Nonstandard Analysis: A new way to compute
Sam Sanders1
CCR2012, July 2, 2012
1
This 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.
Son of a. . .
INT
BISH
CLASS
RUSS
Son of a. . .
INT
BISH
CLASS
RUSS
≈Math. programmable on TM
Son of a. . .
INT
BISH
Classical Math. ≈ CLASS
LEM
(P ∨ ¬P)
RUSS
≈Math. programmable on TM
Son of a. . .
INT
BISH
Classical Math. ≈ CLASS
LEM
(P ∨ ¬P)
(proof by contradiction)
RUSS
≈Math. programmable on TM
Son of a. . .
INT
≈Brouwer’s Intuitionistic Math.
BISH
Classical Math. ≈ CLASS
LEM
(P ∨ ¬P)
(proof by contradiction)
RUSS
≈Math. programmable on TM
Son of a. . .
INT
≈Brouwer’s Intuitionistic Math.
BISH
Classical Math. ≈ CLASS
LEM
MP
(¬¬P → P)
(P ∨ ¬P)
(proof by contradiction)
RUSS
≈Math. programmable on TM
Son of a. . .
INT
≈Brouwer’s Intuitionistic Math.
BISH
Classical Math. ≈ CLASS
LEM
MP
(¬¬P → P)
RUSS
≈Math. programmable on TM
(P ∨ ¬P)
(proof by contradiction)
CPF
Son of a. . .
CONT
INT
≈Brouwer’s Intuitionistic Math.
BISH
Classical Math. ≈ CLASS
LEM
MP
(¬¬P → P)
RUSS
≈Math. programmable on TM
(P ∨ ¬P)
(proof by contradiction)
CPF
Son of a. . .
CONT
INT
≈Brouwer’s Intuitionistic Math.
FAN
BISH
Classical Math. ≈ CLASS
LEM
MP
(¬¬P → P)
RUSS
≈Math. programmable on TM
(P ∨ ¬P)
(proof by contradiction)
CPF
Son of a. . .
CONT
INT
≈Brouwer’s Intuitionistic Math.
CONT0
FAN
BISH
Classical Math. ≈ CLASS
LEM
MP
(¬¬P → P)
RUSS
≈Math. programmable on TM
(P ∨ ¬P)
(proof by contradiction)
CPF
Son of a. . .
Errett Bishop’s Constructive Analysis (also ‘BISH’) is a constructive
redevelopment of Mathematics, consistent with CLASS, RUSS and INT.
CONT
INT
≈Brouwer’s Intuitionistic Math.
CONT0
FAN
BISH
Classical Math. ≈
CLASS
MP
(¬¬P → P)
LEM (P ∨ ¬P)
(proof by contradiction)
RUSS
≈Math. programmable on TM
CPF
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
From BISH to NSA
In BISH, proof and algorithm are central.
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
From BISH to NSA
In BISH, proof and algorithm are central.
We will define a system of Nonstandard Analyis NSA where
Transfer and Ω-invariant procedure play the same role.
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
From BISH to NSA
In BISH, proof and algorithm are central.
We will define a system of Nonstandard Analyis NSA where
Transfer and Ω-invariant procedure play the same role.
NSA is similar to ∗ RCA0 , a nonstandard version of RCA0 (Keisler &
Yokoyama).
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
From BISH to NSA
In BISH, proof and algorithm are central.
We will define a system of Nonstandard Analyis NSA where
Transfer and Ω-invariant procedure play the same role.
NSA is similar to ∗ RCA0 , a nonstandard version of RCA0 (Keisler &
Yokoyama).
Three important features:
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
From BISH to NSA
In BISH, proof and algorithm are central.
We will define a system of Nonstandard Analyis NSA where
Transfer 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 (T) (except for ∆0 ).
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
From BISH to NSA
In BISH, proof and algorithm are central.
We will define a system of Nonstandard Analyis NSA where
Transfer 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 (T) (except for ∆0 ).
2
No ∆01 -CA, but Ω-CA. (CA for Ω-invariant formulas)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
From BISH to NSA
In BISH, proof and algorithm are central.
We will define a system of Nonstandard Analyis NSA where
Transfer 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 (T) (except for ∆0 ).
2
No ∆01 -CA, but Ω-CA. (CA for Ω-invariant formulas)
3
Levels of infinity (Stratified NSA).
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Feature 3: Stratified Nonstandard Analysis
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Feature 3: Stratified Nonstandard Analysis
The usual picture of ∗ N:
∗ N,
the hypernatural numbers
}|
z
ω1 . . .
0 1...
|
{z
} |
N, the natural/finite numbers
ω2
{
. . . ωk . . .
{z
-
Ω = ∗ N \ N, the infinite numbers
}
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Feature 3: Stratified Nonstandard Analysis
The usual picture of ∗ N:
In NSA, the infinite numbers are split into ‘small’ and ‘large’.
∗ N,
the hypernatural numbers
}|
z
ω1 . . .
0 1...
|
{z
} |
N, the natural/finite numbers
ω2
{
. . . ωk . . .
{z
-
Ω = ∗ N \ N, the infinite numbers
}
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Feature 3: Stratified Nonstandard Analysis
The usual picture of ∗ N:
In NSA, the infinite numbers are split into ‘small’ and ‘large’.
∗ N,
the hypernatural numbers
}|
z
the small infinite numbers
0 1...
|
{z
z
} |
}| { z
ω1 . . . ω2
N, the natural/finite numbers
{
the large infinite numbers
}|
. . . ωk . . .
{z
{
-
Ω = ∗ N \ N, the infinite numbers
}
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Feature 3: Stratified Nonstandard Analysis
The usual picture of ∗ N:
In NSA, the infinite numbers are split into ‘small’ and ‘large’.
∗ N,
the hypernatural numbers
}|
z
N1 =N∪ the small infinite numbers
z
0 1...
|
{z
}|
} |
Ω1 =∗ N\N1 , the large infinite numbers
{z
ω1 . . . ω2
N, the natural/finite numbers
{
}|
. . . ωk . . .
{z
{
-
Ω = ∗ N \ N, the infinite numbers
}
Introduction
NSA, BISH and Constructive Reverse Mathematics
Feature 2: Ω-invariance
Conclusion
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 0 ∈ Ω)[ψ(n, ω) ↔ ψ(n, ω 0 )].
