A copy of several Reverse Mathematics Sam Sanders Department of Pure Mathematics Ghent University Belgium
April 23, 2010, Tohoku University
FACULTY OF SCIENCES
Outline
1
Introduction
2
A copy of the RM of WKL0
3
A constructive copy
4
Big questions
5
Future research
Reverse Mathematics
Reverse Mathematics = finding the minimal axioms A that prove a theorem T .
Reverse Mathematics
Reverse Mathematics = finding the minimal axioms A that prove a theorem T . T is a theorem of ordinary mathematics in many cases: A equivalent to T
Big Five: RCA0 , WKL0 , ACA0 , ATR0 and Π11 -CA0
Reverse Mathematics
Reverse Mathematics = finding the minimal axioms A that prove a theorem T . T is a theorem of ordinary mathematics in many cases: A equivalent to T
Big Five: RCA0 , WKL0 , ACA0 , ATR0 and Π11 -CA0 Most theorems of ‘everyday’ mathematics are either provable in RCA0 or equivalent to one of the ‘Big Five’ theories.
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
An example: Reverse Mathematics for WKL0 Central principle:
Theorem (Weak K¨onig’s Lemma) Every infinite binary tree has an infinite path.
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
An example: Reverse Mathematics for WKL0 Central principle:
Theorem (Weak K¨onig’s Lemma) Every infinite binary tree has an infinite path. Assuming the base theory RCA0 , WKL is equivalent to 1
Heine-Borel: every countable covering of [0, 1] has a finite subcovering.
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
An example: Reverse Mathematics for WKL0 Central principle:
Theorem (Weak K¨onig’s Lemma) Every infinite binary tree has an infinite path. Assuming the base theory RCA0 , WKL is equivalent to 1
Heine-Borel: every countable covering of [0, 1] has a finite subcovering.
2
A continuous function on [0, 1] is uniformly continuous.
3
A continuous function on [0, 1] is Riemann integrable.
4
Weierstrass’ theorem: a continuous function on [0, 1] attains its maximum.
5
Peano’s theorem for differential equations y 0 = f (x, y ).
Introduction
7
A copy of the RM of WKL0
A constructive copy
G¨odel’s completeness theorem.
Big questions
Future research
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
7
G¨odel’s completeness theorem.
8
A countable commutative ring has a prime ideal.
9
A countable formally real field is orderable.
10
A countable formally real field has a (unique) closure.
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
7
G¨odel’s completeness theorem.
8
A countable commutative ring has a prime ideal.
9
A countable formally real field is orderable.
10
A countable formally real field has a (unique) closure.
11
Brouwer’s fixed point theorem: A continuous function from [0, 1]n to [0, 1]n has a fixed point.
12
The separable Hahn-Banach theorem.
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
7
G¨odel’s completeness theorem.
8
A countable commutative ring has a prime ideal.
9
A countable formally real field is orderable.
10
A countable formally real field has a (unique) closure.
11
Brouwer’s fixed point theorem: A continuous function from [0, 1]n to [0, 1]n has a fixed point.
12
The separable Hahn-Banach theorem.
13
A continuous function on [0, 1] can be approximated by (Bernstein) polynomials.
14
And many more. . .
References
Introduction
A copy of the RM of WKL0
A constructive copy
A ‘gap’ in Reverse Mathematics
Big questions
Future research
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
A ‘gap’ in Reverse Mathematics The ’Big Five’ of RM have first-order strength at least I Σ1 ([5]).
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
A ‘gap’ in Reverse Mathematics The ’Big Five’ of RM have first-order strength at least I Σ1 ([5]). Weak
Strong -
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
A ‘gap’ in Reverse Mathematics The ’Big Five’ of RM have first-order strength at least I Σ1 ([5]). Weak
Strong I Σ1 I Σ2 . . . PA
Π11 -CA0 -
RCA0 WKL0 |
ACA0 ATR0 {z
Big Five
}
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
A ‘gap’ in Reverse Mathematics The ’Big Five’ of RM have first-order strength at least I Σ1 ([5]). Weak Q
|
Strong
S21 S22 . . . S2
{z
Bounded Arithmetic
I Σ1 I Σ2 . . . PA
Π11 -CA0 -
}
RCA0 WKL0 |
ACA0 ATR0 {z
Big Five
}
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
A ‘gap’ in Reverse Mathematics The ’Big Five’ of RM have first-order strength at least I Σ1 ([5]). Weak Q
S21 S22 . . . S2
Riemann integral is not definable. . . | {z } Bounded Arithmetic
Strong I Σ1 I Σ2 . . . PA
Π11 -CA0 -
RCA0 WKL0 |
ACA0 ATR0 {z
Big Five
}
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
A ‘gap’ in Reverse Mathematics The ’Big Five’ of RM have first-order strength at least I Σ1 ([5]). GAP
z Q
S21 S22 . . . S2
Riemann integral is not definable. . . | {z } Bounded Arithmetic
}|
{ I Σ1 I Σ2 . . . PA
Π11 -CA0 -
RCA0 WKL0 |
ACA0 ATR0 {z
Big Five
}
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
A ‘gap’ in Reverse Mathematics The ’Big Five’ of RM have first-order strength at least I Σ1 ([5]). GAP
z Q
S21 S22 . . . S2
}| I ∆0 + exp
Riemann integral (and many others) is not definable. . . | {z } Bounded Arithmetic
{ I Σ1 I Σ2 . . . PA
Π11 -CA0 -
RCA0 WKL0 |
ACA0 ATR0 {z
Big Five
}
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
A ‘gap’ in Reverse Mathematics The ’Big Five’ of RM have first-order strength at least I Σ1 ([5]). GAP
z Q
S21 S22 . . . S2
}| I ∆0 + exp
Riemann integral (and many others) is not definable. . . | {z } Bounded Arithmetic
{ I Σ1 I Σ2 . . . PA
Π11 -CA0 -
RCA0 WKL0 |
What about the RM of I ∆0 + exp?
ACA0 ATR0 {z
Big Five
}
Introduction
A copy of the RM of WKL0
RM for I ∆0 + exp
A constructive copy
Big questions
Future research
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
RM for I ∆0 + exp
We work in ERNA, a nonstandard version of I ∆0 + exp.
