.

We define a system of Nonstandard Analyis NSA where Transfer (T) and Ω-invariant procedure play the same role. NSA is similar to ∗RCA0, a nonstandard version of RCA0 (Keisler & Yokoyama).

Three important features:

1 No Transfer Principle, except for ∆0.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

From BISH to NSA

In BISH, proof and algorithm are central.

We define a system of Nonstandard Analyis NSA where Transfer (T) and Ω-invariant procedure play the same role. NSA is similar to ∗RCA0, a nonstandard version of RCA0 (Keisler & Yokoyama).

Three important features:

1 No Transfer Principle, except for ∆0. 2 No ∆0

1-CA, but Ω-CA. (CA for Ω-invariant formulas)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

From BISH to NSA

In BISH, proof and algorithm are central.

We define a system of Nonstandard Analyis NSA where Transfer (T) and Ω-invariant procedure play the same role. NSA is similar to ∗RCA0, a nonstandard version of RCA0 (Keisler & Yokoyama).

Three important features:

1 No Transfer Principle, except for ∆0. 2 No ∆0

1-CA, but Ω-CA. (CA for Ω-invariant formulas)

3 Levels of infinity (Stratified NSA).

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Feature 3: Stratified Nonstandard Analysis

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Feature 3: Stratified Nonstandard Analysis

The usual picture of ∗N:

∗N, the hypernatural numbers

• . . .

ω2 . . . ωk . . .

✲

ω1 0 1 . . .

• N, the natural/finite numbers
• Ω = ∗N \ N, the infinite numbers

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Feature 3: Stratified Nonstandard Analysis

The usual picture of ∗N:

∗N, the hypernatural numbers

• . . .

ω2 . . . ωk . . .

✲

ω1 0 1 . . .

• N, the natural/finite numbers
• Ω = ∗N \ N, the infinite numbers

In NSA, the infinite numbers are split into ‘small’ and ‘large’.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Feature 3: Stratified Nonstandard Analysis

The usual picture of ∗N:

∗N, the hypernatural numbers

• . . .

ω2 . . . ωk . . .

✲

ω1 0 1 . . .

• N, the natural/finite numbers
• Ω = ∗N \ N, the infinite numbers

In NSA, the infinite numbers are split into ‘small’ and ‘large’.

the small infinite numbers

• the large infinite numbers

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Feature 3: Stratified Nonstandard Analysis

The usual picture of ∗N:

∗N, the hypernatural numbers

• . . .

ω2 . . . ωk . . .

✲

ω1 0 1 . . .

• N, the natural/finite numbers
• Ω = ∗N \ N, the infinite numbers

In NSA, the infinite numbers are split into ‘small’ and ‘large’.

N1=N ∪ the small infinite numbers

• Ω1=∗N\N1, the large infinite numbers

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Feature 2: Ω-invariance

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Feature 2: Ω-invariance

Ω-invariance ≈ algorithm ≈ finite procedure

Definition (Ω-invariance)

For ψ(n, m) ∈ ∆0 and ω ∈ Ω, the formula ψ(n, ω) is Ω-invariant if

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Feature 2: Ω-invariance

Ω-invariance ≈ algorithm ≈ finite procedure

Definition (Ω-invariance)

For ψ(n, m) ∈ ∆0 and ω ∈ Ω, the formula ψ(n, ω) is Ω-invariant if (∀n ∈ N)(∀ω′ ∈ Ω)[ψ(n, ω) ↔ ψ(n, ω′)].

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Feature 2: Ω-invariance

Ω-invariance ≈ algorithm ≈ finite procedure

Definition (Ω-invariance)

For ψ(n, m) ∈ ∆0 and ω ∈ Ω, the formula ψ(n, ω) is Ω-invariant if (∀n ∈ N)(∀ω′ ∈ Ω)[ψ(n, ω) ↔ ψ(n, ω′)].

Note that ψ(n, ω) depends on ω ∈ Ω, but not on the choice of ω ∈ Ω.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Feature 2: Ω-invariance

Ω-invariance ≈ algorithm ≈ finite procedure

Definition (Ω-invariance)

For ψ(n, m) ∈ ∆0 and ω ∈ Ω, the formula ψ(n, ω) is Ω-invariant if (∀n ∈ N)(∀ω′ ∈ Ω)[ψ(n, ω) ↔ ψ(n, ω′)].

Note that ψ(n, ω) depends on ω ∈ Ω, but not on the choice of ω ∈ Ω.

NSA has Ω-CA instead of ∆1-CA.

Principle (Ω-CA)

For all Ω-invariant ψ(n, ω), we have (∃X ⊂ N)(∀n ∈ N)(n ∈ X ↔ ψ(n, ω)).

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

BISH (based on BHK)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

BISH (based on BHK) NSA (based on CL)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

BISH (based on BHK) NSA (based on CL)

Central: algorithm and proof

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

BISH (based on BHK) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

BISH (based on BHK) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and Transfer (T)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

BISH (based on BHK) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and Transfer (T)

A V B:

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

BISH (based on BHK) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and Transfer (T)

A V B: There is Ω-invariant ψ( x, ω) s.t. ψ( x, ω) → [A( x) ∧ [A( x) ∈ T]] ∧ ¬ψ( x, ω) → [B( x) ∧ [B( x) ∈ T]]

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

BISH (based on BHK) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and Transfer (T)

A V B: There is Ω-invariant ψ( x, ω) s.t. ψ( x, ω) → [A( x) ∧ [A( x) ∈ T]] ∧ ¬ψ( x, ω) → [B( x) ∧ [B( x) ∈ T]] A → B: an algo converts a proof of A to a proof of B

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

BISH (based on BHK) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and Transfer (T)

A V B: There is Ω-invariant ψ( x, ω) s.t. ψ( x, ω) → [A( x) ∧ [A( x) ∈ T]] ∧ ¬ψ( x, ω) → [B( x) ∧ [B( x) ∈ T]] A → B: an algo converts a proof of A to a proof of B A ⇛ B:

• A ∧ [A ∈ T]
• →
• B ∧ [B ∈ T]

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

BISH (based on BHK) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and Transfer (T)

A V B: There is Ω-invariant ψ( x, ω) s.t. ψ( x, ω) → [A( x) ∧ [A( x) ∈ T]] ∧ ¬ψ( x, ω) → [B( x) ∧ [B( x) ∈ T]] A → B: an algo converts a proof of A to a proof of B A ⇛ B:

• A ∧ [A ∈ T]
• →
• B ∧ [B ∈ T]
• ‘A ∈ T’ means ‘A satisfies Transfer’.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

BISH (based on BHK) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and Transfer (T)

A V B: There is Ω-invariant ψ( x, ω) s.t. ψ( x, ω) → [A( x) ∧ [A( x) ∈ T]] ∧ ¬ψ( x, ω) → [B( x) ∧ [B( x) ∈ T]] A → B: an algo converts a proof of A to a proof of B A ⇛ B:

• A ∧ [A ∈ T]
• →
• B ∧ [B ∈ T]
• ‘A ∈ T’ means ‘A satisfies Transfer’.