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 0 ∈ Ω)[ψ(n, ω) ↔ ψ(n, ω 0 )].
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 0 ∈ Ω)[ψ(n, ω) ↔ ψ(n, ω 0 )].
Note that ψ(n, ω) depends on ω ∈ Ω, but not on the choice of ω ∈ Ω.
This is the way infinitesimals are used in Physics!
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 0 ∈ Ω)[ψ(n, ω) ↔ ψ(n, ω 0 )].
Note that ψ(n, ω) depends on ω ∈ Ω, but not on the choice of ω ∈ Ω.
This is the way infinitesimals are used in Physics!
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
Lost in translation
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Lost in translation
BISH (based on BHK)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Lost in translation
BISH (based on BHK)
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Lost in translation
BISH (based on BHK)
Central: algorithm and proof
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Lost in translation
BISH (based on BHK)
Central: algorithm and proof
A ∨ B:
an algo yields a proof of A or of B
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Lost in translation
BISH (based on BHK)
Central: algorithm and proof
A ∨ B:
an algo yields a proof of A or of B
NSA (based on CL)
Central: Ω-invariance and Transfer (T)
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Lost in translation
BISH (based on BHK)
NSA (based on CL)
Central: algorithm and proof
Central: Ω-invariance and Transfer (T)
A ∨ B:
A V B:
an algo yields a proof of A or of B
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Lost in translation
BISH (based on BHK)
NSA (based on CL)
Central: algorithm and proof
Central: Ω-invariance and Transfer (T)
A ∨ B:
A V B: There is Ω-invariant ψ(~x , ω) s.t.
an algo yields a proof of A or of B
ψ(~x , ω) → [A(~x ) ∧ [A(~x ) ∈ T]]
∧
¬ψ(~x , ω) → [B(~x ) ∧ [B(~x ) ∈ T]]
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Lost in translation
BISH (based on BHK)
NSA (based on CL)
Central: algorithm and proof
Central: Ω-invariance and Transfer (T)
A ∨ B:
A V B: There is Ω-invariant ψ(~x , ω) s.t.
an algo yields a proof of A or of B
A → B: an algo converts a proof of A
to a proof of B
ψ(~x , ω) → [A(~x ) ∧ [A(~x ) ∈ T]]
∧
¬ψ(~x , ω) → [B(~x ) ∧ [B(~x ) ∈ T]]
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Lost in translation
BISH (based on BHK)
NSA (based on CL)
Central: algorithm and proof
Central: Ω-invariance and Transfer (T)
A ∨ B:
A V B: There is Ω-invariant ψ(~x , ω) s.t.
an algo yields a proof of A or of B
A → B: an algo converts a proof of A
to a proof of B
ψ(~x , ω) → [A(~x ) ∧ [A(~x ) ∈ T]]
∧
¬ψ(~x , ω) → [B(~x ) ∧ [B(~x ) ∈ T]]
A V B: A ∧ [A ∈ T] → B ∧ [B ∈ T]
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Lost in translation
BISH (based on BHK)
NSA (based on CL)
Central: algorithm and proof
Central: Ω-invariance and Transfer (T)
A ∨ B:
A V B: There is Ω-invariant ψ(~x , ω) s.t.
an algo yields a proof of A or of B
A → B: an algo converts a proof of A
to a proof of B
ψ(~x , ω) → [A(~x ) ∧ [A(~x ) ∈ T]]
∧
¬ψ(~x , ω) → [B(~x ) ∧ [B(~x ) ∈ T]]
A V B: A ∧ [A ∈ T] → B ∧ [B ∈ T]
‘A ∈ T’ means ‘A satisfies Transfer’.
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Lost in translation
BISH (based on BHK)
NSA (based on CL)
Central: algorithm and proof
Central: Ω-invariance and Transfer (T)
A ∨ B:
A V B: There is Ω-invariant ψ(~x , ω) s.t.
an algo yields a proof of A or of B
A → B: an algo converts a proof of A
to a proof of B
ψ(~x , ω) → [A(~x ) ∧ [A(~x ) ∈ T]]
∧
¬ψ(~x , ω) → [B(~x ) ∧ [B(~x ) ∈ T]]
A V 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
Lost in translation
BISH (based on BHK)
NSA (based on CL)
Central: algorithm and proof
Central: Ω-invariance and Transfer (T)
A ∨ B:
A V B: There is Ω-invariant ψ(~x , ω) s.t.
an algo yields a proof of A or of B
A → B: an algo converts a proof of A
to a proof of B
ψ(~x , ω) → [A(~x ) ∧ [A(~x ) ∈ T]]
∧
¬ψ(~x , ω) → [B(~x ) ∧ [B(~x ) ∈ T]]
A V 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
Lost in translation
BISH (based on BHK)
NSA (based on CL)
Central: algorithm and proof
Central: Ω-invariance and Transfer (T)
A ∨ B:
A V B: There is Ω-invariant ψ(~x , ω) s.t.
an algo yields a proof of A or of B
A → B: an algo converts a proof of A
to a proof of B
¬A: A → (0 = 1)
ψ(~x , ω) → [A(~x ) ∧ [A(~x ) ∈ T]]
∧
¬ψ(~x , ω) → [B(~x ) ∧ [B(~x ) ∈ T]]
A V B: A ∧ [A ∈ T] → B ∧ [B ∈ T]
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Lost in translation
BISH (based on BHK)
NSA (based on CL)
Central: algorithm and proof
Central: Ω-invariance and Transfer (T)
A ∨ B:
A V B: There is Ω-invariant ψ(~x , ω) s.t.
an algo yields a proof of A or of B
A → B: an algo converts a proof of A
to a proof of B
¬A: A → (0 = 1)
ψ(~x , ω) → [A(~x ) ∧ [A(~x ) ∈ T]]
∧
¬ψ(~x , ω) → [B(~x ) ∧ [B(~x ) ∈ T]]
A V B: A ∧ [A ∈ T] → B ∧ [B ∈ T]
∼A: A V (0 = 1)
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Lost in translation
BISH (based on BHK)
NSA (based on CL)
Central: algorithm and proof
Central: Ω-invariance and Transfer (T)
A ∨ B:
A V B: There is Ω-invariant ψ(~x , ω) s.t.