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
RM for I ∆0 + exp
We work in ERNA, a nonstandard version of I ∆0 + exp. ERNA has a field ∗ Q:
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
RM for I ∆0 + exp
We work in ERNA, a nonstandard version of I ∆0 + exp. ERNA has a field ∗ Q:
. . . −1 0
1 2
1
5 3
2 ... -
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
RM for I ∆0 + exp
We work in ERNA, a nonstandard version of I ∆0 + exp. ERNA has a field ∗ Q:
1
. . . −1 0 2 1 | {z
5 3
2 ... }
Q, the rational numbers
-
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
RM for I ∆0 + exp
We work in ERNA, a nonstandard version of I ∆0 + exp. ERNA has a field ∗ Q: new numbers, not in Q 1 2
1 . . . −1 0 | {z
5 3
2 ... }
Q, the rational numbers
z
}|
{ -
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
RM for I ∆0 + exp
We work in ERNA, a nonstandard version of I ∆0 + exp. ERNA has a field ∗ Q: new numbers, not in Q 1 2
1 . . . −1 0 | {z
5 3
2 ... }
Q, the rational numbers
z
}| ω
{ -
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
RM for I ∆0 + exp
We work in ERNA, a nonstandard version of I ∆0 + exp. ERNA has a field ∗ Q: new numbers, not in Q 1 2
1 . . . −1 0 | {z
5 3
2 ... }
Q, the rational numbers
z
}| ω/2 ω
{ -
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
RM for I ∆0 + exp
We work in ERNA, a nonstandard version of I ∆0 + exp. ERNA has a field ∗ Q: new numbers, not in Q 1 2
1 . . . −1 0 | {z
5 3
2 ... }
Q, the rational numbers
z
}| ω/2 ω
{ 2ω-
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
RM for I ∆0 + exp
We work in ERNA, a nonstandard version of I ∆0 + exp. ERNA has a field ∗ Q: new numbers, not in Q 1 2
1 . . . −1 0 | {z
5 3
z √
}| 2 . . .d ωe ω/2 ω }
Q, the rational numbers
{ 2ω-
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
RM for I ∆0 + exp
We work in ERNA, a nonstandard version of I ∆0 + exp. ERNA has a field ∗ Q:
1
. . . −1 0 2 1 | {z
5 3
√ 2 . . .d ωe ω/2 ω } | {z
Q, the rational numbers
infinite numbers
2ω}
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
RM for I ∆0 + exp
We work in ERNA, a nonstandard version of I ∆0 + exp. ERNA has a field ∗ Q:
1
. . . −1 0 2 1 | {z
5 3
√ 2 . . .d ωe ω/2 ω } | {z
Q, the rational numbers
2ω}
positive infinite numbers
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
RM for I ∆0 + exp
We work in ERNA, a nonstandard version of I ∆0 + exp. ERNA has a field ∗ Q:
√ 1 −2ω −ω −d ωe . . . −1 0 2 1 | {z }| {z negative infinite numbers
5 3
√ 2 . . .d ωe ω/2 ω } | {z
Q, the rational numbers
2ω}
positive infinite numbers
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
RM for I ∆0 + exp
We work in ERNA, a nonstandard version of I ∆0 + exp. ERNA has a field ∗ Q:
√ 1 −2ω −ω −d ωe . . . −1 0 2 1 | {z }| {z negative infinite numbers
5 3
√ 2 . . .d ωe ω/2 ω } | {z
Q, the rational numbers
x is infinite iff |x| > q, for all q ∈ Q+
2ω}
positive infinite numbers
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
RM for I ∆0 + exp
We work in ERNA, a nonstandard version of I ∆0 + exp. ERNA has a field ∗ Q:
√ 1 −2ω −ω −d ωe . . . −1 0 2 1 | {z }| {z negative infinite numbers
5 3
the finite numbers
√ 2 . . .d ωe ω/2 ω } | {z
x is infinite iff |x| > q, for all q ∈ Q+
2ω}
positive infinite numbers
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
RM for I ∆0 + exp
We work in ERNA, a nonstandard version of I ∆0 + exp. ERNA has a field ∗ Q:
√ 1 −2ω −ω −d ωe . . . −1 0 2 1 | {z }| {z negative infinite numbers
5 3
the finite numbers
√ 2 . . .d ωe ω/2 ω } | {z
2ω}
positive infinite numbers
x is infinite iff |x| > q, for all q ∈ Q+ x is infinitely small iff |x| < q, for all q ∈ Q+
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
RM for I ∆0 + exp
We work in ERNA, a nonstandard version of I ∆0 + exp. ERNA has a field ∗ Q:
1 1 √ −2ω −ω −d ωe . . . −1 0 ω 2 1 | {z }| {z negative infinite numbers
5 3
the finite numbers
√ 2 . . .d ωe ω/2 ω } | {z
2ω}
positive infinite numbers
x is infinite iff |x| > q, for all q ∈ Q+ x is infinitely small iff |x| < q, for all q ∈ Q+
(e.g. 1/ω)
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
RM for I ∆0 + exp
We work in ERNA, a nonstandard version of I ∆0 + exp. ERNA has a field ∗ Q:
1 1 √ −2ω −ω −d ωe . . . −1 0 ω 2 1 | {z }| {z negative infinite numbers
5 3
the finite numbers
√ 2 . . .d ωe ω/2 ω } | {z
2ω}
positive infinite numbers
x is infinite iff |x| > q, for all q ∈ Q+ x is infinitely small iff |x| < q, for all q ∈ Q+ (also ‘x ≈ 0’ or ‘x is infinitesimal’)
(e.g. 1/ω)
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
RM for I ∆0 + exp
We work in ERNA, a nonstandard version of I ∆0 + exp. ERNA has a field ∗ Q: ∗ Q,
z −2ω |
the hyperrational numbers
}| 1 1 √ −ω −d ωe . . . −1 0 ω 2 1 {z }| {z
negative infinite numbers
5 3
the finite numbers
{
√
2 . . .d ωe ω/2 ω } | {z
2ω}
positive infinite numbers
x is infinite iff |x| > q, for all q ∈ Q+ x is infinitely small iff |x| < q, for all q ∈ Q+ (also ‘x ≈ 0’ or ‘x is infinitesimal’)
(e.g. 1/ω)
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Central principle for ERNA
The following principle, called ‘Π1 -transfer’, plays the role of WKL.
Axiom schema (Π1 -TRANS)
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Central principle for ERNA
The following principle, called ‘Π1 -transfer’, plays the role of WKL.
Axiom schema (Π1 -TRANS) For all standard quantifier-free ϕ, (∀x ∈ Q)ϕ(x) → (∀x ∈ ∗ Q)ϕ(x) standard = no infinite numbers† , no ‘x ≈ y ’, no ‘x is (in)finite’.
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Central principle for ERNA
The following principle, called ‘Π1 -transfer’, plays the role of WKL.
Axiom schema (Π1 -TRANS) For all standard quantifier-free ϕ, (∀x ∈ Q)ϕ(x) → (∀x ∈ ∗ Q)ϕ(x) standard = no infinite numbers† , no ‘x ≈ y ’, no ‘x is (in)finite’. († except in terms like
xn n=0 n! ,
Pω
n x 2n n=0 (−1) (2n)! ,
Pω
...)
Introduction
A copy of the RM of WKL0
Continuity
A constructive copy
Big questions
Future research
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Continuity A function f is standard continuous (or ε-δ-continuous) if (∀ε ∈ Q)(∃δ ∈ Q)(∀x, y ∈ Q)(|x − y | < δ → |f (x) − f (y )| < ε).
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Continuity A function f is standard continuous if (∀ε ∈ Q)(∃δ ∈ Q)(∀x, y ∈ Q)(|x − y | < δ → |f (x) − f (y )| < ε). A function f is nonstandard continuous if (∀x, y ∈ ∗ Q)(x ≈ y → f (x) ≈ f (y )).
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Continuity A function f is standard continuous if (∀ε ∈ Q)(∃δ ∈ Q)(∀x, y ∈ Q)(|x − y | < δ → |f (x) − f (y )| < ε). A function f is nonstandard continuous if (∀x, y ∈ ∗ Q)(x ≈ y → f (x) ≈ f (y )).
Principle (Continuity principle) A standard function is standard cont. iff it is nonstandard cont. standard= no infinite numbers† , no ‘x ≈ y ’, no ‘x is (in)finite’.
(1)
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Continuity A function f is standard continuous if (∀ε ∈ Q)(∃δ ∈ Q)(∀x, y ∈ Q)(|x − y | < δ → |f (x) − f (y )| < ε). A function f is nonstandard continuous if (∀x, y ∈ ∗ Q)(x ≈ y → f (x) ≈ f (y )).
Principle (Continuity principle) A standard function is standard cont. iff it is nonstandard cont. The Continuity Principle is used throughout Physics: While modeling reality, physicists use the intuitive definition (1) and not ε-δ-continuity.
(1)
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Continuity A function f is standard continuous if (∀ε ∈ Q)(∃δ ∈ Q)(∀x, y ∈ Q)(|x − y | < δ → |f (x) − f (y )| < ε). A function f is nonstandard continuous if (∀x, y ∈ ∗ Q)(x ≈ y → f (x) ≈ f (y )).
Principle (Continuity principle) A standard function is standard cont. iff it is nonstandard cont.
Theorem (S.) In ERNA, the Continuity principle is equivalent to Π1 -transfer.
(1)
Introduction
A copy of the RM of WKL0
A constructive copy
Weierstrass extremum principle 6
f (x)
a
b-
Big questions
Future research
References
Introduction
A copy of the RM of WKL0
A constructive copy
Weierstrass extremum principle 6
f (x)
a
b-
Big questions
Future research
References
We find the maximum over [a, b] of the continuous function f (x).
Introduction
A copy of the RM of WKL0
A constructive copy
Weierstrass extremum principle 6
Big questions
Future research
References
We find the maximum over [a, b] of the continuous function f (x).