E.g. ‘(∀n ∈ N)ϕ(n) ∈ T’ is [(∀n ∈ N)ϕ(n) → (∀n ∈ ∗N)ϕ(n)]

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

BISH (based on BHK) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and Transfer (T)

A V B: There is Ω-invariant ψ( x, ω) s.t. ψ( x, ω) → [A( x) ∧ [A( x) ∈ T]] ∧ ¬ψ( x, ω) → [B( x) ∧ [B( x) ∈ T]] A → B: an algo converts a proof of A to a proof of B A ⇛ B:

• A ∧ [A ∈ T]
• →
• B ∧ [B ∈ T]
• ‘A ∈ T’ means ‘A satisfies Transfer’.

E.g. ‘(∀n ∈ N)ϕ(n) ∈ T’ is [(∀n ∈ N)ϕ(n) → (∀n ∈ ∗N)ϕ(n)] E.g. ‘(∃n ∈ ∗N)ϕ(n) ∈ T’ is [(∃n ∈ ∗N)ϕ(n) → (∃n ∈ N1)ϕ(n)]

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

BISH (based on BHK) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and Transfer (T)

A V B: There is Ω-invariant ψ( x, ω) s.t. ψ( x, ω) → [A( x) ∧ [A( x) ∈ T]] ∧ ¬ψ( x, ω) → [B( x) ∧ [B( x) ∈ T]] A → B: an algo converts a proof of A to a proof of B A ⇛ B:

• A ∧ [A ∈ T]
• →
• B ∧ [B ∈ T]
• ¬A: A → (0 = 1)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

BISH (based on BHK) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and Transfer (T)

A V B: There is Ω-invariant ψ( x, ω) s.t. ψ( x, ω) → [A( x) ∧ [A( x) ∈ T]] ∧ ¬ψ( x, ω) → [B( x) ∧ [B( x) ∈ T]] A → B: an algo converts a proof of A to a proof of B A ⇛ B:

• A ∧ [A ∈ T]
• →
• B ∧ [B ∈ T]
• ¬A: A → (0 = 1)

∼A: A ⇛ (0 = 1)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

BISH (based on BHK) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and Transfer (T)

A V B: There is Ω-invariant ψ( x, ω) s.t. ψ( x, ω) → [A( x) ∧ [A( x) ∈ T]] ∧ ¬ψ( x, ω) → [B( x) ∧ [B( x) ∈ T]] A → B: an algo converts a proof of A to a proof of B A ⇛ B:

• A ∧ [A ∈ T]
• →
• B ∧ [B ∈ T]
• ¬A: A → (0 = 1)

∼A: A ⇛ (0 = 1)

(∃x)A(x): an algo computes x0 such that A(x0)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

BISH (based on BHK) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and Transfer (T)

A V B: There is Ω-invariant ψ( x, ω) s.t. ψ( x, ω) → [A( x) ∧ [A( x) ∈ T]] ∧ ¬ψ( x, ω) → [B( x) ∧ [B( x) ∈ T]] A → B: an algo converts a proof of A to a proof of B A ⇛ B:

• A ∧ [A ∈ T]
• →
• B ∧ [B ∈ T]
• ¬A: A → (0 = 1)

∼A: A ⇛ (0 = 1)

(∃x)A(x): an algo computes x0 such that A(x0) (∃x)A(x): “an Ω-inv. proc. computes x0 such that A(x0)”

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

BISH (based on BHK) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and Transfer (T)

A V B: There is Ω-invariant ψ( x, ω) s.t. ψ( x, ω) → [A( x) ∧ [A( x) ∈ T]] ∧ ¬ψ( x, ω) → [B( x) ∧ [B( x) ∈ T]] A → B: an algo converts a proof of A to a proof of B A ⇛ B:

• A ∧ [A ∈ T]
• →
• B ∧ [B ∈ T]
• ¬A: A → (0 = 1)

∼A: A ⇛ (0 = 1)

(∃x)A(x): an algo computes x0 such that A(x0) (∃x)A(x): “an Ω-inv. proc. computes x0 such that A(x0)”

∼[(∀n ∈ N)A(n)]

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

BISH (based on BHK) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and Transfer (T)

A V B: There is Ω-invariant ψ( x, ω) s.t. ψ( x, ω) → [A( x) ∧ [A( x) ∈ T]] ∧ ¬ψ( x, ω) → [B( x) ∧ [B( x) ∈ T]] A → B: an algo converts a proof of A to a proof of B A ⇛ B:

• A ∧ [A ∈ T]
• →
• B ∧ [B ∈ T]
• ¬A: A → (0 = 1)

∼A: A ⇛ (0 = 1)

(∃x)A(x): an algo computes x0 such that A(x0) (∃x)A(x): “an Ω-inv. proc. computes x0 such that A(x0)”

∼[(∀n ∈ N)A(n)] ≡ (∃n ∈ N1)∼A(n) WEAKER than (∃n ∈ N)∼A(n).

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

BISH (based on BHK) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and Transfer (T)

A V B: There is Ω-invariant ψ( x, ω) s.t. ψ( x, ω) → [A( x) ∧ [A( x) ∈ T]] ∧ ¬ψ( x, ω) → [B( x) ∧ [B( x) ∈ T]] A → B: an algo converts a proof of A to a proof of B A ⇛ B:

• A ∧ [A ∈ T]
• →
• B ∧ [B ∈ T]
• ¬A: A → (0 = 1)

∼A: A ⇛ (0 = 1)

(∃x)A(x): an algo computes x0 such that A(x0) (∃x)A(x): “an Ω-inv. proc. computes x0 such that A(x0)”

∼[(∀n ∈ N)A(n)] ≡ (∃n ∈ N1)∼A(n) WEAKER than (∃n ∈ N)∼A(n). ¬[(∀n ∈ N)A(n)] is WEAKER than (∃n ∈ N)¬A(n).

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

BISH (based on BHK) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and Transfer (T)

A V B: There is Ω-invariant ψ( x, ω) s.t. ψ( x, ω) → [A( x) ∧ [A( x) ∈ T]] ∧ ¬ψ( x, ω) → [B( x) ∧ [B( x) ∈ T]] A → B: an algo converts a proof of A to a proof of B A ⇛ B:

• A ∧ [A ∈ T]
• →
• B ∧ [B ∈ T]
• ¬A: A → (0 = 1)

∼A: A ⇛ (0 = 1)

(∃x)A(x): an algo computes x0 such that A(x0) (∃x)A(x): “an Ω-inv. proc. computes x0 such that A(x0)”

We know: If BISH ⊢ X then X→LPO, LLPO, MP, . . . (princ. rejected in BISH)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

BISH (based on BHK) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and Transfer (T)

A V B: There is Ω-invariant ψ( x, ω) s.t. ψ( x, ω) → [A( x) ∧ [A( x) ∈ T]] ∧ ¬ψ( x, ω) → [B( x) ∧ [B( x) ∈ T]] A → B: an algo converts a proof of A to a proof of B A ⇛ B:

• A ∧ [A ∈ T]
• →
• B ∧ [B ∈ T]
• ¬A: A → (0 = 1)

∼A: A ⇛ (0 = 1)

(∃x)A(x): an algo computes x0 such that A(x0) (∃x)A(x): “an Ω-inv. proc. computes x0 such that A(x0)”

We know: If BISH ⊢ X then X→LPO, LLPO, MP, . . . (princ. rejected in BISH) We show: If NSA ⊢ Y then Y ⇛ LPO, LLPO, MP, . . .