an algo yields a proof of A or of B
A → B: an algo converts a proof of A
to a proof of B
¬A: A → (0 = 1)
(∃x)A(x): an algo computes x0
such that A(x0 )
ψ(~x , ω) → [A(~x ) ∧ [A(~x ) ∈ T]]
∧
¬ψ(~x , ω) → [B(~x ) ∧ [B(~x ) ∈ T]]
A V B: A ∧ [A ∈ T] → B ∧ [B ∈ T]
∼A: A V (0 = 1)
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Lost in translation
BISH (based on BHK)
NSA (based on CL)
Central: algorithm and proof
Central: Ω-invariance and Transfer (T)
A ∨ B:
A V B: There is Ω-invariant ψ(~x , ω) s.t.
an algo yields a proof of A or of B
A → B: an algo converts a proof of A
to a proof of B
¬A: A → (0 = 1)
ψ(~x , ω) → [A(~x ) ∧ [A(~x ) ∈ T]]
∧
¬ψ(~x , ω) → [B(~x ) ∧ [B(~x ) ∈ T]]
A V B: A ∧ [A ∈ T] → B ∧ [B ∈ T]
∼A: A V (0 = 1)
(∃x)A(x): an algo computes x0
(∃x)A(x): an Ω-inv. proc. computes x0
such that A(x0 )
such that A(x0 )
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Lost in translation
BISH (based on BHK)
NSA (based on CL)
Central: algorithm and proof
Central: Ω-invariance and Transfer (T)
A ∨ B:
A V B: There is Ω-invariant ψ(~x , ω) s.t.
an algo yields a proof of A or of B
A → B: an algo converts a proof of A
to a proof of B
¬A: A → (0 = 1)
ψ(~x , ω) → [A(~x ) ∧ [A(~x ) ∈ T]]
∧
¬ψ(~x , ω) → [B(~x ) ∧ [B(~x ) ∈ T]]
A V B: A ∧ [A ∈ T] → B ∧ [B ∈ T]
∼A: A V (0 = 1)
(∃x)A(x): an algo computes x0
(∃x)A(x): an Ω-inv. proc. computes x0
such that A(x0 )
such that A(x0 )
∼[(∀n ∈ N)A(n)]
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Lost in translation
BISH (based on BHK)
NSA (based on CL)
Central: algorithm and proof
Central: Ω-invariance and Transfer (T)
A ∨ B:
A V B: There is Ω-invariant ψ(~x , ω) s.t.
an algo yields a proof of A or of B
A → B: an algo converts a proof of A
to a proof of B
¬A: A → (0 = 1)
ψ(~x , ω) → [A(~x ) ∧ [A(~x ) ∈ T]]
∧
¬ψ(~x , ω) → [B(~x ) ∧ [B(~x ) ∈ T]]
A V B: A ∧ [A ∈ T] → B ∧ [B ∈ T]
∼A: A V (0 = 1)
(∃x)A(x): an algo computes x0
(∃x)A(x): an Ω-inv. proc. computes x0
such that A(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
Lost in translation
BISH (based on BHK)
NSA (based on CL)
Central: algorithm and proof
Central: Ω-invariance and Transfer (T)
A ∨ B:
A V B: There is Ω-invariant ψ(~x , ω) s.t.
an algo yields a proof of A or of B
A → B: an algo converts a proof of A
to a proof of B
¬A: A → (0 = 1)
ψ(~x , ω) → [A(~x ) ∧ [A(~x ) ∈ T]]
∧
¬ψ(~x , ω) → [B(~x ) ∧ [B(~x ) ∈ T]]
A V B: A ∧ [A ∈ T] → B ∧ [B ∈ T]
∼A: A V (0 = 1)
(∃x)A(x): an algo computes x0
(∃x)A(x): an Ω-inv. proc. computes x0
such that A(x0 )
such that A(x0 )
¬[(∀n ∈ N)A(n)] is WEAKER
than (∃n ∈ N)¬A(n).
∼[(∀n ∈ N)A(n)] ≡ (∃n ∈ N1 )∼A(n)
WEAKER than (∃n ∈ N)∼A(n).
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Lost in translation
BISH (based on BHK)
NSA (based on CL)
Central: algorithm and proof
Central: Ω-invariance and Transfer (T)
A ∨ B:
A V B: There is Ω-invariant ψ(~x , ω) s.t.
an algo yields a proof of A or of B
A → B: an algo converts a proof of A
to a proof of B
¬A: A → (0 = 1)
ψ(~x , ω) → [A(~x ) ∧ [A(~x ) ∈ T]]
∧
¬ψ(~x , ω) → [B(~x ) ∧ [B(~x ) ∈ T]]
A V B: A ∧ [A ∈ T] → B ∧ [B ∈ T]
∼A: A V (0 = 1)
(∃x)A(x): an algo computes x0
(∃x)A(x): an Ω-inv. proc. computes x0
such that A(x0 )
such that A(x0 )
WHY is this a good/faithful/reasonable/. . . translation?
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Lost in translation
BISH (based on BHK)
NSA (based on CL)
Central: algorithm and proof
Central: Ω-invariance and Transfer (T)
A ∨ B:
A V B: There is Ω-invariant ψ(~x , ω) s.t.
an algo yields a proof of A or of B
A → B: an algo converts a proof of A
to a proof of B
¬A: A → (0 = 1)
ψ(~x , ω) → [A(~x ) ∧ [A(~x ) ∈ T]]
∧
¬ψ(~x , ω) → [B(~x ) ∧ [B(~x ) ∈ T]]
A V B: A ∧ [A ∈ T] → B ∧ [B ∈ T]
∼A: A V (0 = 1)
(∃x)A(x): an algo computes x0
(∃x)A(x): an Ω-inv. proc. computes x0
such that A(x0 )
such that A(x0 )
WHY is this a good/faithful/reasonable/. . . translation?
BECAUSE the non-algorithmic/non-constructive principles behave the same!