1) Define xi = a + ε(b − a)i. (i ≤ ω) f (x)
a xi xi+1 -
ε≈0
b-
Introduction
A copy of the RM of WKL0
A constructive copy
Weierstrass extremum principle 6
Big questions
Future research
References
We find the maximum over [a, b] of the continuous function f (x).
1) Define xi = a + ε(b − a)i. (i ≤ ω) 2) Every x ∈ [a, b] is in some [xj , xj+1 ].
f (x)
a xi xi+1 -
ε≈0
b-
Introduction
A copy of the RM of WKL0
A constructive copy
Weierstrass extremum principle 6
Big questions
Future research
References
We find the maximum over [a, b] of the continuous function f (x).
1) Define xi = a + ε(b − a)i. (i ≤ ω) 2) Every x ∈ [a, b] is in some [xj , xj+1 ].
f (x)
3) x ≈ xj ≈ xj+1 implies f (x) ≈ f (xj ) a xi xi+1 -
ε≈0
b-
Introduction
A copy of the RM of WKL0
A constructive copy
Weierstrass extremum principle M •
6
Big questions
Future research
References
We find the maximum over [a, b] of the continuous function f (x).
1) Define xi = a + ε(b − a)i. (i ≤ ω) 2) Every x ∈ [a, b] is in some [xj , xj+1 ].
f (x)
3) x ≈ xj ≈ xj+1 implies f (x) ≈ f (xj ) 4) Calculate M = maxi≤ω f (xi ). a xi xi+1 -
ε≈0
b-
Introduction
A copy of the RM of WKL0
A constructive copy
Weierstrass extremum principle M •
6
Big questions
Future research
References
We find the maximum over [a, b] of the continuous function f (x).
1) Define xi = a + ε(b − a)i. (i ≤ ω) 2) Every x ∈ [a, b] is in some [xj , xj+1 ].
f (x)
3) x ≈ xj ≈ xj+1 implies f (x) ≈ f (xj ) 4) Calculate M = maxi≤ω f (xi ). a xi xi+1 -
ε≈0
b- 5) We have (∀x ∈ [a, b])(f (x) / M).
Introduction
A copy of the RM of WKL0
A constructive copy
Weierstrass extremum principle M •
6
Big questions
Future research
References
We find the maximum over [a, b] of the continuous function f (x).
1) Define xi = a + ε(b − a)i. (i ≤ ω) 2) Every x ∈ [a, b] is in some [xj , xj+1 ].
f (x)
3) x ≈ xj ≈ xj+1 implies f (x) ≈ f (xj ) 4) Calculate M = maxi≤ω f (xi ). a xi xi+1 -
b- 5) We have (∀x ∈ [a, b])(f (x) / M).
ε ≈ 0 extremum principle) Theorem (Weierstrass A standard function which is standard continuous on [a, b] attains its maximum there, up to infinitesimals.
Introduction
A copy of the RM of WKL0
A constructive copy
Weierstrass extremum principle M •
6
Big questions
Future research
References
We find the maximum over [a, b] of the continuous function f (x).
1) Define xi = a + ε(b − a)i. (i ≤ ω) 2) Every x ∈ [a, b] is in some [xj , xj+1 ].
f (x)
3) x ≈ xj ≈ xj+1 implies f (x) ≈ f (xj ) 4) Calculate M = maxi≤ω f (xi ). a xi xi+1 -
b- 5) We have (∀x ∈ [a, b])(f (x) / M).
ε ≈ 0 extremum principle) Theorem (Weierstrass A standard function which is standard continuous on [a, b] attains its maximum there, up to infinitesimals. In Physics, results also hold up to a degree of accuracy.
Introduction
A copy of the RM of WKL0
A constructive copy
Weierstrass extremum principle M •
6
Big questions
Future research
References
We find the maximum over [a, b] of the continuous function f (x).
1) Define xi = a + ε(b − a)i. (i ≤ ω) 2) Every x ∈ [a, b] is in some [xj , xj+1 ].
f (x)
3) x ≈ xj ≈ xj+1 implies f (x) ≈ f (xj ) 4) Calculate M = maxi≤ω f (xi ). a xi xi+1 -
b- 5) We have (∀x ∈ [a, b])(f (x) / M).
ε ≈ 0 extremum principle) Theorem (Weierstrass A standard function which is standard continuous on [a, b] attains its maximum there, up to infinitesimals.
Theorem (S.) In ERNA, the Weierstrass ext. prin. is equivalent to Π1 -transfer.
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Brouwer fixed point theorem Theorem (Brouwer fixed point theorem) A continuous [0, 1] → [0, 1]-function has a fixed point ?
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Brouwer fixed point theorem Theorem (Brouwer fixed point theorem) A continuous [0, 1] → [0, 1]-function has a fixed point ? Y y = 1 − x2
y=x
√ • −1 + 5 2
•
−1 + 2
√
5 •
X
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Brouwer fixed point theorem Theorem (Brouwer fixed point theorem) A continuous [0, 1] → [0, 1]-function has a fixed point ? Y y = 1 − x2
y=x
√ • −1 + 5 2
•
−1 + 2
But
√ −1+ 5 2
6∈ ∗ Q because
√
√
5
5 6∈ ∗ Q!
•
X
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Brouwer fixed point theorem Theorem (Brouwer fixed point theorem) A continuous [0, 1] → [0, 1]-function has a fixed point ? Y y = 1 − x2
y=x
√ • −1 + 5 2
•
−1 + 2
But
√ −1+ 5 2
6∈ ∗ Q because
and we have
−1+β 2
√
√
5 •
X
5 6∈ ∗ Q! There is β ∈ ∗ Q with β 2 ≈ 5
2 ≈ 1 − ( −1+β 2 ) .
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Brouwer fixed point theorem Theorem (Brouwer fixed point theorem) A continuous [0, 1] → [0, 1]-function has a fixed point ? Y y = 1 − x2
y=x
√ • −1 + 5 2
•
−1 + 2
But
√ −1+ 5 2
6∈ ∗ Q because
and we have −1+β 2
−1+β 2
√
√
5 •
X
5 6∈ ∗ Q! There is β ∈ ∗ Q with β 2 ≈ 5
2 ≈ 1 − ( −1+β 2 ) .
is fixed point ‘up to infinitesimals’ for y = 1 − x 2 .
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Brouwer fixed point theorem Definition x0 is a fixed point up to infinitesimals if f (x0 ) ≈ x0 .
Future research
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Brouwer fixed point theorem Definition x0 is a fixed point up to infinitesimals if f (x0 ) ≈ x0 .
Theorem (Brouwer fixed point principle) A standard continuous [0, 1]2 → [0, 1]2 -function has a fixed point up to infinitesimals.
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Brouwer fixed point theorem Definition x0 is a fixed point up to infinitesimals if f (x0 ) ≈ x0 .
Theorem (Brouwer fixed point principle) A standard continuous [0, 1]2 → [0, 1]2 -function has a fixed point up to infinitesimals. As in Physics, we only have approximations (of e.g. fixed points).
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Brouwer fixed point theorem Definition x0 is a fixed point up to infinitesimals if f (x0 ) ≈ x0 .
Theorem (Brouwer fixed point principle) A standard continuous [0, 1]2 → [0, 1]2 -function has a fixed point up to infinitesimals. As in Physics, we only have approximations (of e.g. fixed points).
Theorem (S.) In ERNA, the Brouwer fixed point principle is equivalent to Π1 -transfer.