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

The translation B from BISH to NSA

BISH (based on BHK) NSA (based on CL)

Central: algorithm and proof A ∨ B:

an algo yields a proof of A or of B Central: Ω-invariance and Transfer (T)

A V B: There is Ω-invariant ψ( x, ω) s.t. ψ( x, ω) → [A( x) ∧ [A( x) ∈ T]] ∧ ¬ψ( x, ω) → [B( x) ∧ [B( x) ∈ T]] A → B: an algo converts a proof of A to a proof of B A ⇛ B:

• A ∧ [A ∈ T]
• →
• B ∧ [B ∈ T]
• ¬A: A → (0 = 1)

∼A: A ⇛ (0 = 1)

(∃x)A(x): an algo computes x0 such that A(x0) (∃x)A(x): “an Ω-inv. proc. computes x0 such that A(x0)”

We know: If BISH ⊢ X then X→LPO, LLPO, MP, . . . (princ. rejected in BISH) We show: If NSA ⊢ Y then Y ⇛ LPO, LLPO, MP, . . . (e.g. LPO is B(LPO), unprovable in NSA

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

• LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

• LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
• MCT: monotone convergence thm

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

• LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
• MCT: monotone convergence thm
• CIT: Cantor intersection thm

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

• LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
• MCT: monotone convergence thm
• CIT: Cantor intersection thm

non-Ω-invariant

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

• LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
• MCT: monotone convergence thm
• CIT: Cantor intersection thm

non-Ω-invariant LPO: For P ∈ Σ1, P V ∼P ⇚ ⇛

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

• LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
• MCT: monotone convergence thm
• CIT: Cantor intersection thm

non-Ω-invariant LPO: For P ∈ Σ1, P V ∼P ⇚ ⇛ LPR: (∀x ∈ R)(x > 0 V ∼(x > 0)) ⇚ ⇛

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

• LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
• MCT: monotone convergence thm
• CIT: Cantor intersection thm

non-Ω-invariant LPO: For P ∈ Σ1, P V ∼P ⇚ ⇛ LPR: (∀x ∈ R)(x > 0 V ∼(x > 0)) ⇚ ⇛ MCT: monotone convergence thm ⇚ ⇛

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

• LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
• MCT: monotone convergence thm
• CIT: Cantor intersection thm

non-Ω-invariant LPO: For P ∈ Σ1, P V ∼P ⇚ ⇛ LPR: (∀x ∈ R)(x > 0 V ∼(x > 0)) ⇚ ⇛ MCT: monotone convergence thm ⇚ ⇛ CIT: Cantor intersection thm

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

• LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
• MCT: monotone convergence thm
• CIT: Cantor intersection thm

non-Ω-invariant LPO: For P ∈ Σ1, P V ∼P ⇚ ⇛ LPR: (∀x ∈ R)(x > 0 V ∼(x > 0)) ⇚ ⇛ MCT: monotone convergence thm ⇚ ⇛ CIT: Cantor intersection thm (limit computed by algo)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

• LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
• MCT: monotone convergence thm
• CIT: Cantor intersection thm

non-Ω-invariant LPO: For P ∈ Σ1, P V ∼P ⇚ ⇛ LPR: (∀x ∈ R)(x > 0 V ∼(x > 0)) ⇚ ⇛ MCT: monotone convergence thm ⇚ ⇛ CIT: Cantor intersection thm (limit computed by algo) (limit computed by Ω-inv. proc.)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

• LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
• MCT: monotone convergence thm
• CIT: Cantor intersection thm

non-Ω-invariant LPO: For P ∈ Σ1, P V ∼P ⇚ ⇛ LPR: (∀x ∈ R)(x > 0 V ∼(x > 0)) ⇚ ⇛ MCT: monotone convergence thm ⇚ ⇛ CIT: Cantor intersection thm (limit computed by algo) (limit computed by Ω-inv. proc.) (point in intersection computed by algo)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

• LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
• MCT: monotone convergence thm
• CIT: Cantor intersection thm

non-Ω-invariant LPO: For P ∈ Σ1, P V ∼P ⇚ ⇛ LPR: (∀x ∈ R)(x > 0 V ∼(x > 0)) ⇚ ⇛ MCT: monotone convergence thm ⇚ ⇛ CIT: Cantor intersection thm (limit computed by algo) (limit computed by Ω-inv. proc.) (point in intersection computed by algo) (point in intersection computed by Ω-inv. proc.)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

• LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
• MCT: monotone convergence thm
• CIT: Cantor intersection thm

non-Ω-invariant LPO: For P ∈ Σ1, P V ∼P ⇚ ⇛ LPR: (∀x ∈ R)(x > 0 V ∼(x > 0)) ⇚ ⇛ MCT: monotone convergence thm ⇚ ⇛ CIT: Cantor intersection thm (limit computed by algo) (limit computed by Ω-inv. proc.) ⇚ ⇛ Universal Transfer: For all ϕ ∈ ∆0 (∀n ∈ N)ϕ(n) → (∀n ∈ ∗N)ϕ(n)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LPO: For P ∈ Σ1, P ∨ ¬P

• LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
• MCT: monotone convergence thm
• CIT: Cantor intersection thm

non-Ω-invariant LPO: For P ∈ Σ1, P V ∼P ⇚ ⇛ LPR: (∀x ∈ R)(x > 0 V ∼(x > 0)) ⇚ ⇛ MCT: monotone convergence thm ⇚ ⇛ CIT: Cantor intersection thm (limit computed by algo) (limit computed by Ω-inv. proc.) ⇚ ⇛ Universal Transfer: For all ϕ ∈ ∆0 (∀n ∈ N)ϕ(n) → (∀n ∈ ∗N)ϕ(n) NSA does prove (∀δ ∈ R)

• δ > 0 ⇛ (x > 0) V(x < δ)
• .

BISH does prove (∀δ ∈ R)

• δ > 0 → (x > 0) ∨ (x < δ)
• .