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics
BISH (based on BHK)
non-constructive/non-algorithmic
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics
BISH (based on BHK)
non-constructive/non-algorithmic
LPO: For P ∈ Σ1 , P ∨ ¬P
l
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics
BISH (based on BHK)
non-constructive/non-algorithmic
LPO: For P ∈ Σ1 , P ∨ ¬P
l
LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
l
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics
BISH (based on BHK)
non-constructive/non-algorithmic
LPO: For P ∈ Σ1 , P ∨ ¬P
l
LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
l
MCT: monotone convergence thm
l
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics
BISH (based on BHK)
non-constructive/non-algorithmic
LPO: For P ∈ Σ1 , P ∨ ¬P
l
LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
l
MCT: monotone convergence thm
l
CIT: Cantor intersection thm
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics
BISH (based on BHK)
non-constructive/non-algorithmic
LPO: For P ∈ Σ1 , P ∨ ¬P
l
LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
l
MCT: monotone convergence thm
l
CIT: Cantor intersection thm
NSA (based on CL)
non-Ω-invariant
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics
BISH (based on BHK)
NSA (based on CL)
non-constructive/non-algorithmic
non-Ω-invariant
LPO: For P ∈ Σ1 , P ∨ ¬P
LPO: For P ∈ Σ1 , P V ∼P
l
WV
LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0))
l
MCT: monotone convergence thm
l
CIT: Cantor intersection thm
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics
BISH (based on BHK)
NSA (based on CL)
non-constructive/non-algorithmic
non-Ω-invariant
LPO: For P ∈ Σ1 , P ∨ ¬P
LPO: For P ∈ Σ1 , P V ∼P
l
WV
LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0)) LPR: (∀x ∈ R)(x > 0 V ∼(x > 0))
l
WV
MCT: monotone convergence thm
l
CIT: Cantor intersection thm
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics
BISH (based on BHK)
NSA (based on CL)
non-constructive/non-algorithmic
non-Ω-invariant
LPO: For P ∈ Σ1 , P ∨ ¬P
LPO: For P ∈ Σ1 , P V ∼P
l
WV
LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0)) LPR: (∀x ∈ R)(x > 0 V ∼(x > 0))
l
WV
MCT: monotone convergence thm MCT: monotone convergence thm
l
WV
CIT: Cantor intersection thm
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics
BISH (based on BHK)
NSA (based on CL)
non-constructive/non-algorithmic
non-Ω-invariant
LPO: For P ∈ Σ1 , P ∨ ¬P
LPO: For P ∈ Σ1 , P V ∼P
l
WV
LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0)) LPR: (∀x ∈ R)(x > 0 V ∼(x > 0))
l
WV
MCT: monotone convergence thm MCT: monotone convergence thm
l
WV
CIT: Cantor intersection thm
CIT: Cantor intersection thm
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics
BISH (based on BHK)
NSA (based on CL)
non-constructive/non-algorithmic
non-Ω-invariant
LPO: For P ∈ Σ1 , P ∨ ¬P
LPO: For P ∈ Σ1 , P V ∼P
l
WV
LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0)) LPR: (∀x ∈ R)(x > 0 V ∼(x > 0))
l
WV
MCT: monotone convergence thm MCT: monotone convergence thm
l (limit computed by algo) WV
CIT: Cantor intersection thm
CIT: Cantor intersection thm
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics
BISH (based on BHK)
NSA (based on CL)
non-constructive/non-algorithmic
non-Ω-invariant
LPO: For P ∈ Σ1 , P ∨ ¬P
LPO: For P ∈ Σ1 , P V ∼P
l
WV
LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0)) LPR: (∀x ∈ R)(x > 0 V ∼(x > 0))
l
WV
MCT: monotone convergence thm MCT: monotone convergence thm
l (limit computed by algo) WV (limit computed by Ω-inv. proc.)
CIT: Cantor intersection thm
CIT: Cantor intersection thm
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics
BISH (based on BHK)
NSA (based on CL)
non-constructive/non-algorithmic
non-Ω-invariant
LPO: For P ∈ Σ1 , P ∨ ¬P
LPO: For P ∈ Σ1 , P V ∼P
l
WV
LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0)) LPR: (∀x ∈ R)(x > 0 V ∼(x > 0))
l
WV
MCT: monotone convergence thm MCT: monotone convergence thm
l (limit computed by algo) WV (limit computed by Ω-inv. proc.)
CIT: Cantor intersection thm
CIT: Cantor intersection thm
(point in intersection computed by algo)
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics
BISH (based on BHK)
NSA (based on CL)
non-constructive/non-algorithmic
non-Ω-invariant
LPO: For P ∈ Σ1 , P ∨ ¬P
LPO: For P ∈ Σ1 , P V ∼P
l
WV
LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0)) LPR: (∀x ∈ R)(x > 0 V ∼(x > 0))
l
WV
MCT: monotone convergence thm MCT: monotone convergence thm
l (limit computed by algo) WV (limit computed by Ω-inv. proc.)
CIT: Cantor intersection thm
CIT: Cantor intersection thm
(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
BISH (based on BHK)
NSA (based on CL)
non-constructive/non-algorithmic
non-Ω-invariant
LPO: For P ∈ Σ1 , P ∨ ¬P
LPO: For P ∈ Σ1 , P V ∼P
l
WV
LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0)) LPR: (∀x ∈ R)(x > 0 V ∼(x > 0))
l
WV
MCT: monotone convergence thm MCT: monotone convergence thm
l (limit computed by algo) WV (limit computed by Ω-inv. proc.)
CIT: Cantor intersection thm
CIT: Cantor intersection thm
WV
Universal Transfer
(∀n ∈ N)ϕ(n) → (∀n ∈ ∗ N)ϕ(n)
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics
BISH (based on BHK)
NSA (based on CL)
non-constructive/non-algorithmic
non-Ω-invariant
LPO: For P ∈ Σ1 , P ∨ ¬P
LPO: For P ∈ Σ1 , P V ∼P
l
WV
LPR: (∀x ∈ R)(x > 0 ∨ ¬(x > 0)) LPR: (∀x ∈ R)(x > 0 V ∼(x > 0))
l
WV
MCT: monotone convergence thm MCT: monotone convergence thm
l (limit computed by algo) WV (limit computed by Ω-inv. proc.)
CIT: Cantor intersection thm
CIT: Cantor intersection thm
WV
Universal Transfer
(∀n ∈ N)ϕ(n) → (∀n ∈ ∗ N)ϕ(n)
NSA does prove (∀δ ∈ R) δ > 0 V (x > 0) V(x < δ) .
BISH does prove (∀δ ∈ R) δ > 0 → (x > 0) ∨ (x < δ) .