Introduction
A copy of the RM of WKL0
A constructive copy
Riemann integration
6
f (x)
a
b -
Big questions
Future research
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Riemann integration
6
f (x)
a
b -
Rb a
f (x) dx = the surface under f (x)
Future research
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Riemann integration
6
f (x)
a
b -
Rb a
f (x) dx = the surface under f (x)
Future research
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Riemann integration 1) Hyperfine partition π of [a, b] = set of ω subintervals [xi , xi+1 ] of length εi ≈ 0 6
f (x)
a
b -
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Riemann integration 1) Hyperfine partition π of [a, b] = set of ω subintervals [xi , xi+1 ] of length εi ≈ 0 6
f (x)
ω intervals [xi , xi+1 ]
a
z
}| -
εi ≈ 0
{
b -
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Riemann integration 1) Hyperfine partition π of [a, b] = set of ω subintervals [xi , xi+1 ] of length εi ≈ 0 6
f (x)
a
b -
εi ≈ 0
-
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Riemann integration 1) Hyperfine partition π of [a, b] = set of ω subintervals [xi , xi+1 ] of length εi ≈ 0 6
f (x)
a
xi • εi ≈ 0
b -
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Riemann integration 1) Hyperfine partition π of [a, b] = set of ω subintervals [xi , xi+1 ] of length εi ≈ 0 6
f (xi ) •
a
xi • εi ≈ 0
f (x)
b -
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Riemann integration 1) Hyperfine partition π of [a, b] = set of ω subintervals [xi , xi+1 ] of length εi ≈ 0 6
2) Surface of rectangle is f (xi )εi
f (xi ) •
a
xi • εi ≈ 0
f (x)
b -
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Riemann integration 1) Hyperfine partition π of [a, b] = set of ω subintervals [xi , xi+1 ] of length εi ≈ 0 6
2) Surface of rectangle is f (xi )εi
f (xi ) •
a
xi • εi ≈ 0
f (x)
b -
3) Construct all rectangles
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Riemann integration 1) Hyperfine partition π of [a, b] = set of ω subintervals [xi , xi+1 ] of length εi ≈ 0 6
2) Surface of rectangle is f (xi )εi
f (xi ) •
a
xi • εi ≈ 0
f (x)
b -
3) Construct all rectangles
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Riemann integration 1) Hyperfine partition π of [a, b] = set of ω subintervals [xi , xi+1 ] of length εi ≈ 0 6
2) Surface of rectangle is f (xi )εi
f (xi ) •
a
xi • εi ≈ 0
f (x)
b -
3) Construct all rectangles P 4) Total surface is ωi=1 f (xi )εi
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Riemann integration 1) Hyperfine partition π of [a, b] = set of ω subintervals [xi , xi+1 ] of length εi ≈ 0 6
2) Surface of rectangle is f (xi )εi
f (xi ) •
f (x)
3) Construct all rectangles P 4) Total surface is ωi=1 f (xi )εi = Sπ , the Riemann sum of π
a
xi • εi ≈ 0
b -
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Riemann integration 1) Hyperfine partition π of [a, b] = set of ω subintervals [xi , xi+1 ] of length εi ≈ 0 6
2) Surface of rectangle is f (xi )εi
f (xi ) •
a
xi • εi ≈ 0
f (x)
b -
3) Construct all rectangles P 4) Total surface is ωi=1 f (xi )εi = Sπ , the Riemann sum of π Rb ≈ a f (x) dx
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Riemann integration 1) Hyperfine partition π of [a, b] = set of ω subintervals [xi , xi+1 ] of length εi ≈ 0 6
2) Surface of rectangle is f (xi )εi
f (xi ) •
a
xi • εi ≈ 0
f (x)
b -
3) Construct all rectangles P 4) Total surface is ωi=1 f (xi )εi = Sπ , the Riemann sum of π Rb ≈ a f (x) dx
f is integrable if Sπ ≈ q ≈ Sπ0 , for all hyperfine π, π 0
(q ∈ ∗ Q is finite)
Introduction
A copy of the RM of WKL0
Riemann integration II
A constructive copy
Big questions
Future research
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Riemann integration II
Principle (Integrability principle) A standard continuous function is Riemann integrable.
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Riemann integration II
Principle (Integrability principle) A standard continuous function is Riemann integrable.
Theorem (S.) In ERNA, the Integrability principle is equivalent to Π1 -transfer.
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Riemann integration II
Principle (Integrability principle) A standard continuous function is Riemann integrable.
Theorem (S.) In ERNA, the Integrability principle is equivalent to Π1 -transfer. Rb The integral a f (x)dx is only defined up to infinitesimals, i.e. only approximately.
Introduction
A copy of the RM of WKL0
A constructive copy
Peano existence theorem
Big questions
Future research
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Peano existence theorem Principle (Peano existence principle) Let f (x, y ) be standard continuous on [−a, a] × [−b, b] with maximum M. Then there exists φ(x), cont. differentiable for |x| ≤ α such that φ0 (x) ≈ f (x, φ(x)),
φ(0) = 0
holds for |x| < α, with α = min(a, b/M).
(2)
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Peano existence theorem Principle (Peano existence principle) Let f (x, y ) be standard continuous on [−a, a] × [−b, b] with maximum M. Then there exists φ(x), cont. differentiable for |x| ≤ α such that φ0 (x) ≈ f (x, φ(x)),
φ(0) = 0
holds for |x| < α, with α = min(a, b/M). = the ‘real’ Peano existence theorem, up to infinitesimals
(2)
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Peano existence theorem Principle (Peano existence principle) Let f (x, y ) be standard continuous on [−a, a] × [−b, b] with maximum M. Then there exists φ(x), cont. differentiable for |x| ≤ α such that φ0 (x) ≈ f (x, φ(x)),
φ(0) = 0
holds for |x| < α, with α = min(a, b/M). = the ‘real’ Peano existence theorem, up to infinitesimals
Theorem In ERNA, the Peano exist. princ. is equivalent to Π1 -transfer.
(2)
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Isomorphism Theorem (inspired by work of Sommer and Suppes [6])
Theorem (Isomorphism theorem) Every ‘R-like’ model of a physical problem has an isomorphic ‘Q-like’ model.
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Isomorphism Theorem (inspired by work of Sommer and Suppes [6])
Theorem (Isomorphism theorem) Every ‘R-like’ model of a physical problem has an isomorphic ‘Q-like’ model. R-like model = irrational objects (e.g. π and e x ) are used Q-like model = only rational numbers and functions are used
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Isomorphism Theorem (inspired by work of Sommer and Suppes [6])
Theorem (Isomorphism theorem) Every ‘R-like’ model of a physical problem has an isomorphic ‘Q-like’ model. R-like model = irrational objects (e.g. π and e x ) are used Q-like model = only rational numbers and functions are used
Theorem In ERNA, the Isomorphism theorem is equivalent to Π1 -transfer.
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Isomorphism Theorem (inspired by work of Sommer and Suppes [6])
Theorem (Isomorphism theorem) Every ‘R-like’ model of a physical problem has an isomorphic ‘Q-like’ model. R-like model = irrational objects (e.g. π and e x ) are used Q-like model = only rational numbers and functions are used
Theorem In ERNA, the Isomorphism theorem is equivalent to Π1 -transfer. The Isomorphism theorem implies that no physical experiment can decide whether reality is discrete or continuous.
Introduction
A copy of the RM of WKL0
A constructive copy
And many more principles. . .
Big questions
Future research
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
And many more principles. . . In ERNA, the following are equivalent to Π1 -transfer.
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
And many more principles. . . In ERNA, the following are equivalent to Π1 -transfer. 1
Continuity principle (NS-cont. is equivalent to S-cont. )
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
And many more principles. . . In ERNA, the following are equivalent to Π1 -transfer. 1
Continuity principle (NS-cont. is equivalent to S-cont. )
2
Weierstrass extremum theorem (‘up to infinitesimals’)
3
Brouwer fixpoint theorem (‘up to infinitesimals’)
4
Peano existence theorem (‘up to infinitesimals’)
5
Integrability principle
6
Isomorphism theorem
7
Completeness up to infinitesimals of ∗ Q (Cauchy, Dedekind, Cantor)
8
Fundamental theorem of calculus.
9
And many more. . .
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
And many more principles. . . In ERNA, the following are equivalent to Π1 -transfer. 1
Continuity principle (NS-cont. is equivalent to S-cont. )
2
Weierstrass extremum theorem (‘up to infinitesimals’)
3
Brouwer fixpoint theorem (‘up to infinitesimals’)
4
Peano existence theorem (‘up to infinitesimals’)
5
Integrability principle
6
Isomorphism theorem
7
Completeness up to infinitesimals of ∗ Q (Cauchy, Dedekind, Cantor)
8
Fundamental theorem of calculus.
