A copy of several Reverse Mathematics - Google Sites

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 constructive copy

Big questions

Future research

An example: Reverse Mathematics for WKL0 Central principle:

Theorem (Weak König’s Lemma) Every infinite binary tree has an infinite path.

References

Introduction


A constructive copy

Big questions

Future research


Theorem (Weak König’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 constructive copy

Big questions

Future research

References


Theorem (Weak König’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 constructive copy

Gödel’s completeness theorem.

Big questions

Future research

References

Introduction


A constructive copy

Big questions

Future research

7


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 constructive copy

Big questions

Future research

7


8


9


10


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 constructive copy

Big questions

Future research

7


8


9


10


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 constructive copy

A ‘gap’ in Reverse Mathematics

Big questions

Future research

References

Introduction


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 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 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 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 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 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 constructive copy

Big questions

Future research

References


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 constructive copy

Big questions

Future research

References


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


RM for I ∆0 + exp

A constructive copy

Big questions

Future research

References

Introduction


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 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 constructive copy

Big questions

Future research

References

RM for I ∆0 + exp


. . . −1 0

1 2

1

5 3

2 ... -

Introduction


A constructive copy

Big questions

Future research

References

RM for I ∆0 + exp


1

. . . −1 0 2 1 | {z

5 3

2 ... }

Q, the rational numbers

-

Introduction


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 ... }


z

}|

{ -

Introduction


A constructive copy

Big questions

Future research

References

RM for I ∆0 + exp


1 . . . −1 0 | {z

5 3

2 ... }


z

}| ω

{ -

Introduction


A constructive copy

Big questions

Future research

References

RM for I ∆0 + exp


1 . . . −1 0 | {z

5 3

2 ... }


z

}| ω/2 ω

{ -

Introduction


A constructive copy

Big questions

Future research

References

RM for I ∆0 + exp


1 . . . −1 0 | {z

5 3

2 ... }


z

}| ω/2 ω

{ 2ω-

Introduction


A constructive copy

Big questions

Future research

References

RM for I ∆0 + exp


1 . . . −1 0 | {z

5 3

z √

}| 2 . . .d ωe ω/2 ω }


{ 2ω-

Introduction


A constructive copy

Big questions

Future research

References

RM for I ∆0 + exp


1

. . . −1 0 2 1 | {z

5 3

√ 2 . . .d ωe ω/2 ω } | {z


infinite numbers

2ω}

Introduction


A constructive copy

Big questions

Future research

References

RM for I ∆0 + exp


1

. . . −1 0 2 1 | {z

5 3

√ 2 . . .d ωe ω/2 ω } | {z


2ω}

positive infinite numbers

Introduction


A constructive copy

Big questions

Future research

References

RM for I ∆0 + exp


√ 1 −2ω −ω −d ωe . . . −1 0 2 1 | {z }| {z negative infinite numbers

5 3

√ 2 . . .d ωe ω/2 ω } | {z


2ω}


Introduction


A constructive copy

Big questions

Future research

References

RM for I ∆0 + exp



5 3

√ 2 . . .d ωe ω/2 ω } | {z


x is infinite iff |x| > q, for all q ∈ Q+

2ω}


Introduction


A constructive copy

Big questions

Future research

References

RM for I ∆0 + exp



5 3

the finite numbers

√ 2 . . .d ωe ω/2 ω } | {z

x is infinite iff |x| > q, for all q ∈ Q+

2ω}


Introduction


A constructive copy

Big questions

Future research

References

RM for I ∆0 + exp



5 3

the finite numbers

√ 2 . . .d ωe ω/2 ω } | {z

2ω}


x is infinite iff |x| > q, for all q ∈ Q+ x is infinitely small iff |x| < q, for all q ∈ Q+

Introduction


A constructive copy

Big questions

Future research

References

RM for I ∆0 + exp


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}


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 constructive copy

Big questions

Future research

References

RM for I ∆0 + exp


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}


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 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}


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 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 constructive copy

Big questions

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References



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 constructive copy

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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


Continuity

A constructive copy

Big questions

Future research

References

Introduction


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 constructive copy

Big questions

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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 constructive copy

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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 constructive copy

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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 constructive copy

Big questions

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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 constructive copy

Weierstrass extremum principle 6

f (x)

a

b-

Big questions

Future research

References

Introduction


A constructive copy


f (x)

a

b-

Big questions

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References

We find the maximum over [a, b] of the continuous function f (x).

Introduction


A constructive copy


Big questions

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References


1) Define xi = a + ε(b − a)i. (i ≤ ω) f (x)

a xi xi+1 -

ε≈0

b-

Introduction


A constructive copy


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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 constructive copy


Big questions

Future research

References



f (x)

3) x ≈ xj ≈ xj+1 implies f (x) ≈ f (xj ) a xi xi+1 -

ε≈0

b-

Introduction


A constructive copy

Weierstrass extremum principle M •

6

Big questions

Future research

References



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 constructive copy


6

Big questions

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References



f (x)


ε≈0

b- 5) We have (∀x ∈ [a, b])(f (x) / M).