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics II
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics II
BISH (based on BHK)
non-constructive/non-algorithmic
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics II
BISH (based on BHK)
non-constructive/non-algorithmic
LLPO
For P, Q ∈ Σ1 , ¬(P ∧ Q) → ¬P ∨ ¬Q
l
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics II
BISH (based on BHK)
non-constructive/non-algorithmic
LLPO
For P, Q ∈ Σ1 , ¬(P ∧ Q) → ¬P ∨ ¬Q
l
LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
l
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics II
BISH (based on BHK)
non-constructive/non-algorithmic
LLPO
For P, Q ∈ Σ1 , ¬(P ∧ Q) → ¬P ∨ ¬Q
l
LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
l
NIL
(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)
l
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics II
BISH (based on BHK)
non-constructive/non-algorithmic
LLPO
For P, Q ∈ Σ1 , ¬(P ∧ Q) → ¬P ∨ ¬Q
l
LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
l
NIL
(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)
l
IVT: Intermediate value theorem
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics II
BISH (based on BHK)
non-constructive/non-algorithmic
LLPO
For P, Q ∈ Σ1 , ¬(P ∧ Q) → ¬P ∨ ¬Q
l
LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
l
NIL
(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)
l
IVT: Intermediate value theorem
NSA (based on CL)
non-Ω-invariant
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics II
BISH (based on BHK)
non-constructive/non-algorithmic
LLPO
For P, Q ∈ Σ1 , ¬(P ∧ Q) → ¬P ∨ ¬Q
l
LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
l
NIL
(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)
l
IVT: Intermediate value theorem
NSA (based on CL)
non-Ω-invariant
LLPO
For P, Q ∈ Σ1 , ∼(P ∧ Q) V ∼P V ∼Q
WV
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics II
BISH (based on BHK)
non-constructive/non-algorithmic
LLPO
For P, Q ∈ Σ1 , ¬(P ∧ Q) → ¬P ∨ ¬Q
l
LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
l
NIL
(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)
l
IVT: Intermediate value theorem
NSA (based on CL)
non-Ω-invariant
LLPO
For P, Q ∈ Σ1 , ∼(P ∧ Q) V ∼P V ∼Q
WV
LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0)
WV
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics II
BISH (based on BHK)
non-constructive/non-algorithmic
LLPO
For P, Q ∈ Σ1 , ¬(P ∧ Q) → ¬P ∨ ¬Q
l
LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
l
NIL
(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)
l
IVT: Intermediate value theorem
NSA (based on CL)
non-Ω-invariant
LLPO
For P, Q ∈ Σ1 , ∼(P ∧ Q) V ∼P V ∼Q
WV
LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0)
WV
NIL
(∀x, y ∈ R)(xy = 0 V x = 0 V y = 0)
WV
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics II
BISH (based on BHK)
non-constructive/non-algorithmic
LLPO
For P, Q ∈ Σ1 , ¬(P ∧ Q) → ¬P ∨ ¬Q
l
LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
l
NIL
(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)
NSA (based on CL)
non-Ω-invariant
LLPO
For P, Q ∈ Σ1 , ∼(P ∧ Q) V ∼P V ∼Q
WV
LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0)
WV
NIL
(∀x, y ∈ R)(xy = 0 V x = 0 V y = 0)
l
WV
IVT: Intermediate value theorem IVT: Intermediate value theorem
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics II
BISH (based on BHK)
non-constructive/non-algorithmic
LLPO
For P, Q ∈ Σ1 , ¬(P ∧ Q) → ¬P ∨ ¬Q
l
LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
l
NIL
(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)
NSA (based on CL)
non-Ω-invariant
LLPO
For P, Q ∈ Σ1 , ∼(P ∧ Q) V ∼P V ∼Q
WV
LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0)
WV
NIL
(∀x, y ∈ R)(xy = 0 V x = 0 V y = 0)
l
WV
IVT: Intermediate value theorem IVT: Intermediate value theorem
(int. value computed by algo)
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics II
BISH (based on BHK)
non-constructive/non-algorithmic
LLPO
For P, Q ∈ Σ1 , ¬(P ∧ Q) → ¬P ∨ ¬Q
l
LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
l
NIL
(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)
NSA (based on CL)
non-Ω-invariant
LLPO
For P, Q ∈ Σ1 , ∼(P ∧ Q) V ∼P V ∼Q
WV
LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0)
WV
NIL
(∀x, y ∈ R)(xy = 0 V x = 0 V y = 0)
l
WV
IVT: Intermediate value theorem 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 II
BISH (based on BHK)
non-constructive/non-algorithmic
LLPO
For P, Q ∈ Σ1 , ¬(P ∧ Q) → ¬P ∨ ¬Q
l
LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
l
NIL
(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)
NSA (based on CL)
non-Ω-invariant
LLPO
For P, Q ∈ Σ1 , ∼(P ∧ Q) V ∼P V ∼Q
WV
LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0)
WV
NIL
(∀x, y ∈ R)(xy = 0 V x = 0 V y = 0)
l
WV
IVT: Intermediate value theorem IVT: Intermediate value theorem
l (int. value computed by algo) (int. value computed by Ω-inv. proc.)
WKL
WV WKL
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics II
BISH (based on BHK)
non-constructive/non-algorithmic
LLPO
For P, Q ∈ Σ1 , ¬(P ∧ Q) → ¬P ∨ ¬Q
l
LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
l
NIL
(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)
NSA (based on CL)
non-Ω-invariant
LLPO
For P, Q ∈ Σ1 , ∼(P ∧ Q) V ∼P V ∼Q
WV
LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0)
WV
NIL
(∀x, y ∈ R)(xy = 0 V x = 0 V y = 0)
l
WV
IVT: Intermediate value theorem IVT: Intermediate value theorem
l (int. value computed by algo) (int. value computed by Ω-inv. proc.)
WKL
WV WKL WV ∨-Transfer
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics II
BISH (based on BHK)
non-constructive/non-algorithmic
LLPO
For P, Q ∈ Σ1 , ¬(P ∧ Q) → ¬P ∨ ¬Q
l
LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
l
NIL
(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)
NSA (based on CL)
non-Ω-invariant
LLPO
For P, Q ∈ Σ1 , ∼(P ∧ Q) V ∼P V ∼Q
WV
LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0)
WV
NIL
(∀x, y ∈ R)(xy = 0 V x = 0 V y = 0)
l
WV
IVT: Intermediate value theorem IVT: Intermediate value theorem
l (int. value computed by algo) (int. value computed by Ω-inv. proc.)