9
And many more. . .
The RM of ERNA + Π1 -TRANS is a copy up to infinitesimals of the RM of WKL0 .
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
And many more principles. . . In ERNA, the following are equivalent to Π1 -transfer. 1
Continuity principle (NS-cont. is equivalent to S-cont. )
2
Weierstrass extremum theorem (‘up to infinitesimals’)
3
Brouwer fixpoint theorem (‘up to infinitesimals’)
4
Peano existence theorem (‘up to infinitesimals’)
5
Integrability principle
6
Isomorphism theorem
7
Completeness up to infinitesimals of ∗ Q (Cauchy, Dedekind, Cantor)
8
Fundamental theorem of calculus.
9
And many more. . .
However, (7) belongs to the RM of ACA0 , NOT to that of WKL0 .
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
And many more principles. . . In ERNA, the following are equivalent to Π1 -transfer. 1
Continuity principle (NS-cont. is equivalent to S-cont. )
2
Weierstrass extremum theorem (‘up to infinitesimals’)
3
Brouwer fixpoint theorem (‘up to infinitesimals’)
4
Peano existence theorem (‘up to infinitesimals’)
5
Integrability principle
6
Isomorphism theorem
7
Completeness up to infinitesimals of ∗ Q (Cauchy, Dedekind, Cantor)
8
Fundamental theorem of calculus.
9
And many more. . .
However, (7) belongs to the RM of ACA0 , NOT to that of WKL0 . Why is this so?
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Similarities between Constructivism and ERNA
Constructivism
ERNA
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Similarities between Constructivism and ERNA
√
Constructivism x is not always defined
√
ERNA x is not always defined
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Similarities between Constructivism and ERNA
Constructivism x is not always defined no standard part function √
ERNA x is not always defined no standard part function √
standard part function: st(x + ε) = x, with x standard and ε ≈ 0
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Similarities between Constructivism and ERNA
Constructivism x is not always defined no standard part function uniform cont. and diff. √
ERNA x is not always defined no standard part function uniform cont. and diff. √
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Similarities between Constructivism and ERNA
Constructivism x is not always defined no standard part function uniform cont. and diff. finite set property √
ERNA x is not always defined no standard part function uniform cont. and diff. hyperfinite set property √
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Similarities between Constructivism and ERNA
Constructivism x is not always defined no standard part function uniform cont. and diff. finite set property √
ERNA x is not always defined no standard part function uniform cont. and diff. hyperfinite set property √
finite set property: not every subset of a finite set is finite hyperfinite set property: not every subset of a hyperfinite set is hyperfinite
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Similarities between Constructivism and ERNA
Constructivism x is not always defined no standard part function uniform cont. and diff. finite set property √
ERNA x is not always defined no standard part function uniform cont. and diff. hyperfinite set property √
finite set property: not every subset of a finite set is finite hyperfinite set property: not every subset of a hyperfinite set is hyperfinite (hyperfinite sets are of the form {0, 1, . . . , N} for (in)finite N.)
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Similarities between Constructivism and ERNA
Constructivism x is not always defined no standard part function uniform cont. and diff. finite set property computable witnesses √
ERNA x is not always defined no standard part function uniform cont. and diff. hyperfinite set property computable witnesses √
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Similarities between Constructivism and ERNA
Constructivism x is not always defined no standard part function uniform cont. and diff. finite set property computable witnesses √
ERNA x is not always defined no standard part function uniform cont. and diff. hyperfinite set property computable witnesses √
Constructivism (∃n)ϕ(x) means ‘there is an algorithm to compute n0 s.t. ϕ(n0 )’.
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Similarities between Constructivism and ERNA
Constructivism x is not always defined no standard part function uniform cont. and diff. finite set property computable witnesses √
ERNA x is not always defined no standard part function uniform cont. and diff. hyperfinite set property computable witnesses √
Constructivism (∃n)ϕ(x) means ‘there is an algorithm to compute n0 s.t. ϕ(n0 )’. ERNA If (∃n ∈ N)ϕ(n), then n0 := (µn ≤ ω)ϕ(n) satisfies ϕ(n0 ).
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Similarities between Constructivism and ERNA
Constructivism x is not always defined no standard part function uniform cont. and diff. finite set property computable witnesses √
ERNA x is not always defined no standard part function uniform cont. and diff. hyperfinite set property computable witnesses √
Constructivism (∃n)ϕ(x) means ‘there is an algorithm to compute n0 s.t. ϕ(n0 )’. ERNA + Π1 -transfer If (∃n ∈ ∗ N)ϕ(n), then n0 := (µn ≤ ω)ϕ(n) satisfies ϕ(n0 ).
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Constructive Reverse Mathematics (CRM)
Future research
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Constructive Reverse Mathematics (CRM) In CRM, an important principle (related to Completeness) is
Principle (Σ1 -excluded middle) For every q.f. formula ϕ, we have (∃n)ϕ(n) ∨ (∀n)¬ϕ(n).
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Constructive Reverse Mathematics (CRM) In CRM, an important principle (related to Completeness) is
Principle (Σ1 -excluded middle) For every q.f. formula ϕ, we have (∃n)ϕ(n) ∨ (∀n)¬ϕ(n). In constructive math., (∃n)ϕ(n) means ‘we can compute a number n0 s.t. ϕ(n0 )’.
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Constructive Reverse Mathematics (CRM) In CRM, an important principle (related to Completeness) is
Principle (Σ1 -excluded middle) For every q.f. formula ϕ, we have (∃n)ϕ(n) ∨ (∀n)¬ϕ(n). In constructive math., (∃n)ϕ(n) means ‘we can compute a number n0 s.t. ϕ(n0 )’.
Principle (Π1 -transfer) For every q.f. formula ϕ, we have (∃n ∈ N)ϕ(n) ∨ (∀n ∈ ∗ N)¬ϕ(n).
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Constructive Reverse Mathematics (CRM) In CRM, an important principle (related to Completeness) is
Principle (Σ1 -excluded middle) For every q.f. formula ϕ, we have (∃n)ϕ(n) ∨ (∀n)¬ϕ(n). In constructive math., (∃n)ϕ(n) means ‘we can compute a number n0 s.t. ϕ(n0 )’.
Principle (Π1 -transfer) For every q.f. formula ϕ, we have (∃n ∈ N)ϕ(n) ∨ (∀n ∈ ∗ N)¬ϕ(n). To find a witness for (∃n ∈ ∗ N)ϕ(n), calculate (µn ≤ ω)ϕ(n).
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Constructive Reverse Mathematics (CRM) In CRM, an important principle (related to Completeness) is
Principle (Σ1 -excluded middle) For every q.f. formula ϕ, we have (∃n)ϕ(n) ∨ (∀n)¬ϕ(n). In constructive math., (∃n)ϕ(n) means ‘we can compute a number n0 s.t. ϕ(n0 )’.
Principle (Π1 -transfer) For every q.f. formula ϕ, we have (∃n ∈ N)ϕ(n) ∨ (∀n ∈ ∗ N)¬ϕ(n). To find a witness for (∃n ∈ ∗ N)ϕ(n), calculate (µn ≤ ω)ϕ(n). Thus, Π1 -transfer is ‘hyperexcluded middle’: it excludes the possibility ‘(∀n ∈ N)¬ϕ(n) ∧ (∃n ∈ ∗ N)ϕ(n)’.
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Constructive Reverse Mathematics
Conjecture: CRM with Σ1 -PEM is similar to the RM of ERNA + Π1 -TRANS.
References
Introduction
A copy of the RM of WKL0
Big Questions
A constructive copy
Big questions
Future research
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Big Questions
(#1) Is physical reality continuous or discrete?
Future research
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Big Questions
(#1) Is physical reality continuous or discrete? (#2) What are good foundations for Mathematics?
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Big Questions
(#1) Is physical reality continuous or discrete? (#2) What are good foundations for Mathematics? (#3) Do infinitesimals exist?