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B II

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B II

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B II

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B II

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

• LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B II

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

• LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
• NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B II

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

• LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
• NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

• IVT: Intermediate value theorem

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B II

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

• LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
• NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

• IVT: Intermediate value theorem

non-Ω-invariant

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B II

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

• LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
• NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

• IVT: Intermediate value theorem

non-Ω-invariant LLPO

For P, Q ∈ Σ1, ∼(P ∧ Q) ⇛ ∼P V ∼Q

⇚ ⇛

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B II

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

• LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
• NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

• IVT: Intermediate value theorem

non-Ω-invariant LLPO

For P, Q ∈ Σ1, ∼(P ∧ Q) ⇛ ∼P V ∼Q

⇚ ⇛ LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0) ⇚ ⇛

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B II

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

• LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
• NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

• IVT: Intermediate value theorem

non-Ω-invariant LLPO

For P, Q ∈ Σ1, ∼(P ∧ Q) ⇛ ∼P V ∼Q

⇚ ⇛ LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0) ⇚ ⇛ NIL

(∀x, y ∈ R)(xy = 0 ⇛ x = 0 V y = 0)

⇚ ⇛

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B II

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

• LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
• NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

• IVT: Intermediate value theorem

non-Ω-invariant LLPO

For P, Q ∈ Σ1, ∼(P ∧ Q) ⇛ ∼P V ∼Q

⇚ ⇛ LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0) ⇚ ⇛ NIL

(∀x, y ∈ R)(xy = 0 ⇛ x = 0 V y = 0)

⇚ ⇛ IVT: Intermediate value theorem

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B II

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

• LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
• NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

• IVT: Intermediate value theorem

non-Ω-invariant LLPO

For P, Q ∈ Σ1, ∼(P ∧ Q) ⇛ ∼P V ∼Q

⇚ ⇛ LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0) ⇚ ⇛ NIL

(∀x, y ∈ R)(xy = 0 ⇛ x = 0 V y = 0)

⇚ ⇛ IVT: Intermediate value theorem (int. value computed by algo)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B II

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

• LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
• NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

• IVT: Intermediate value theorem

non-Ω-invariant LLPO

For P, Q ∈ Σ1, ∼(P ∧ Q) ⇛ ∼P V ∼Q

⇚ ⇛ LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0) ⇚ ⇛ NIL

(∀x, y ∈ R)(xy = 0 ⇛ x = 0 V y = 0)

⇚ ⇛ IVT: Intermediate value theorem (int. value computed by algo) (int. value computed by Ω-inv. proc.)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B II

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

• LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
• NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

• IVT: Intermediate value theorem

non-Ω-invariant LLPO

For P, Q ∈ Σ1, ∼(P ∧ Q) ⇛ ∼P V ∼Q

⇚ ⇛ LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0) ⇚ ⇛ NIL

(∀x, y ∈ R)(xy = 0 ⇛ x = 0 V y = 0)

⇚ ⇛ IVT: Intermediate value theorem (int. value computed by algo) (int. value computed by Ω-inv. proc.)

• WKL

⇚ ⇛ WKL

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B II

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

• LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
• NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

• IVT: Intermediate value theorem

non-Ω-invariant LLPO

For P, Q ∈ Σ1, ∼(P ∧ Q) ⇛ ∼P V ∼Q

⇚ ⇛ LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0) ⇚ ⇛ NIL

(∀x, y ∈ R)(xy = 0 ⇛ x = 0 V y = 0)

⇚ ⇛ IVT: Intermediate value theorem (int. value computed by algo) (int. value computed by Ω-inv. proc.)

• WKL

⇚ ⇛ WKL ⇚ ⇛ ∨-Transfer

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B II

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

• LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
• NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

• IVT: Intermediate value theorem

non-Ω-invariant LLPO

For P, Q ∈ Σ1, ∼(P ∧ Q) ⇛ ∼P V ∼Q

⇚ ⇛ LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0) ⇚ ⇛ NIL

(∀x, y ∈ R)(xy = 0 ⇛ x = 0 V y = 0)

⇚ ⇛ IVT: Intermediate value theorem (int. value computed by algo) (int. value computed by Ω-inv. proc.)

• WKL

⇚ ⇛ WKL ⇚ ⇛ ∨-Transfer Axioms of R: ¬(x > 0 ∧ x < 0)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B II

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic LLPO

For P, Q ∈ Σ1, ¬(P ∧ Q) → ¬P ∨ ¬Q

• LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
• NIL

(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)

• IVT: Intermediate value theorem

non-Ω-invariant LLPO

For P, Q ∈ Σ1, ∼(P ∧ Q) ⇛ ∼P V ∼Q

⇚ ⇛ LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0) ⇚ ⇛ NIL

(∀x, y ∈ R)(xy = 0 ⇛ x = 0 V y = 0)

⇚ ⇛ IVT: Intermediate value theorem (int. value computed by algo) (int. value computed by Ω-inv. proc.)

• WKL

⇚ ⇛ WKL ⇚ ⇛ ∨-Transfer Axioms of R: ¬(x > 0 ∧ x < 0) Axioms of R: ∼(x > 0 ∧ x < 0)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B III

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B III

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B III

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B III

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

• MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B III

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

• MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
• EXT: the extensionality theorem

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B III

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

• MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
• EXT: the extensionality theorem

non-Ω-invariant

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B III

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

• MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
• EXT: the extensionality theorem

non-Ω-invariant MP: For P ∈ Σ1, ∼∼P ⇛ P ⇚ ⇛

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B III

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

• MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
• EXT: the extensionality theorem

non-Ω-invariant MP: For P ∈ Σ1, ∼∼P ⇛ P ⇚ ⇛ MPR: (∀x ∈ R)(∼∼(x > 0) ⇛ x > 0) ⇚ ⇛

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B III

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

• MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
• EXT: the extensionality theorem

non-Ω-invariant MP: For P ∈ Σ1, ∼∼P ⇛ P ⇚ ⇛ MPR: (∀x ∈ R)(∼∼(x > 0) ⇛ x > 0) ⇚ ⇛ EXT: the extensionality theorem

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B III

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

• MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
• EXT: the extensionality theorem

non-Ω-invariant MP: For P ∈ Σ1, ∼∼P ⇛ P ⇚ ⇛ MPR: (∀x ∈ R)(∼∼(x > 0) ⇛ x > 0) ⇚ ⇛ EXT: the extensionality theorem WLPO: For P ∈ Σ1, ¬¬P ∨ ¬P

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B III

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

• MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
• EXT: the extensionality theorem

non-Ω-invariant MP: For P ∈ Σ1, ∼∼P ⇛ P ⇚ ⇛ MPR: (∀x ∈ R)(∼∼(x > 0) ⇛ x > 0) ⇚ ⇛ EXT: the extensionality theorem WLPO: For P ∈ Σ1, ¬¬P ∨ ¬P

• WLPR: (∀x ∈ R)
• ¬¬(x > 0) ∨ ¬(x > 0)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B III

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

• MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
• EXT: the extensionality theorem

non-Ω-invariant MP: For P ∈ Σ1, ∼∼P ⇛ P ⇚ ⇛ MPR: (∀x ∈ R)(∼∼(x > 0) ⇛ x > 0) ⇚ ⇛ EXT: the extensionality theorem WLPO: For P ∈ Σ1, ¬¬P ∨ ¬P

• WLPR: (∀x ∈ R)
• ¬¬(x > 0) ∨ ¬(x > 0)
• DISC:

A discontinuous 2N → N-function exists.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B III