WKL
WV WKL WV ∨-Transfer
Axioms of R: ¬(x > 0 ∧ x < 0)
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics II
BISH (based on BHK)
non-constructive/non-algorithmic
LLPO
For P, Q ∈ Σ1 , ¬(P ∧ Q) → ¬P ∨ ¬Q
l
LLPR: (∀x ∈ R)(x ≥ 0 ∨ x ≤ 0)
l
NIL
(∀x, y ∈ R)(xy = 0 → x = 0 ∨ y = 0)
NSA (based on CL)
non-Ω-invariant
LLPO
For P, Q ∈ Σ1 , ∼(P ∧ Q) V ∼P V ∼Q
WV
LLPR: (∀x ∈ R)(x ≥ 0 V x ≤ 0)
WV
NIL
(∀x, y ∈ R)(xy = 0 V x = 0 V y = 0)
l
WV
IVT: Intermediate value theorem IVT: Intermediate value theorem
l (int. value computed by algo) (int. value computed by Ω-inv. proc.)
WKL
WV WKL WV ∨-Transfer
Axioms of R: ¬(x > 0 ∧ x < 0)
Axioms of R: ∼(x > 0 ∧ x < 0)
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics III
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics III
BISH (based on BHK)
non-constructive/non-algorithmic
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics III
BISH (based on BHK)
non-constructive/non-algorithmic
MP: For P ∈ Σ1 , ¬¬P → P
l
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics III
BISH (based on BHK)
non-constructive/non-algorithmic
MP: For P ∈ Σ1 , ¬¬P → P
l
MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
l
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics III
BISH (based on BHK)
non-constructive/non-algorithmic
MP: For P ∈ Σ1 , ¬¬P → P
l
MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
l
EXT: the extensionality theorem
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics III
BISH (based on BHK)
non-constructive/non-algorithmic
MP: For P ∈ Σ1 , ¬¬P → P
l
MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
l
EXT: the extensionality theorem
NSA (based on CL)
non-Ω-invariant
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics III
BISH (based on BHK)
non-constructive/non-algorithmic
NSA (based on CL)
non-Ω-invariant
MP: For P ∈ Σ1 , ¬¬P → P
MP: For P ∈ Σ1 , ∼∼P V P
l
WV
MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0)
l
EXT: the extensionality theorem
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics III
BISH (based on BHK)
non-constructive/non-algorithmic
NSA (based on CL)
non-Ω-invariant
MP: For P ∈ Σ1 , ¬¬P → P
MP: For P ∈ Σ1 , ∼∼P V P
l
WV
MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0) MPR: (∀x ∈ R)(∼∼(x > 0) V x > 0)
l
WV
EXT: the extensionality theorem
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics III
BISH (based on BHK)
non-constructive/non-algorithmic
NSA (based on CL)
non-Ω-invariant
MP: For P ∈ Σ1 , ¬¬P → P
MP: For P ∈ Σ1 , ∼∼P V P
l
WV
MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0) MPR: (∀x ∈ R)(∼∼(x > 0) V x > 0)
l
WV
EXT: the extensionality theorem
EXT: the extensionality theorem
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics III
BISH (based on BHK)
non-constructive/non-algorithmic
NSA (based on CL)
non-Ω-invariant
MP: For P ∈ Σ1 , ¬¬P → P
MP: For P ∈ Σ1 , ∼∼P V P
l
WV
MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0) MPR: (∀x ∈ R)(∼∼(x > 0) V x > 0)
l
WV
EXT: the extensionality theorem
EXT: the extensionality theorem
WLPO: For P ∈ Σ1 , ¬¬P ∨ ¬P
l
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics III
BISH (based on BHK)
non-constructive/non-algorithmic
NSA (based on CL)
non-Ω-invariant
MP: For P ∈ Σ1 , ¬¬P → P
MP: For P ∈ Σ1 , ∼∼P V P
l
WV
MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0) MPR: (∀x ∈ R)(∼∼(x > 0) V x > 0)
l
WV
EXT: the extensionality theorem
EXT: the extensionality theorem
WLPO: For P ∈ Σ1 , ¬¬P ∨ ¬P
l
WLPR: (∀x ∈ R) ¬¬(x > 0) ∨ ¬(x > 0)
l
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics III
BISH (based on BHK)
non-constructive/non-algorithmic
NSA (based on CL)
non-Ω-invariant
MP: For P ∈ Σ1 , ¬¬P → P
MP: For P ∈ Σ1 , ∼∼P V P
l
WV
MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0) MPR: (∀x ∈ R)(∼∼(x > 0) V x > 0)
l
WV
EXT: the extensionality theorem
EXT: the extensionality theorem
WLPO: For P ∈ Σ1 , ¬¬P ∨ ¬P
l
WLPR: (∀x ∈ R) ¬¬(x > 0) ∨ ¬(x > 0)
l
DISC:
A discontinuous 2N → N-function exists.
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics III
BISH (based on BHK)
non-constructive/non-algorithmic
NSA (based on CL)
non-Ω-invariant
MP: For P ∈ Σ1 , ¬¬P → P
MP: For P ∈ Σ1 , ∼∼P V P
l
WV
MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0) MPR: (∀x ∈ R)(∼∼(x > 0) V x > 0)
l
WV
EXT: the extensionality theorem
EXT: the extensionality theorem
WLPO: For P ∈ Σ1 , ¬¬P ∨ ¬P
WLPO: For P ∈ Σ1 , ∼∼P V ∼P
l
WV
WLPR: (∀x ∈ R) ¬¬(x > 0) ∨ ¬(x > 0)
l
DISC:
A discontinuous 2N → N-function exists.
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics III
BISH (based on BHK)
non-constructive/non-algorithmic
NSA (based on CL)
non-Ω-invariant
MP: For P ∈ Σ1 , ¬¬P → P
MP: For P ∈ Σ1 , ∼∼P V P
l
WV
MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0) MPR: (∀x ∈ R)(∼∼(x > 0) V x > 0)
l
WV
EXT: the extensionality theorem
EXT: the extensionality theorem
WLPO: For P ∈ Σ1 , ¬¬P ∨ ¬P
WLPO: For P ∈ Σ1 , ∼∼P V ∼P
l
WV
WLPR: (∀x ∈ R) ¬¬(x > 0) ∨ ¬(x > 0) WLPR: (∀x ∈ R) ∼∼(x > 0) V ∼(x > 0)
l
WV
DISC:
A discontinuous 2N → N-function exists.