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Small answers (#2) The RM of ERNA + Π1 -TRANS is a copy up to infinitesimals of the RM of WKL0 .
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Small answers (#2) The RM of ERNA + Π1 -TRANS is a copy up to infinitesimals of the RM of WKL0 . Observation: RM is ‘robust for infinitesimal error’.
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Small answers (#2) The RM of ERNA + Π1 -TRANS is a copy up to infinitesimals of the RM of WKL0 . Observation: RM is ‘robust for infinitesimal error’. ‘Robust statistics’ attempts to produce estimators that are not particularly affected by small departures from model assumptions ([1]).
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Small answers (#2) The RM of ERNA + Π1 -TRANS is a copy up to infinitesimals of the RM of WKL0 . Observation: RM is ‘robust for infinitesimal error’. ‘Robust statistics’ attempts to produce estimators that are not particularly affected by small departures from model assumptions ([1]). In CS, robust refers to an algorithm that performs well not only under ordinary conditions but also under unusual conditions that stress its designers’ assumptions’ ([3]).
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Small answers (#2) The RM of ERNA + Π1 -TRANS is a copy up to infinitesimals of the RM of WKL0 . Observation: RM is ‘robust for infinitesimal error’. ‘Robust statistics’ attempts to produce estimators that are not particularly affected by small departures from model assumptions ([1]). In CS, robust refers to an algorithm that performs well not only under ordinary conditions but also under unusual conditions that stress its designers’ assumptions’ ([3]). Thus, ‘robust’ methods are reasonably resistant to errors in the results, produced by deviations from assumptions. Robustness is important throughout the exact sciences.
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Small answers (#2) The RM of ERNA + Π1 -TRANS is a copy up to infinitesimals of the RM of WKL0 . Observation: RM is ‘robust for infinitesimal error’. ‘Robust statistics’ attempts to produce estimators that are not particularly affected by small departures from model assumptions ([1]). In CS, robust refers to an algorithm that performs well not only under ordinary conditions but also under unusual conditions that stress its designers’ assumptions’ ([3]). Thus, ‘robust’ methods are reasonably resistant to errors in the results, produced by deviations from assumptions. Robustness is important throughout the exact sciences.
(#2) As RM is robust, it is a good foundation for Mathematics!
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Small answers (#3)
The RM of ERNA + Π1 -TRANS is a copy up to infinitesimals of the RM of WKL0 .
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Small answers (#3)
The RM of ERNA + Π1 -TRANS is a copy up to infinitesimals of the RM of WKL0 . The RM of ERNA + Π1 -TRANS is ‘better’ because the base theory (I ∆0 + exp) is weaker than that of WKL0 (I Σ1 ).
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Small answers (#3)
The RM of ERNA + Π1 -TRANS is a copy up to infinitesimals of the RM of WKL0 . The RM of ERNA + Π1 -TRANS is ‘better’ because the base theory (I ∆0 + exp) is weaker than that of WKL0 (I Σ1 ). (#3) ERNA + Π1 -TRANS is ‘more real’ than WKL0 .
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Small answers (#3)
The RM of ERNA + Π1 -TRANS is a copy up to infinitesimals of the RM of WKL0 . The RM of ERNA + Π1 -TRANS is ‘better’ because the base theory (I ∆0 + exp) is weaker than that of WKL0 (I Σ1 ). (#3) ERNA + Π1 -TRANS is ‘more real’ than WKL0 . Thus, infinitesimals are ‘more real’ than subsets of N.
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Small answers (#1) Recall that in ERNA, the following are equivalent to Π1 -transfer. 1
Continuity principle (NS-cont. is equivalent to S-cont. )
2
Weierstrass extremum theorem (‘up to infinitesimals’)
3
Brouwer fixpoint theorem (‘up to infinitesimals’)
4
Peano existence theorem (‘up to infinitesimals’)
5
Integrability principle
6
Isomorphism theorem
7
8
Completeness up to infinitesimals of ∗ Q (Cauchy, Dedekind, Cantor) And many more. . .
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Small answers (#1) The Continuity principle is equivalent to the Isomorphism theorem.
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Small answers (#1) The Continuity principle is equivalent to the Isomorphism theorem.
Principle (Continuity principle) A standard function is standard cont. iff it is nonstandard cont.
Theorem (Isomorphism theorem) Every ‘R-like’ model of a physical problem has an isomorphic ‘Q-like’ model.
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Small answers (#1) The Continuity principle is equivalent to the Isomorphism theorem.
Principle (Continuity principle) A standard function is standard cont. iff it is nonstandard cont.
Theorem (Isomorphism theorem) Every ‘R-like’ model of a physical problem has an isomorphic ‘Q-like’ model. Recall: R-like model = irrational objects (e.g. π and e x ) are used Q-like model = only rational numbers and functions are used A function f is standard continuous if (∀ε ∈ Q)(∃δ ∈ Q)(∀x, y ∈ Q)(|x − y | < δ → |f (x) − f (y )| < ε). A function f is nonstandard continuous if (∀x, y ∈ ∗ Q)(x ≈ y → f (x) ≈ f (y )).
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Small answers (#1) The Continuity principle is equivalent to the Isomorphism theorem.
Principle (Continuity principle) A standard function is standard cont. iff it is nonstandard cont.
Theorem (Isomorphism theorem) Every ‘R-like’ model of a physical problem has an isomorphic ‘Q-like’ model. The Continuity principle is an essential part of Physics.
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Small answers (#1) The Continuity principle is equivalent to the Isomorphism theorem.
Principle (Continuity principle) A standard function is standard cont. iff it is nonstandard cont.
Theorem (Isomorphism theorem) Every ‘R-like’ model of a physical problem has an isomorphic ‘Q-like’ model. The Continuity principle is an essential part of Physics. Thus, so is the Isomorphism theorem.
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Small answers (#1) The Continuity principle is equivalent to the Isomorphism theorem.
Principle (Continuity principle) A standard function is standard cont. iff it is nonstandard cont.
Theorem (Isomorphism theorem) Every ‘R-like’ model of a physical problem has an isomorphic ‘Q-like’ model. The Continuity principle is an essential part of Physics. Thus, so is the Isomorphism theorem. But the Isomorphism theorem implies that no physical experiment can decide whether reality is discrete or continuous!
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Small answers (#1)
(# 1) Is physical reality discrete or continuous?
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Small answers (#1)
(# 1) Is physical reality discrete or continuous? Answer to (#1): this is undecidable because of the nature of mathematical modeling in Physics.
Proofs are available in [2] and [4].
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future Research and open problems
Future research
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Future Research and open problems
The RM of ERNA + Π1 -TRANS is a copy up to infinitesimals of the RM of WKL0 .
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Future Research and open problems
The RM of ERNA + Π1 -TRANS is a copy up to infinitesimals of the RM of WKL0 . 1) What system T corresponds to ACA0 ?
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Future Research and open problems
The RM of ERNA + Π1 -TRANS is a copy up to infinitesimals of the RM of WKL0 . 1) What system T corresponds to ACA0 ? T should involve Π1 -transfer and Πst 2 -IND.
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Future Research and open problems
The RM of ERNA + Π1 -TRANS is a copy up to infinitesimals of the RM of WKL0 . 1) What system T corresponds to ACA0 ? T should involve Π1 -transfer and Πst 2 -IND. 2) What about Π2 -transfer?
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Future Research and open problems
The RM of ERNA + Π1 -TRANS is a copy up to infinitesimals of the RM of WKL0 . 1) What system T corresponds to ACA0 ? T should involve Π1 -transfer and Πst 2 -IND. 2) What about Π2 -transfer? Results with arbitrary precision instead of ≈.
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Future Research and open problems
The RM of ERNA + Π1 -TRANS is a copy up to infinitesimals of the RM of WKL0 . 1) What system T corresponds to ACA0 ? T should involve Π1 -transfer and Πst 2 -IND. 2) What about Π2 -transfer? Results with arbitrary precision instead of ≈. 3) What about Σn -PEM and Πn -transfer for n > 1?