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

• MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
• EXT: the extensionality theorem

non-Ω-invariant MP: For P ∈ Σ1, ∼∼P ⇛ P ⇚ ⇛ MPR: (∀x ∈ R)(∼∼(x > 0) ⇛ x > 0) ⇚ ⇛ EXT: the extensionality theorem WLPO: For P ∈ Σ1, ¬¬P ∨ ¬P

• WLPR: (∀x ∈ R)
• ¬¬(x > 0) ∨ ¬(x > 0)
• DISC:

A discontinuous 2N → N-function exists. WLPO: For P ∈ Σ1, ∼∼P V ∼P ⇚ ⇛

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B III

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

• MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
• EXT: the extensionality theorem

non-Ω-invariant MP: For P ∈ Σ1, ∼∼P ⇛ P ⇚ ⇛ MPR: (∀x ∈ R)(∼∼(x > 0) ⇛ x > 0) ⇚ ⇛ EXT: the extensionality theorem WLPO: For P ∈ Σ1, ¬¬P ∨ ¬P

• WLPR: (∀x ∈ R)
• ¬¬(x > 0) ∨ ¬(x > 0)
• DISC:

A discontinuous 2N → N-function exists. WLPO: For P ∈ Σ1, ∼∼P V ∼P ⇚ ⇛ WLPR: (∀x ∈ R)

• ∼∼(x > 0) V ∼(x > 0)
• ⇚

⇛

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B III

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

• MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
• EXT: the extensionality theorem

non-Ω-invariant MP: For P ∈ Σ1, ∼∼P ⇛ P ⇚ ⇛ MPR: (∀x ∈ R)(∼∼(x > 0) ⇛ x > 0) ⇚ ⇛ EXT: the extensionality theorem WLPO: For P ∈ Σ1, ¬¬P ∨ ¬P

• WLPR: (∀x ∈ R)
• ¬¬(x > 0) ∨ ¬(x > 0)
• DISC:

A discontinuous 2N → N-function exists. WLPO: For P ∈ Σ1, ∼∼P V ∼P ⇚ ⇛ WLPR: (∀x ∈ R)

• ∼∼(x > 0) V ∼(x > 0)
• ⇚

⇛ DISC: A discontinuous 2N → N-function exists.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B III

BISH (based on BHK) NSA (based on CL) non-constructive/non-algorithmic MP: For P ∈ Σ1, ¬¬P → P

• MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
• EXT: the extensionality theorem

non-Ω-invariant MP: For P ∈ Σ1, ∼∼P ⇛ P ⇚ ⇛ MPR: (∀x ∈ R)(∼∼(x > 0) ⇛ x > 0) ⇚ ⇛ EXT: the extensionality theorem WLPO: For P ∈ Σ1, ¬¬P ∨ ¬P

• WLPR: (∀x ∈ R)
• ¬¬(x > 0) ∨ ¬(x > 0)
• DISC:

A discontinuous 2N → N-function exists. WLPO: For P ∈ Σ1, ∼∼P V ∼P ⇚ ⇛ WLPR: (∀x ∈ R)

• ∼∼(x > 0) V ∼(x > 0)
• ⇚

⇛ DISC: A discontinuous 2N → N-function exists. (Four Remarks)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Ω-invariance is weaker than Recursive

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Ω-invariance is weaker than Recursive

Markov’s principle MP can be reformulated as If it is impossible that a TM runs forever, then it must halt.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Ω-invariance is weaker than Recursive

Markov’s principle MP can be reformulated as If it is impossible that a TM runs forever, then it must halt. As no algorithmic upper bound on the halting time of the TM is given, MP is rejected in BISH.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Ω-invariance is weaker than Recursive

Markov’s principle MP can be reformulated as If it is impossible that a TM runs forever, then it must halt. As no algorithmic upper bound on the halting time of the TM is given, MP is rejected in BISH. The notion of algorithm in BISH is not identical to ‘recursive’.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Ω-invariance is weaker than Recursive

Markov’s principle MP can be reformulated as If it is impossible that a TM runs forever, then it must halt. As no algorithmic upper bound on the halting time of the TM is given, MP is rejected in BISH. The notion of algorithm in BISH is not identical to ‘recursive’.

Definition (In NSA)

A formula ψ is ∆1 if ψ ⇚ ⇛ (∃n ∈ N)ϕ1(n) ⇚ ⇛ (∀m ∈ N)ϕ2(m).

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Ω-invariance is weaker than Recursive

Markov’s principle MP can be reformulated as If it is impossible that a TM runs forever, then it must halt. As no algorithmic upper bound on the halting time of the TM is given, MP is rejected in BISH. The notion of algorithm in BISH is not identical to ‘recursive’.

Definition (In NSA)

A formula ψ is ∆1 if ψ ⇚ ⇛ (∃n ∈ N)ϕ1(n) ⇚ ⇛ (∀m ∈ N)ϕ2(m).

Theorem (In NSA)

Only given MP, every ∆1-formula is decidable.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Ω-invariance is weaker than Recursive

Markov’s principle MP can be reformulated as If it is impossible that a TM runs forever, then it must halt. As no algorithmic upper bound on the halting time of the TM is given, MP is rejected in BISH. The notion of algorithm in BISH is not identical to ‘recursive’.

Definition (In NSA)

A formula ψ is ∆1 if ψ ⇚ ⇛ (∃n ∈ N)ϕ1(n) ⇚ ⇛ (∀m ∈ N)ϕ2(m).

Theorem (In NSA)

Only given MP, every ∆1-formula is decidable. But MP is not available in NSA!

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Fannying about: FAN∆ vs WKL

FAN∆ (Every detachable bar is uniform) is accepted in INT.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Fannying about: FAN∆ vs WKL

FAN∆ (Every detachable bar is uniform) is accepted in INT. WKL (Every infinite tree T ⊂ 2N has a path)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Fannying about: FAN∆ vs WKL

FAN∆ (Every detachable bar is uniform) is accepted in INT. WKL (Every infinite tree T ⊂ 2N has a path) is the classical contraposition of FAN∆ and rejected in INT.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Fannying about: FAN∆ vs WKL

FAN∆ (Every detachable bar is uniform) is accepted in INT. WKL (Every infinite tree T ⊂ 2N has a path) is the classical contraposition of FAN∆ and rejected in INT. In BISH, we have WKL → FAN∆, and both are rejected.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Fannying about: FAN∆ vs WKL

FAN∆ (Every detachable bar is uniform) is accepted in INT. WKL (Every infinite tree T ⊂ 2N has a path) is the classical contraposition of FAN∆ and rejected in INT. In BISH, we have WKL → FAN∆, and both are rejected. What happens in NSA?