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics III
BISH (based on BHK)
non-constructive/non-algorithmic
NSA (based on CL)
non-Ω-invariant
MP: For P ∈ Σ1 , ¬¬P → P
MP: For P ∈ Σ1 , ∼∼P V P
l
WV
MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0) MPR: (∀x ∈ R)(∼∼(x > 0) V x > 0)
l
WV
EXT: the extensionality theorem
EXT: the extensionality theorem
WLPO: For P ∈ Σ1 , ¬¬P ∨ ¬P
WLPO: For P ∈ Σ1 , ∼∼P V ∼P
l
WV
WLPR: (∀x ∈ R) ¬¬(x > 0) ∨ ¬(x > 0) WLPR: (∀x ∈ R) ∼∼(x > 0) V ∼(x > 0)
l
WV
DISC:
DISC: A discontinuous
A discontinuous 2N → N-function exists.
2N → N-function exists.
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Constructive Reverse Mathematics III
BISH (based on BHK)
non-constructive/non-algorithmic
NSA (based on CL)
non-Ω-invariant
MP: For P ∈ Σ1 , ¬¬P → P
MP: For P ∈ Σ1 , ∼∼P V P
l
WV
MPR: (∀x ∈ R)(¬¬(x > 0) → x > 0) MPR: (∀x ∈ R)(∼∼(x > 0) V x > 0)
l
WV
EXT: the extensionality theorem
EXT: the extensionality theorem
WLPO: For P ∈ Σ1 , ¬¬P ∨ ¬P
WLPO: For P ∈ Σ1 , ∼∼P V ∼P
l
WV
WLPR: (∀x ∈ R) ¬¬(x > 0) ∨ ¬(x > 0) WLPR: (∀x ∈ R) ∼∼(x > 0) V ∼(x > 0)
l
WV
DISC:
DISC: A discontinuous
A discontinuous 2N → N-function exists.
2N → N-function exists.
(Four Remarks)
Introduction
NSA, BISH and Constructive Reverse Mathematics
Ω-invariance is weaker than Recursive
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
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.
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
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.
Conclusion
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 WV (∃n ∈ N)ϕ1 (n) WV (∀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 WV (∃n ∈ N)ϕ1 (n) WV (∀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 WV (∃n ∈ N)ϕ1 (n) WV (∀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
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 WV (∃n ∈ N)ϕ1 (n) WV (∀m ∈ N)ϕ2 (m).
Theorem (In NSA)
Only given MP, every ∆1 -formula is decidable.
But MP is not available in NSA!
Examples of non-Ω-invariant procedures?
Introduction
NSA, BISH and Constructive Reverse Mathematics
Fannying about: FAN∆ vs WKL
FAN∆ (Every detachable bar is uniform) is accepted in INT.
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Fannying about: FAN∆ vs WKL
FAN∆ (Every detachable bar is uniform) is accepted in INT.
WKL (Every infinite tree T ⊂ 2N has a path)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
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.
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
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.
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
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 about NSA?
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
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 about NSA?
WKL(∀n ∈ N)(∃α ∈ 2N )(αn ∈ T ) V (∃α ∈ 2N )(∀n ∈ N)(αn ∈ T )
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
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 about NSA?
WKL(∀n ∈ N)(∃α ∈ 2N )(αn ∈ T ) V (∃α ∈ 2N )(∀n ∈ N)(αn ∈ T )
≈ If the trees T and ∗ T are (hyper)infinite, they share a path.
Conclusion
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 about NSA?
WKL(∀n ∈ N)(∃α ∈ 2N )(αn ∈ T ) V (∃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) V (∃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 about NSA?
WKL(∀n ∈ N)(∃α ∈ 2N )(αn ∈ T ) V (∃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) V (∃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 about NSA?
WKL(∀n ∈ N)(∃α ∈ 2N )(αn ∈ T ) V (∃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) V (∃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 V FAN∆ , and both are unavailable.
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics IV
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics IV
Same for WMP, FAN∆ , BD-N, and MP∨ .
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics IV
Same for WMP, FAN∆ , BD-N, and MP∨ .
Same for ‘mixed’ theorems:
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics IV
Same for WMP, FAN∆ , BD-N, and MP∨ .
Same for ‘mixed’ theorems:
BISH (based on BHK)
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics IV
Same for WMP, FAN∆ , BD-N, and MP∨ .
Same for ‘mixed’ theorems:
BISH (based on BHK)
LPO ↔ MP+WLPO
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics IV
Same for WMP, FAN∆ , BD-N, and MP∨ .
Same for ‘mixed’ theorems:
BISH (based on BHK)
LPO ↔ MP+WLPO
MP ↔ WMP + MP∨
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics IV
Same for WMP, FAN∆ , BD-N, and MP∨ .
Same for ‘mixed’ theorems:
BISH (based on BHK)
LPO ↔ MP+WLPO
MP ↔ WMP + MP∨
WLPO → LLPO
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics IV
Same for WMP, FAN∆ , BD-N, and MP∨ .
Same for ‘mixed’ theorems:
BISH (based on BHK)
LPO ↔ MP+WLPO
MP ↔ WMP + MP∨
WLPO → LLPO
LLPO → MP∨
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics IV
Same for WMP, FAN∆ , BD-N, and MP∨ .
Same for ‘mixed’ theorems:
BISH (based on BHK)
LPO ↔ MP+WLPO
MP ↔ WMP + MP∨
WLPO → LLPO
LLPO → MP∨
LPO → BD-N
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics IV
Same for WMP, FAN∆ , BD-N, and MP∨ .
Same for ‘mixed’ theorems:
BISH (based on BHK)
LPO ↔ MP+WLPO
MP ↔ WMP + MP∨
WLPO → LLPO
LLPO → MP∨
LPO → BD-N
LLPO → FAN∆
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics IV
Same for WMP, FAN∆ , BD-N, and MP∨ .