Introduction
A copy of the RM of WKL0
Stratified NSA
A constructive copy
Big questions
Future research
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Stratified NSA In classical NSA, a number is either finite or infinite.
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Stratified NSA In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity.
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Stratified NSA In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider NA :
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Stratified NSA In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider NA :
0 1 ... | {z } N
ωα
...
ωβ
...
ωγ
... -
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Stratified NSA In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider NA :
0 1 ... | {z }
ωα
...
ωβ
N
where A = {0, α, β, γ, . . . }
...
ωγ
... -
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Stratified NSA In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider NA :
0 1 ... | {z }
ωα
...
ωβ
...
ωγ
...
N
where A = {0, α, β, γ, . . . } (e.g. A is finite or N)
-
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Stratified NSA In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider NA :
0 1 ... | {z }
ωα
...
ωβ
N, 0-finite
where A = {0, α, β, γ, . . . }
...
ωγ
... -
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Stratified NSA In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider NA :
0 1 ... | {z } |
ωα
...
ωβ
ωγ
...
-
{z
N, 0-finite
0-infinite
where A = {0, α, β, γ, . . . }
...
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Stratified NSA In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider NA :
0 1 ... | {z } |
ωα
...
ωβ
ωγ
...
-
{z
N, finite
infinite
where A = {0, α, β, γ, . . . }
...
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Stratified NSA In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider NA :
α-finite
z
0 1 ... | {z } |
}|
ωα
{
...
ωβ
ωγ
...
-
{z
N, finite
infinite
where A = {0, α, β, γ, . . . }
...
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Stratified NSA In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider NA :
α-finite
z
0 1 ... | {z } |
}|
α-infinite
ωα
{ z
...
ωβ
N, finite
...
}| ωγ {z
infinite
where A = {0, α, β, γ, . . . }
... -
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Stratified NSA In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider NA : Fuzzy border: α-finite
z
0 1 ... | {z } |
}|
α-infinite
ωα
{ z
...
ωβ
N, finite
...
}| ωγ {z
infinite
where A = {0, α, β, γ, . . . }
... -
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Stratified NSA In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider NA : Fuzzy border: no least α-infinite number α-finite
z
0 1 ... | {z } |
}|
α-infinite
ωα
{ z
...
ωβ
N, finite
...
}| ωγ {z
infinite
where A = {0, α, β, γ, . . . }
... -
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Stratified NSA In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider NA :
α-finite
z
0 1 ... | {z } |
}|
α-infinite
ωα
{ z
...
ωβ
N, finite
...
}| ωγ {z
infinite
where A = {0, α, β, γ, . . . }
... -
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Stratified NSA In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider NA :
α-finite
z
0 1 ... | {z } |
}|
α-infinite
ωα
{ z
...
ωβ
N, finite
...
}| ωγ {z
infinite
where A = {0, α, β, γ, . . . }
... -
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Stratified NSA In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider NA : β-finite
z
}|
{
α-finite
z
0 1 ... | {z } |
}|
α-infinite
ωα
{ z
...
ωβ
N, finite
...
}| ωγ {z
infinite
where A = {0, α, β, γ, . . . }
... -
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Stratified NSA In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider NA : β-finite
z
β-infinite
}|
{ z
α-finite
z
0 1 ... | {z } |
}|
}| α-infinite
ωα
{ z
...
ωβ
N, finite
...
}| ωγ {z
infinite
where A = {0, α, β, γ, . . . }
... -
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Stratified NSA In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider NA : β-finite
z
β-infinite
}|
{ z
α-finite
z
0 1 ... | {z } |
}|
}| α-infinite
ωα
{ z
...
ωβ
N, finite
...
}| ωγ {z
...
infinite
where A = {0, α, β, γ, . . . } The infinite numbers are ‘stratified’ in |A| many levels of infinity.
-
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Stratified NSA In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider NA : β-finite
z
β-infinite
}|
{ z
α-finite
z
0 1 ... | {z } |
}|
}| α-infinite
ωα
{ z
...
ωβ
N, finite
...
}| ωγ {z
...
infinite
where A = {0, α, β, γ, . . . } The infinite numbers are ‘stratified’ in |A| many levels of infinity. Then ωβ is infinite ‘relative’ to ωα
-
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Stratified NSA In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider NA : β-finite
z
β-infinite
}|
{ z
α-finite
z
0 1 ... | {z } |
}|
}| α-infinite
ωα
{ z
...
ωβ
N, finite
...
}| ωγ {z
...
infinite
where A = {0, α, β, γ, . . . } The infinite numbers are ‘stratified’ in |A| many levels of infinity. Then ωβ is infinite ‘relative’ to ωα and finite ‘relative’ to ωγ .
-
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Stratified NSA In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider NA : β-finite
z
β-infinite
}|
{ z
α-finite
z
0 1 ... | {z } |
}|
}| α-infinite
ωα
{ z
...
ωβ
N, finite
...
}| ωγ {z
...
infinite
where A = {0, α, β, γ, . . . } The infinite numbers are ‘stratified’ in |A| many levels of infinity. Then ωβ is infinite ‘relative’ to ωα and finite ‘relative’ to ωγ . We write ωα ωβ ωγ .
-
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Transfer =‘All levels α ∈ A have the same properties as NA .’
Future research
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Transfer =‘All levels α ∈ A have the same properties as NA .’ Define Nα = {n ∈ NA |n is α-finite}.
Future research
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Transfer =‘All levels α ∈ A have the same properties as NA .’ Define Nα = {n ∈ NA |n is α-finite}.
Axiom schema (Σn -transfer) For all α-standard ϕ ∈ ∆0 and α ∈ A (∃x1 ∈ NA )(∀x2 ∈ NA ) . . . (Qxn ∈ NA )ϕ(x1 , . . . , xn ) is equivalent to (∃x1 ∈ Nα )(∀x2 ∈ Nα ) . . . (Qxn ∈ Nα )ϕ(x1 , . . . , xn )
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Transfer =‘All levels α ∈ A have the same properties as NA .’ Define Nα = {n ∈ NA |n is α-finite}.
Axiom schema (Σn -transfer) For all α-standard ϕ ∈ ∆0 and α ∈ A (∃x1 ∈ NA )(∀x2 ∈ NA ) . . . (Qxn ∈ NA )ϕ(x1 , . . . , xn ) is equivalent to (∃x1 ∈ Nα )(∀x2 ∈ Nα ) . . . (Qxn ∈ Nα )ϕ(x1 , . . . , xn ) α-standard
= no α-infinite numbers
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Transfer =‘All levels α ∈ A have the same properties as NA .’ Define Nα = {n ∈ NA |n is α-finite}.
Axiom schema (Σn -transfer) For all α-standard ϕ ∈ ∆0 and α ∈ A (∃x1 ∈ NA )(∀x2 ∈ NA ) . . . (Qxn ∈ NA )ϕ(x1 , . . . , xn ) is equivalent to (∃x1 ∈ Nα )(∀x2 ∈ Nα ) . . . (Qxn ∈ Nα )ϕ(x1 , . . . , xn ) α-standard
= no α-infinite numbers = no predicate ‘x is γ-(in)finite’ for any γ ∈ A.
References
Introduction
A copy of the RM of WKL0
From I ∆0 to PA
A constructive copy
Big questions
Future research
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
From I ∆0 to PA The weak axioms NSA introduce NA . The former contains e.g. If x and y are α-finite, then so is x + y .
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
From I ∆0 to PA The weak axioms NSA introduce NA . The former contains e.g. If x and y are α-finite, then so is x + y .
Theorem (S.) I ∆0 + NSA + Σn -transfer proves I Σn
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
From I ∆0 to PA The weak axioms NSA introduce NA . The former contains e.g. If x and y are α-finite, then so is x + y .
Theorem (S.) I ∆0 + NSA + Σn -transfer proves I Σn TRANS := ∪n∈N Σn -transfer
Corollary I ∆0 + NSA + TRANS proves PA
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
From I ∆0 to PA The weak axioms NSA introduce NA . The former contains e.g. If x and y are α-finite, then so is x + y .