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Fannying about: FAN∆ vs WKL

FAN∆ (Every detachable bar is uniform) is accepted in INT. WKL (Every infinite tree T ⊂ 2N has a path) is the classical contraposition of FAN∆ and rejected in INT. In BISH, we have WKL → FAN∆, and both are rejected. What happens in NSA? WKL(∀n ∈N)(∃α ∈2N)(αn ∈T) ⇛ (∃α ∈2N)(∀n ∈N)(αn ∈ T)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Fannying about: FAN∆ vs WKL

FAN∆ (Every detachable bar is uniform) is accepted in INT. WKL (Every infinite tree T ⊂ 2N has a path) is the classical contraposition of FAN∆ and rejected in INT. In BISH, we have WKL → FAN∆, and both are rejected. What happens in NSA? WKL(∀n ∈N)(∃α ∈2N)(αn ∈T) ⇛ (∃α ∈2N)(∀n ∈N)(αn ∈ T) ≈ If the trees T and ∗T are (hyper)infinite, they share a path.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Fannying about: FAN∆ vs WKL

FAN∆ (Every detachable bar is uniform) is accepted in INT. WKL (Every infinite tree T ⊂ 2N has a path) is the classical contraposition of FAN∆ and rejected in INT. In BISH, we have WKL → FAN∆, and both are rejected. What happens in NSA? WKL(∀n ∈N)(∃α ∈2N)(αn ∈T) ⇛ (∃α ∈2N)(∀n ∈N)(αn ∈ T) ≈ If the trees T and ∗T are (hyper)infinite, they share a path. FAN∆ (∀α ∈2N)(∃n ∈N)(αn ∈B)⇛(∃k ∈N)(∀α ∈2N)(∃n ≤ k)(αn∈B)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Fannying about: FAN∆ vs WKL

FAN∆ (Every detachable bar is uniform) is accepted in INT. WKL (Every infinite tree T ⊂ 2N has a path) is the classical contraposition of FAN∆ and rejected in INT. In BISH, we have WKL → FAN∆, and both are rejected. What happens in NSA? WKL(∀n ∈N)(∃α ∈2N)(αn ∈T) ⇛ (∃α ∈2N)(∀n ∈N)(αn ∈ T) ≈ If the trees T and ∗T are (hyper)infinite, they share a path. FAN∆ (∀α ∈2N)(∃n ∈N)(αn ∈B)⇛(∃k ∈N)(∀α ∈2N)(∃n ≤ k)(αn∈B) ≈ If a tree T is infinite, it has a path (∗T can be hyperfinite).

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Fannying about: FAN∆ vs WKL

FAN∆ (Every detachable bar is uniform) is accepted in INT. WKL (Every infinite tree T ⊂ 2N has a path) is the classical contraposition of FAN∆ and rejected in INT. In BISH, we have WKL → FAN∆, and both are rejected. What happens in NSA? WKL(∀n ∈N)(∃α ∈2N)(αn ∈T) ⇛ (∃α ∈2N)(∀n ∈N)(αn ∈ T) ≈ If the trees T and ∗T are (hyper)infinite, they share a path. FAN∆ (∀α ∈2N)(∃n ∈N)(αn ∈B)⇛(∃k ∈N)(∀α ∈2N)(∃n ≤ k)(αn∈B) ≈ If a tree T is infinite, it has a path (∗T can be hyperfinite). In NSA, we have WKL ⇛ FAN∆.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

A note on Coding and Assymetry

Recall that ‘A ∈ T’ means ‘A satisfies Transfer’.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

A note on Coding and Assymetry

Recall that ‘A ∈ T’ means ‘A satisfies Transfer’. E.g. ‘(∀n ∈ N)ϕ(n) ∈ T’ is [(∀n ∈ N)ϕ(n) → (∀n ∈ ∗N)ϕ(n)]

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

A note on Coding and Assymetry

Recall that ‘A ∈ T’ means ‘A satisfies Transfer’. E.g. ‘(∀n ∈ N)ϕ(n) ∈ T’ is [(∀n ∈ N)ϕ(n) → (∀n ∈ ∗N)ϕ(n)] E.g. ‘(∃n ∈ ∗N)ϕ(n) ∈ T’ is [(∃n ∈ ∗N)ϕ(n) → (∃n ∈ N1)ϕ(n)]

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

A note on Coding and Assymetry

Recall that ‘A ∈ T’ means ‘A satisfies Transfer’. E.g. ‘(∀n ∈ N)ϕ(n) ∈ T’ is [(∀n ∈ N)ϕ(n) → (∀n ∈ ∗N)ϕ(n)] E.g. ‘(∃n ∈ ∗N)ϕ(n) ∈ T’ is [(∃n ∈ ∗N)ϕ(n) → (∃n ∈ N1)ϕ(n)] Transfer is clearly asymmetric.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

A note on Coding and Assymetry

Recall that ‘A ∈ T’ means ‘A satisfies Transfer’. E.g. ‘(∀n ∈ N)ϕ(n) ∈ T’ is [(∀n ∈ N)ϕ(n) → (∀n ∈ ∗N)ϕ(n)] E.g. ‘(∃n ∈ ∗N)ϕ(n) ∈ T’ is [(∃n ∈ ∗N)ϕ(n) → (∃n ∈ N1)ϕ(n)] Transfer is clearly asymmetric. First, to make hypernegation ‘∼’ work like intuitionistic negation.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

A note on Coding and Assymetry

Recall that ‘A ∈ T’ means ‘A satisfies Transfer’. E.g. ‘(∀n ∈ N)ϕ(n) ∈ T’ is [(∀n ∈ N)ϕ(n) → (∀n ∈ ∗N)ϕ(n)] E.g. ‘(∃n ∈ ∗N)ϕ(n) ∈ T’ is [(∃n ∈ ∗N)ϕ(n) → (∃n ∈ N1)ϕ(n)] Transfer is clearly asymmetric. First, to make hypernegation ‘∼’ work like intuitionistic negation. Secondly, for fundamental reasons:

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

A note on Coding and Assymetry

Recall that ‘A ∈ T’ means ‘A satisfies Transfer’. E.g. ‘(∀n ∈ N)ϕ(n) ∈ T’ is [(∀n ∈ N)ϕ(n) → (∀n ∈ ∗N)ϕ(n)] E.g. ‘(∃n ∈ ∗N)ϕ(n) ∈ T’ is [(∃n ∈ ∗N)ϕ(n) → (∃n ∈ N1)ϕ(n)] Transfer is clearly asymmetric. First, to make hypernegation ‘∼’ work like intuitionistic negation. Secondly, for fundamental reasons:

In ‘(∃n0 ∈ ∗N)ϕ(n0)’, the number n0 could be a code for some f : N → N (Keisler).