Introduction


A constructive copy


6

Big questions

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f (x)


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 constructive copy


6

Big questions

Future research

References



f (x)


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 constructive copy


6

Big questions

Future research

References



f (x)


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 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 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 constructive copy

Big questions

Future research


y=x

√ • −1 + 5 2

•

−1 + 2

But

√ −1+ 5 2

6∈ ∗ Q because

√

√

5

5 6∈ ∗ Q!

•

X

References

Introduction


A constructive copy

Big questions

Future research

References


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 constructive copy

Big questions

Future research

References


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 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 constructive copy

Big questions

Future research

References


Theorem (Brouwer fixed point principle) A standard continuous [0, 1]2 → [0, 1]2 -function has a fixed point up to infinitesimals.

Introduction


A constructive copy

Big questions

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References


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 constructive copy

Big questions

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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 constructive copy

Riemann integration

6

f (x)

a

b -

Big questions

Future research

References

Introduction


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 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 constructive copy

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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 constructive copy

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f (x)

ω intervals [xi , xi+1 ]

a

z

}| -

εi ≈ 0

{

b -

Introduction


A constructive copy

Big questions

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References


f (x)

a

b -

εi ≈ 0

-

Introduction


A constructive copy

Big questions

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References


f (x)

a

xi • εi ≈ 0

b -

Introduction


A constructive copy

Big questions

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References


f (xi ) •

a

xi • εi ≈ 0

f (x)

b -

Introduction


A constructive copy

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2) Surface of rectangle is f (xi )εi

f (xi ) •

a

xi • εi ≈ 0

f (x)

b -

Introduction


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f (xi ) •

a

xi • εi ≈ 0

f (x)

b -

3) Construct all rectangles

Introduction


A constructive copy

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f (xi ) •

a

xi • εi ≈ 0

f (x)

b -

3) Construct all rectangles

Introduction


A constructive copy

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f (xi ) •

a

xi • εi ≈ 0

f (x)

b -

3) Construct all rectangles P 4) Total surface is ωi=1 f (xi )εi

Introduction


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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


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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 constructive copy

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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


Riemann integration II

A constructive copy

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References

Introduction


A constructive copy

Big questions

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Principle (Integrability principle) A standard continuous function is Riemann integrable.

References

Introduction


A constructive copy

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Theorem (S.) In ERNA, the Integrability principle is equivalent to Π1 -transfer.

References

Introduction


A constructive copy

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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 constructive copy

Peano existence theorem

Big questions

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References

Introduction


A constructive copy

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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 constructive copy

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φ(0) = 0

holds for |x| < α, with α = min(a, b/M). = the ‘real’ Peano existence theorem, up to infinitesimals

(2)

Introduction


A constructive copy

Big questions

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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 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 constructive copy

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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 constructive copy

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Theorem In ERNA, the Isomorphism theorem is equivalent to Π1 -transfer.

References

Introduction


A constructive copy

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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 constructive copy

And many more principles. . .

Big questions

Future research

References

Introduction


A constructive copy

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And many more principles. . . In ERNA, the following are equivalent to Π1 -transfer.

References

Introduction


A constructive copy

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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 constructive copy

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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 constructive copy

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2


3


4


5


6

Isomorphism theorem

7


8


9

And many more. . .

The RM of ERNA + Π1 -TRANS is a copy up to infinitesimals of the RM of WKL0 .

References

Introduction


A constructive copy

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2


3


4


5


6

Isomorphism theorem

7


8


9

And many more. . .

However, (7) belongs to the RM of ACA0 , NOT to that of WKL0 .

Introduction


A constructive copy

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2


3


4


5


6

Isomorphism theorem

7


8


9

And many more. . .

However, (7) belongs to the RM of ACA0 , NOT to that of WKL0 . Why is this so?

Introduction


A constructive copy

Big questions

Future research

Similarities between Constructivism and ERNA

Constructivism

ERNA

References

Introduction


A constructive copy

Big questions

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√

Constructivism x is not always defined

√

ERNA x is not always defined

References

Introduction


A constructive copy

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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 constructive copy

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Future research


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 constructive copy

Big questions

Future research


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 constructive copy

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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


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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.)

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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 √

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Constructivism (∃n)ϕ(x) means ‘there is an algorithm to compute n0 s.t. ϕ(n0 )’.

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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 ).

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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 ).

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Constructive Reverse Mathematics (CRM)

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A constructive copy

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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).

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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 )’.

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Principle (Π1 -transfer) For every q.f. formula ϕ, we have (∃n ∈ N)ϕ(n) ∨ (∀n ∈ ∗ N)¬ϕ(n).

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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).

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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


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Constructive Reverse Mathematics

Conjecture: CRM with Σ1 -PEM is similar to the RM of ERNA + Π1 -TRANS.