Theorem (S.) I ∆0 + NSA + Σn -transfer proves I Σn TRANS := ∪n∈N Σn -transfer
Corollary I ∆0 + NSA + TRANS proves PA By MacDowell-Specker theorem, I ∆0 + NSA + TRANS is conservative over PA. (Hrbacek)
References
Introduction
A copy of the RM of WKL0
The reduction theorem
A constructive copy
Big questions
Future research
References
Introduction
A copy of the RM of WKL0
A constructive copy
The reduction theorem In I ∆0 + NSA + Σn -transfer, we prove
Big questions
Future research
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
The reduction theorem In I ∆0 + NSA + Σn -transfer, we prove
Theorem (Σn -reduction, S.) For every α-standard ϕ ∈ ∆0 , (∃x1 ∈ NA )(∀x2 ∈ NA ) . . . (Qxn ∈ NA )ϕ(x1 , . . . , xn )
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
The reduction theorem In I ∆0 + NSA + Σn -transfer, we prove
Theorem (Σn -reduction, S.) For every α-standard ϕ ∈ ∆0 , (∃x1 ∈ NA )(∀x2 ∈ NA ) . . . (Qxn ∈ NA )ϕ(x1 , . . . , xn ) is equivalent to (∃x1 ≤ ωβ )(∀x2 ≤ ωγ ) . . . (Qxn ≤ ωη )ϕ(x1 , . . . , xn ), with ωα ωβ ωγ . . . ωη .
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
The reduction theorem In I ∆0 + NSA + Σn -transfer, we prove
Theorem (Σn -reduction, S.) For every α-standard ϕ ∈ ∆0 , (∃x1 ∈ NA )(∀x2 ∈ NA ) . . . (Qxn ∈ NA )ϕ(x1 , . . . , xn ) is equivalent to (∃x1 ≤ ωβ )(∀x2 ≤ ωγ ) . . . (Qxn ≤ ωη )ϕ(x1 , . . . , xn ), with ωα ωβ ωγ . . . ωη . Thus, every Σn -formula ‘reduces’ to a ∆0 -formula.
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
The reduction theorem In I ∆0 + NSA + Σn -transfer, we prove
Theorem (Σn -reduction, S.) For every α-standard ϕ ∈ ∆0 , (∃x1 ∈ NA )(∀x2 ∈ NA ) . . . (Qxn ∈ NA )ϕ(x1 , . . . , xn ) is equivalent to (∃x1 ≤ ωβ )(∀x2 ≤ ωγ ) . . . (Qxn ≤ ωη )ϕ(x1 , . . . , xn ), with ωα ωβ ωγ . . . ωη . Thus, every Σn -formula ‘reduces’ to a ∆0 -formula. No arithm. formula can capture properties of all numbers! (Tarski)
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
The reduction theorem In I ∆0 + NSA + Σn -transfer, we prove
Theorem (Σn -reduction, S.) For every α-standard ϕ ∈ ∆0 , (∃x1 ∈ NA )(∀x2 ∈ NA ) . . . (Qxn ∈ NA )ϕ(x1 , . . . , xn ) is equivalent to (∃x1 ≤ ωβ )(∀x2 ≤ ωγ ) . . . (Qxn ≤ ωη )ϕ(x1 , . . . , xn ), with ωα ωβ ωγ . . . ωη . Thus, every Σn -formula ‘reduces’ to a ∆0 -formula. No arithm. formula can capture properties of all numbers! (Tarski) Generalizes smoothly to second-order arithmetic (Keita Yokoyama).
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
Reversal
Surprisingly, there holds
Theorem (S.) In I ∆0 + NSA , the Σn -reduction theorem is equivalent to Σn -transfer.
Proof. Uses overspill (form one level into the next).
References
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Second-order arithmetic Theorem (Π1n -reduction) For every standard ϕ ∈ ∆0 , (∀X1 ∈ P(N))(∃X2 ∈ P(N)) . . . (QXn ∈ P(N))
~ , ~y ) (∀y1 ∈ N)(∃y2 ∈ N) . . . (Qym ∈ N)ϕ(X
is equivalent to
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Second-order arithmetic Theorem (Π1n -reduction) For every standard ϕ ∈ ∆0 , (∀X1 ∈ P(N))(∃X2 ∈ P(N)) . . . (QXn ∈ P(N))
~ , ~y ) (∀y1 ∈ N)(∃y2 ∈ N) . . . (Qym ∈ N)ϕ(X
is equivalent to (∀x1 ≤ ωα )(∃x2 ≤ ωβ ) . . . (Qxn ≤ ωη )
(∀y1 ≤ ωσ )(∃y2 ≤ ωτ ) . . . (Qym ≤ ωζ )ϕ(~x , ~y )
with α < β < · · · < η < σ < τ < · · · < ζ.
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Second-order arithmetic Theorem (Π1n -reduction) For every standard ϕ ∈ ∆0 , (∀X1 ∈ P(N))(∃X2 ∈ P(N)) . . . (QXn ∈ P(N))
~ , ~y ) (∀y1 ∈ N)(∃y2 ∈ N) . . . (Qym ∈ N)ϕ(X
is equivalent to (∀x1 ≤ ωα )(∃x2 ≤ ωβ ) . . . (Qxn ≤ ωη )
(∀y1 ≤ ωσ )(∃y2 ≤ ωτ ) . . . (Qym ≤ ωζ )ϕ(~x , ~y )
with α < β < · · · < η < σ < τ < · · · < ζ. The number xi codes the set Xi (STP, Yokoyama, Keisler).
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Second-order arithmetic Theorem (Π1n -reduction) For every standard ϕ ∈ ∆0 , (∀X1 ∈ P(N))(∃X2 ∈ P(N)) . . . (QXn ∈ P(N))
~ , ~y ) (∀y1 ∈ N)(∃y2 ∈ N) . . . (Qym ∈ N)ϕ(X
is equivalent to (∀x1 ≤ ωα )(∃x2 ≤ ωβ ) . . . (Qxn ≤ ωη )
(∀y1 ≤ ωσ )(∃y2 ≤ ωτ ) . . . (Qym ≤ ωζ )ϕ(~x , ~y )
with α < β < · · · < η < σ < τ < · · · < ζ. The number xi codes the set Xi (STP, Yokoyama, Keisler). We need to quantify over the levels α ∈ A.
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Final Thoughts
...there are good reasons to believe that Nonstandard Analysis, in some version or other, will be the analysis of the future. Kurt G¨ odel
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Final Thoughts
...there are good reasons to believe that Nonstandard Analysis, in some version or other, will be the analysis of the future. Kurt G¨ odel Und wenn du lange in einen Abgrund blickst, blickt der Abgrund auch in dich hinein. (And when you stare in the abyss for long, the abyss stares into you.)
Friedrich Nietzsche
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
Final Thoughts
...there are good reasons to believe that Nonstandard Analysis, in some version or other, will be the analysis of the future. Kurt G¨ odel Und wenn du lange in einen Abgrund blickst, blickt der Abgrund auch in dich hinein. (And when you stare in the abyss for long, the abyss stares into you.)
Friedrich Nietzsche
Thanks for you attention! Any questions?
Introduction
A copy of the RM of WKL0
A constructive copy
Big questions
Future research
References
[1] Peter J. Huber and Elvezio M. Ronchetti, Robust statistics, 2nd ed., Wiley Series in Probability and Statistics, John Wiley & Sons Inc., Hoboken, NJ, 2009. [2] Chris Impens and Sam Sanders, Transfer and a supremum principle for ERNA, Journal of Symbolic Logic 73 (2008), 689-710. [3] The Linux Information Project, Robustness, 2005. http://www.linfo.org/robust.html. [4] Sam Sanders, Reverse Mathematics and ERNA, To appear (2010). [5] Stephen G. Simpson, Subsystems of second order arithmetic, 2nd ed., Perspectives in Logic, Cambridge University Press, Cambridge, 2009. [6] Richard Sommer and Patrick Suppes, Finite Models of Elementary Recursive Nonstandard Analysis, Notas de la Sociedad Mathematica de Chile 15 (1996), 73-95.