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

A note on Coding and Assymetry

Recall that ‘A ∈ T’ means ‘A satisfies Transfer’. E.g. ‘(∀n ∈ N)ϕ(n) ∈ T’ is [(∀n ∈ N)ϕ(n) → (∀n ∈ ∗N)ϕ(n)] E.g. ‘(∃n ∈ ∗N)ϕ(n) ∈ T’ is [(∃n ∈ ∗N)ϕ(n) → (∃n ∈ N1)ϕ(n)] Transfer is clearly asymmetric. First, to make hypernegation ‘∼’ work like intuitionistic negation. Secondly, for fundamental reasons:

In ‘(∃n0 ∈ ∗N)ϕ(n0)’, the number n0 could be a code for some f : N → N (Keisler). If ‘(∃n0 ∈ ∗N)ϕ(n0)’ implies ‘(∃n1 ∈ N)ϕ(n1)’, then f has a finite code n1 ∈ N, making its graph ∆0.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B IV

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B IV

Same for WMP, FAN∆, BD-N, and MP∨.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B IV

Same for WMP, FAN∆, BD-N, and MP∨. Same for ‘mixed’ theorems:

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B IV

Same for WMP, FAN∆, BD-N, and MP∨. Same for ‘mixed’ theorems: BISH (based on BHK) NSA (based on CL)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B IV

Same for WMP, FAN∆, BD-N, and MP∨. Same for ‘mixed’ theorems: BISH (based on BHK) NSA (based on CL) LPO ↔ MP+WLPO

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B IV

Same for WMP, FAN∆, BD-N, and MP∨. Same for ‘mixed’ theorems: BISH (based on BHK) NSA (based on CL) LPO ↔ MP+WLPO MP ↔ WMP + MP∨

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B IV

Same for WMP, FAN∆, BD-N, and MP∨. Same for ‘mixed’ theorems: BISH (based on BHK) NSA (based on CL) LPO ↔ MP+WLPO MP ↔ WMP + MP∨ WLPO → LLPO

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B IV

Same for WMP, FAN∆, BD-N, and MP∨. Same for ‘mixed’ theorems: BISH (based on BHK) NSA (based on CL) LPO ↔ MP+WLPO MP ↔ WMP + MP∨ WLPO → LLPO LLPO → MP∨

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B IV

Same for WMP, FAN∆, BD-N, and MP∨. Same for ‘mixed’ theorems: BISH (based on BHK) NSA (based on CL) LPO ↔ MP+WLPO MP ↔ WMP + MP∨ WLPO → LLPO LLPO → MP∨ LPO → BD-N

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B IV

Same for WMP, FAN∆, BD-N, and MP∨. Same for ‘mixed’ theorems: BISH (based on BHK) NSA (based on CL) LPO ↔ MP+WLPO MP ↔ WMP + MP∨ WLPO → LLPO LLPO → MP∨ LPO → BD-N LLPO → FAN∆

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B IV

Same for WMP, FAN∆, BD-N, and MP∨. Same for ‘mixed’ theorems: BISH (based on BHK) NSA (based on CL) LPO ↔ MP+WLPO MP ↔ WMP + MP∨ WLPO → LLPO LLPO → MP∨ LPO → BD-N LLPO → FAN∆ LLPO ↔ WKL

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Constructive Reverse Mathematics under B IV

Same for WMP, FAN∆, BD-N, and MP∨. Same for ‘mixed’ theorems: BISH (based on BHK) NSA (based on CL) LPO ↔ MP+WLPO MP ↔ WMP + MP∨ WLPO → LLPO LLPO → MP∨ LPO → BD-N LLPO → FAN∆ LLPO ↔ WKL LPO ⇚ ⇛ MP + WLPO MP ⇚ ⇛ WMP + MP∨ WLPO ⇛ LLPO LLPO ⇛ MP∨ LPO ⇛ BD-N LLPO ⇛ FAN∆ LLPO ⇚ ⇛ WKL

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Conclusion: NSA ≈ BISH

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Conclusion: NSA ≈ BISH

If BISH ⊢ X then X→LPO, LLPO, MP, . . . (princ. rejected in BISH)

If NSA ⊢ Y then Y ⇛ LPO, LLPO, MP, . . . (not provable in NSA)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Conclusion: NSA ≈ BISH

If BISH ⊢ X then X→LPO, LLPO, MP, . . . (princ. rejected in BISH)

If NSA ⊢ Y then Y ⇛ LPO, LLPO, MP, . . . (not provable in NSA)

Reuniting the antipodes (Palmgren & Moerdijk).

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Conclusion: NSA ≈ BISH

If BISH ⊢ X then X→LPO, LLPO, MP, . . . (princ. rejected in BISH)

If NSA ⊢ Y then Y ⇛ LPO, LLPO, MP, . . . (not provable in NSA)

Reuniting the antipodes (Palmgren & Moerdijk). Reverse-engineering Reverse Mathematics (Fuchino-sensei)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Future work: Type Theory

Martin-L¨

• f intended his type theory as a foundation for BISH.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Future work: Type Theory

Martin-L¨

• f intended his type theory as a foundation for BISH.

Can Ω-invariance help capture e.g. Type Theory?

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Future work: Type Theory

Martin-L¨

• f intended his type theory as a foundation for BISH.

Can Ω-invariance help capture e.g. Type Theory? Homotopy: f (x) g(x)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Future work: Type Theory

Martin-L¨

• f intended his type theory as a foundation for BISH.

Can Ω-invariance help capture e.g. Type Theory? Homotopy: f (x) g(x)

• continuous transformation ht of f to g (t ∈ [0, 1]).

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Future work: Type Theory

Martin-L¨

• f intended his type theory as a foundation for BISH.

Can Ω-invariance help capture e.g. Type Theory? Homotopy: f (x) g(x)

• continuous transformation ht of f to g (t ∈ [0, 1]).

ht1(x) ht2(x) ht3(x) . . . . . . . . .

• h1(x) =

h0(x) =

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Future work: Type Theory

Martin-L¨

• f intended his type theory as a foundation for BISH.

Can Ω-invariance help capture e.g. Type Theory? Homotopy: f (x) g(x)

• ✑✑

✑

• ✏✏✏✏

✏

• PPPP

P ❅ ❅ ❅ ❅ ❅

mω(x) ≈ kω(x) ≈

• ❅

❅

• ◗◗

◗

• PPPP

P

• PPPP

P✚✚✚✚✚

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Future work: Type Theory

Martin-L¨

• f intended his type theory as a foundation for BISH.

Can Ω-invariance help capture e.g. Type Theory? Homotopy: f (x) g(x)

• ✑✑

✑

• ✏✏✏✏

✏

• PPPP

P ❅ ❅ ❅ ❅ ❅

mω(x) ≈ kω(x) ≈

• ❅

❅

• ◗◗

◗

• PPPP

P

• PPPP

P✚✚✚✚✚

• PPPP

P

• ✏✏✏✏

✏

• ⇓

increment is multiple of 1

ω

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Future work: Type Theory

Martin-L¨

• f intended his type theory as a foundation for BISH.

Can Ω-invariance help capture e.g. Type Theory? Homotopy: f (x) g(x)

• ✑✑

✑

• ✏✏✏✏

✏

• PPPP

P ❅ ❅ ❅ ❅ ❅

mω(x) ≈ kω(x) ≈

• ❅

❅

• ◗◗

◗

• PPPP

P

• PPPP

P✚✚✚✚✚

• PPPP

P

• ✏✏✏✏

✏

• ⇓

increment is multiple of 1

ω

ONE basic step

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Future work: Type Theory

Martin-L¨

• f intended his type theory as a foundation for BISH.