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Big Questions

A constructive copy

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A constructive copy

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Big Questions

(#1) Is physical reality continuous or discrete?

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A constructive copy

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Big Questions

(#1) Is physical reality continuous or discrete? (#2) What are good foundations for Mathematics?

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A constructive copy

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Big Questions

(#1) Is physical reality continuous or discrete? (#2) What are good foundations for Mathematics? (#3) Do infinitesimals exist?

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Small answers (#2) The RM of ERNA + Π1 -TRANS is a copy up to infinitesimals of the RM of WKL0 .

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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’.

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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]).

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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]).

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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.

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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!

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Small answers (#3)


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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 ).

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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 .

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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.

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Small answers (#1) Recall that in ERNA, the following are equivalent to Π1 -transfer. 1


2


3


4


5


6

Isomorphism theorem

7

8

Completeness up to infinitesimals of ∗ Q (Cauchy, Dedekind, Cantor) And many more. . .

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Small answers (#1) The Continuity principle is equivalent to the Isomorphism theorem.

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Theorem (Isomorphism theorem) Every ‘R-like’ model of a physical problem has an isomorphic ‘Q-like’ model.

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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 )).

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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.

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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.

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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!

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Small answers (#1)

(# 1) Is physical reality discrete or continuous?

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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].

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A constructive copy

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Introduction


A constructive copy

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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


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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


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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


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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


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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


Stratified NSA

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A constructive copy

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Stratified NSA In classical NSA, a number is either finite or infinite.

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A constructive copy

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Stratified NSA In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity.

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A constructive copy

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Stratified NSA In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider NA :

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0 1 ... | {z } N

ωα

...

ωβ

...

ωγ

... -

Introduction


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0 1 ... | {z }

ωα

...

ωβ

N

where A = {0, α, β, γ, . . . }

...

ωγ

... -

Introduction


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0 1 ... | {z }

ωα

...

ωβ

...

ωγ

...

N

where A = {0, α, β, γ, . . . } (e.g. A is finite or N)

-

Introduction


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0 1 ... | {z }

ωα

...

ωβ

N, 0-finite

where A = {0, α, β, γ, . . . }

...

ωγ

... -

Introduction


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0 1 ... | {z } |

ωα

...

ωβ

ωγ

...

-

{z

N, 0-finite

0-infinite

where A = {0, α, β, γ, . . . }

...

Introduction


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0 1 ... | {z } |

ωα

...

ωβ

ωγ

...

-

{z

N, finite

infinite

where A = {0, α, β, γ, . . . }

...

Introduction


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α-finite

z

0 1 ... | {z } |

}|

ωα

{

...

ωβ

ωγ

...

-

{z

N, finite

infinite

where A = {0, α, β, γ, . . . }

...

Introduction


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α-finite

z

0 1 ... | {z } |

}|

α-infinite

ωα

{ z

...

ωβ

N, finite

...

}| ωγ {z

infinite

where A = {0, α, β, γ, . . . }

... -

Introduction


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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


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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


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α-finite

z

0 1 ... | {z } |

}|

α-infinite

ωα

{ z

...

ωβ

N, finite

...

}| ωγ {z

infinite

where A = {0, α, β, γ, . . . }

... -

Introduction


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α-finite

z

0 1 ... | {z } |

}|

α-infinite

ωα

{ z

...

ωβ

N, finite

...

}| ωγ {z

infinite

where A = {0, α, β, γ, . . . }

... -

Introduction


A constructive copy

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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


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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


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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


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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


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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


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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


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Transfer =‘All levels α ∈ A have the same properties as NA .’

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Transfer =‘All levels α ∈ A have the same properties as NA .’ Define Nα = {n ∈ NA |n is α-finite}.

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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 )

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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

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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.

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From I ∆0 to PA

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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 .

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Theorem (S.) I ∆0 + NSA + Σn -transfer proves I Σn

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Theorem (S.) I ∆0 + NSA + Σn -transfer proves I Σn TRANS := ∪n∈N Σn -transfer

Corollary I ∆0 + NSA + TRANS proves PA

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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)

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The reduction theorem

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The reduction theorem In I ∆0 + NSA + Σn -transfer, we prove

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Theorem (Σn -reduction, S.) For every α-standard ϕ ∈ ∆0 , (∃x1 ∈ NA )(∀x2 ∈ NA ) . . . (Qxn ∈ NA )ϕ(x1 , . . . , xn )

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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 .

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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.

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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)

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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


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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).

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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


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~ , ~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.

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~ , ~y ) (∀y1 ∈ N)(∃y2 ∈ N) . . . (Qym ∈ N)ϕ(X


(∀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).

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~ , ~y ) (∀y1 ∈ N)(∃y2 ∈ N) . . . (Qym ∈ N)ϕ(X


(∀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.

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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

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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

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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?

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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.