Same for ‘mixed’ theorems:
BISH (based on BHK)
LPO ↔ MP+WLPO
MP ↔ WMP + MP∨
WLPO → LLPO
LLPO → MP∨
LPO → BD-N
LLPO → FAN∆
LLPO ↔ WKL
NSA (based on CL)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Constructive Reverse Mathematics IV
Same for WMP, FAN∆ , BD-N, and MP∨ .
Same for ‘mixed’ theorems:
BISH (based on BHK)
LPO ↔ MP+WLPO
MP ↔ WMP + MP∨
WLPO → LLPO
LLPO → MP∨
LPO → BD-N
LLPO → FAN∆
LLPO ↔ WKL
NSA (based on CL)
LPO WV MP + WLPO
MP WV WMP + MP∨
WLPO V LLPO
LLPO V MP∨
LPO V BD-N
LLPO V FAN∆
LLPO WV WKL
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion: NSA ≈ BISH
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion: NSA ≈ BISH
For Bishop’s notion of algorithm, we conclude:
Algorithmic ≈ Ω-invariant
because
Non-algorithmic ≈ Non-Ω-invariant.
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion: NSA ≈ BISH
For Bishop’s notion of algorithm, we conclude:
Algorithmic ≈ Ω-invariant
because
Non-algorithmic ≈ Non-Ω-invariant.
Reuniting the antipodes (Palmgren & Moerdijk).
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion: NSA ≈ BISH
For Bishop’s notion of algorithm, we conclude:
Algorithmic ≈ Ω-invariant
because
Non-algorithmic ≈ Non-Ω-invariant.
Reuniting the antipodes (Palmgren & Moerdijk).
Reverse-engineering Reverse Mathematics (Fuchino-sensei)
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
A note on Coding and Assymetry
Recall that ‘A ∈ T’ means ‘A satisfies Transfer’.
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
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)]
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
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)]
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
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.
Conclusion
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
Future work: Type Theory
Martin-Löf intended his type theory as a foundation for BISH.
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Future work: Type Theory
Martin-Löf intended his type theory as a foundation for BISH.
Can Ω-invariance help capture e.g. Type Theory?
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Future work: Type Theory
Martin-Löf intended his type theory as a foundation for BISH.
Can Ω-invariance help capture e.g. Type Theory?
Homotopy:
f (x)
•
g (x)
•
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Future work: Type Theory
Martin-Löf intended his type theory as a foundation for BISH.
Can Ω-invariance help capture e.g. Type Theory?
Homotopy: continuous transformation ht of f to g (t ∈ [0, 1]).
f (x)
•
g (x)
•
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Future work: Type Theory
Martin-Löf intended his type theory as a foundation for BISH.
Can Ω-invariance help capture e.g. Type Theory?
Homotopy: continuous transformation ht of f to g (t ∈ [0, 1]).
h0 (x) = f (x)
••
h1 (x) = g (x)
ht1 (x)
ht2 (x)
..
.
..
.
ht3 (x)
..
.
••
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Future work: Type Theory
Martin-Löf intended his type theory as a foundation for BISH.
Can Ω-invariance help capture e.g. Type Theory?
Homotopy:
•
kω (x) ≈ f (x)
•
•PP
PP
P
•
@
@
•
@
@
@•
@
@•Q
mω (x) ≈ g (x)
QQ
•PP
PP
P•P
PP
P
P•
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Future work: Type Theory
Martin-Löf intended his type theory as a foundation for BISH.
Can Ω-invariance help capture e.g. Type Theory?
Homotopy:
•P
kω (x) ≈ f (x)
•
•
•PP
PP
P•
PP ⇓
@
P
P•
@
increment is multiple of ω1@
@
@•
@
@•Q
mω (x) ≈ g (x)
QQ
•PP
PP
P•P
PP
P
P•
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Future work: Type Theory
Martin-Löf intended his type theory as a foundation for BISH.
Can Ω-invariance help capture e.g. Type Theory?
Homotopy:
•PP
PP
P•
kω (x) ≈ f (x) PP ⇓
@ ONE basic step
P
P
•
•
@
increment is multiple of ω1@
•
@
@
@•Q
@•
mω (x) ≈ g (x) QQ
•PP
PP
P•P
PP
P
P•
•P
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Future work: Type Theory
Martin-Löf intended his type theory as a foundation for BISH.
Can Ω-invariance help capture e.g. Type Theory?
Homotopy:
•PP
PP
P•
kω (x) ≈ f (x) PP ⇓
@ ONE basic step
P
P
•
•
@
..
@
•
.
@
@
ω basic steps
@•Q
@•
mω (x) ≈ g (x) QQ
.
•PP ..
PP
P•P
PP
P
P•
•P
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Future work: Type Theory
Martin-Löf intended his type theory as a foundation for BISH.
Can Ω-invariance help capture e.g. Type Theory?
Homotopy:
•PP
PP
P•
kω (x) ≈ f (x) PP ⇓
@ ONE basic step
P
P
•
•
@
..
@
•
.
@
@
ω basic steps
@•Q
@•
mω (x) ≈ g (x) QQ
.
•PP ..
PP
P•P
PP
P
P•
•P
Independent of the choice of ω
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Conclusion
Future work: Type Theory
Martin-Löf intended his type theory as a foundation for BISH.
Can Ω-invariance help capture e.g. Type Theory?
Homotopy:
≈ Ω-invariant broken-line transformation hω,t of f to g .
•PP
PP
P•
kω (x) ≈ f (x) PP ⇓
@ ONE basic step
P
P
•
•
@
..
@
•
.
@
@
ω basic steps
@•Q
@•
mω (x) ≈ g (x) QQ
.
•PP ..
PP
P•P
PP
P
P•
•P
Independent of the choice of ω
Introduction
NSA, BISH and Constructive Reverse Mathematics
Philosophy of Physics
Conclusion
Introduction
NSA, BISH and Constructive Reverse Mathematics
Philosophy of Physics
Why is Mathematics in Physics so constructive/computable?
Conclusion
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
Final Thoughts
NSA, BISH and Constructive Reverse Mathematics
Conclusion
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 Gödel
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 Gödel
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 Gödel
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 Gödel
We thank the John Templeton Foundation for its generous support!
Thank you for your attention!
Any questions?
Introduction
NSA, BISH and Constructive Reverse Mathematics
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.
Conclusion