Can Ω-invariance help capture e.g. Type Theory? Homotopy: f (x) g(x)

• ✑✑

✑

• ✏✏✏✏

✏

• PPPP

P ❅ ❅ ❅ ❅ ❅

mω(x) ≈ kω(x) ≈

• ❅

❅

• ◗◗

◗

• PPPP

P

• PPPP

P✚✚✚✚✚

• PPPP

P

• ✏✏✏✏

✏

• ⇓

ONE basic step . . . ω basic steps . . .

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Future work: Type Theory

Martin-L¨

• f intended his type theory as a foundation for BISH.

Can Ω-invariance help capture e.g. Type Theory? Homotopy: f (x) g(x)

• ✑✑

✑

• ✏✏✏✏

✏

• PPPP

P ❅ ❅ ❅ ❅ ❅

mω(x) ≈ kω(x) ≈

• ❅

❅

• ◗◗

◗

• PPPP

P

• PPPP

P✚✚✚✚✚

• PPPP

P

• ✏✏✏✏

✏

• ⇓

ONE basic step . . . ω basic steps . . . Independent of the choice of ω

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Future work: Type Theory

Martin-L¨

• f intended his type theory as a foundation for BISH.

Can Ω-invariance help capture e.g. Type Theory? Homotopy: f (x) g(x)

• ✑✑

✑

• ✏✏✏✏

✏

• PPPP

P ❅ ❅ ❅ ❅ ❅

mω(x) ≈ kω(x) ≈

• ❅

❅

• ◗◗

◗

• PPPP

P

• PPPP

P✚✚✚✚✚

• PPPP

P

• ✏✏✏✏

✏

• ⇓

ONE basic step . . . ω basic steps . . . Independent of the choice of ω ≈ Ω-invariant broken-line transformation hω,t of f to g.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Philosophy of Physics

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Philosophy of Physics

Why is Mathematics in Physics so constructive/computable?

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Philosophy of Physics

Why is Mathematics in Physics so constructive/computable? Indeed, most of Physics can be formalized in BISH (e.g. Gleason’s theorem).

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Philosophy of Physics

Why is Mathematics in Physics so constructive/computable? Indeed, most of Physics can be formalized in BISH (e.g. Gleason’s theorem). Yet, in Physics, an informal version of NSA is used to date.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Philosophy of Physics

Why is Mathematics in Physics so constructive/computable? Indeed, most of Physics can be formalized in BISH (e.g. Gleason’s theorem). Yet, in Physics, an informal version of NSA is used to date. (Weierstraß’ notorious ‘ε-δ’ method was never adopted, neither was BISH).

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Philosophy of Physics

Why is Mathematics in Physics so constructive/computable? Indeed, most of Physics can be formalized in BISH (e.g. Gleason’s theorem). Yet, in Physics, an informal version of NSA is used to date. (Weierstraß’ notorious ‘ε-δ’ method was never adopted, neither was BISH). Now, in Physics, the end result of a calculation should have physical meaning (modeling of reality).

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Philosophy of Physics

Why is Mathematics in Physics so constructive/computable? Indeed, most of Physics can be formalized in BISH (e.g. Gleason’s theorem). Yet, in Physics, an informal version of NSA is used to date. (Weierstraß’ notorious ‘ε-δ’ method was never adopted, neither was BISH). Now, in Physics, the end result of a calculation should have physical meaning (modeling of reality). A mathematical result with physical meaning will not depend on the choice of infinite number/infinitesimal used, i.e. it is Ω-invariant. (Alain Connes)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Philosophy of Mathematics: Whither Structuralism?

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Philosophy of Mathematics: Whither Structuralism?

Structuralism ≈ Mathematics is about a single structure.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Philosophy of Mathematics: Whither Structuralism?

Structuralism ≈ Mathematics is about a single structure. E.g. first-order arithmetic is about (models isomorphic to) the standard model N.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Philosophy of Mathematics: Whither Structuralism?

Structuralism ≈ Mathematics is about a single structure. E.g. first-order arithmetic is about (models isomorphic to) the standard model N. Problem: How to exclude the nonstandard models of arithmetic? (Second-order?, Tennenbaum’s Theorem?)

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Philosophy of Mathematics: Whither Structuralism?

Structuralism ≈ Mathematics is about a single structure. E.g. first-order arithmetic is about (models isomorphic to) the standard model N. Problem: How to exclude the nonstandard models of arithmetic? (Second-order?, Tennenbaum’s Theorem?) When life gives you lemons... you make Ω-invariance:

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Philosophy of Mathematics: Whither Structuralism?

Structuralism ≈ Mathematics is about a single structure. E.g. first-order arithmetic is about (models isomorphic to) the standard model N. Problem: How to exclude the nonstandard models of arithmetic? (Second-order?, Tennenbaum’s Theorem?) When life gives you lemons... you make Ω-invariance: Arithmetic is about a computationally robust variety of structures.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Philosophy of Mathematics: Whither Structuralism?

Structuralism ≈ Mathematics is about a single structure. E.g. first-order arithmetic is about (models isomorphic to) the standard model N. Problem: How to exclude the nonstandard models of arithmetic? (Second-order?, Tennenbaum’s Theorem?) When life gives you lemons... you make Ω-invariance: Arithmetic is about a computationally robust variety of structures. Despite Tennenbaum’s Theorem, one can define computability/constructivity via Ω-invariance in each nonstandard model of arithmetic.

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Final Thoughts

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Final Thoughts

And what are these [infinitesimals]? [. . . ] They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?

George Berkeley, The Analyst

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Final Thoughts

And what are these [infinitesimals]? [. . . ] They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?

George Berkeley, The Analyst ...there are good reasons to believe that Nonstandard Analysis, in some version or other, will be the analysis of the future. Kurt G¨

• del

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Final Thoughts

And what are these [infinitesimals]? [. . . ] They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?

George Berkeley, The Analyst ...there are good reasons to believe that Nonstandard Analysis, in some version or other, will be the analysis of the future. Kurt G¨

• del

We thank the John Templeton Foundation for its generous support!

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Final Thoughts

And what are these [infinitesimals]? [. . . ] They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?

George Berkeley, The Analyst ...there are good reasons to believe that Nonstandard Analysis, in some version or other, will be the analysis of the future. Kurt G¨

• del

We thank the John Templeton Foundation for its generous support!

Thank you for your attention!

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Final Thoughts

And what are these [infinitesimals]? [. . . ] They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?

George Berkeley, The Analyst ...there are good reasons to believe that Nonstandard Analysis, in some version or other, will be the analysis of the future. Kurt G¨

• del

We thank the John Templeton Foundation for its generous support!

Thank you for your attention!

Any questions?

Introduction NSA, BISH and Constructive Reverse Mathematics Conclusion

Take-home message

In Nonstandard Analysis, an algorithm is any object whose definition is independent of the choice of infinitesimal (Ω-invariance).

More technically, we define a translation between Constructive Analysis (BISH) and Nonstandard Analysis (NSA):

(Proof and Algorithm) in BISH = (Transfer and Ω-invariance) in NSA Most results from CRM (= RM based on BISH) translate to NSA via a natural translation B.