超实数科普入门读物

 2015年6月，荷兰数学科普专家克拉科夫（Gianni Krako）教授精心撰写的“Hyperreals and a Brief Introduction to Non-Standard Analysis”科普文章（共计16页），简单易懂，很有阅读参考价值。）

 在科普文章中，克拉科夫教授是怎么说的？他说：学习现代数系，尤其是学习超实数，事先“must”学习公理化集合论（AST），进而掌握模型论基本思想。

请见本文附件。 

	克拉科夫

  感谢克拉科夫！

袁萌  陈启清 7月16日

附件：

Hyperreals and a Brief Introduction to Non-Standard Analysis Math 336

Gianni Krako

Abstract Krakoff

 

 

Abstract

The hyperreals are a number system extension of the real number system. With this number system comes many advantages in the use of analysis and applications in calculus. Non-standard analysis refers to the use of innitesimals in doing analysis instead of the usual epsilon and distance functions. The machinery we will build in this papaer will allow us to prove some elementary analytic results. This paer will go through how to construct number systems via equivalence classes, how the hyperreals are constructed, how the hyperreals function, and nally how to use them to prove some theorems about uniform convergence and Riemann Integration.

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Contents

1 Introduction 3

2 Number System Construction 3

2.1 Axiomatic Set Theory . . . . 3

2.2 Integers Z . . . 4

3 Hyperreal Construction   5

3.1 Preliminaries . . . . . . . . 5

3.2 The Equivalence Relation ≡ . . 5

3.3 Transfer Principle . . . . . . . . . . . .6

3.4 Ordered Fiel. . . . . . . 7

3.5 Innitesimals and Unlimiteds . . . . 8

4 Working With *R 9 4.1 Arithmetic in *.9 4.2 The ’Arbitrarily Close’ Equivalence Relation . . . 10

4.3 Set Enlargement . . . . . . . . 10

4.4 Least Upper Bound Property . . . . 10

5 Non-Standard Calculus 11

5.1 Continuity . . . . 11

5.2 Derivative . . . . . . . . . . .  12

5.3 Riemann Sum . .. .12

6 Non-Standard Analysis 12

6.1 Uniform Convergence  . 13 6.2 Adapted Analysis Proofs  . 13

7 Conclusions 14

8 References 16

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1 Introduction

Historically when Leibniz invented calculus the use of innitesimals was somewhat careless, not many people questioned the vailidity of them as mathematical objects. Besides some detractors, namely George Berkeley in his scathing article about innitesimals, no one really questioned the intutive nature of them. For instance Euler proved many theorems innitesimals without much regard for the foundations of the ojects he was working with. For example if one were to calculate the derivative of f(x) = x2, the calculation would be as follows: (x + )2 −x2 = 2x + 2  = 2x +

however  is an innitesiamal and thus disregarded, so the answer is 2x. The quest for absolute rigor lead to the demise of innitesimals in the 19th century, it was during this time that Wierstrass gave his ,δ denition of limit. It wasnt until advances in model theory for Robinson in the 1960s that the full theory of innitesimals was rigorously dened.

2 Number System Construction

2.1 Axiomatic Set Theory

To begin understanding the construction of the non-standard reals we must be familiar with how number systems are constructed. We construct the natural numbers from axiomatic set theory using two axioms, union and the existence of an inductive set. Start with the empty set, ∅ , and dene the sucessor operation S on a set x such that S(x) = x ∪{x}, for instance 0 is associated with the empty set, ∅, and so 1 is associated with {∅,{∅}}. The set with the inductive property, that being of closed under the successor operation, we intentionally call N. Dention 2.1. Equivalence Relation: An Equivalence Relation, ∼ is a relation on sets such that the three following properties hold: 1. Reexivity: a ∼ a 2. Symmetry: a ∼ b implies b ∼ a

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3. Transitivity: a ∼ b and b ∼ c implies a ∼ c Dention 2.2. Equivalence Class An equivalence class is the set of objects satisng some equivalence relation. The integers, Z, is constructed by equivalence classes from orderd pairs of naturals. The rationals, Q, are constructed from equivalence classes on integers.1 The interesting part comes in constructing the reals, R, from the rationals, where we no longer talk about ordered pairs but innite sequences. There are two ways of dening the reals one way is consider Dedekind cuts and the other is to look at equivalence classes of Cauchy Sequences of rational numbers. The Cauchy sequence method is more appropriate for what is being done here, as the hyperreals, *R, are dened in such a way from real numbers.

2.2 Integers Z Example 2.1. Lets constuct the integers from the naturals as simple demonstration of taking an equivalence class. Z = N×N/ ∼, i.e. we identify an integer as an equivalence class or the set of ordered pairs of natural numbers that satisfy the following equivalence relation. Where ∼ is the equivalence relation (a,b) ∼ (c,d) if a + d = b + c. We verify this is an equivalence relation by showing reexivity, symmetricity, and transitivity: (a,b) ∼ (a,b) (a,b) ∼ (c,d) → (c,d) ∼ (a,b) (a,b) ∼ (c,d),(c,d) ∼ (e,f) → (a,b) ∼ (e,f) Obviously these hold by the commutativity and identity properties of the natural numbers, the details are left for the reader.

We construct the hyperreals the same way via the set of all real valued sequences indexed by the natural numbers. Now to the construction

1See Enderton Chapter 5 for details.[2]

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3 Hyperreal Construction

3.1 Preliminaries The construction of hyperreals requires taking an equivalence classes on RN, the set of innite real valued sequences.2 The relation we have to dene for the hyperreals is going to be a special kind of equivalence relation which will need some set theoretic machinery. The rst object we will need is what is called an ultralter. An ultralter, F, on a a set X is a set of subsets of X. The ultralter tests for size of the of subsets of X. When we get to the construction we will use size and real number equality as our equivalence relation. As would be guessed, our intuitive dention of ultralter ts with the formal dention. An ultralter acts like a sieve that lters out small sets, when big sets are desired. Dention 3.1. Power Set :The power set P(X) of a set X is dened to be {x|x ⊂ X}. Dention 3.2. Conite: Given set Y ⊂ X, Y is conite if X\Y is nite. Dention 3.3. Ultralter:An ultralter F on a set X is a collection of subsets of X that satisfy the following properties:

1. X ∈F 2. ∅ /∈F 3. If A,B ∈F then A∩B ∈F 4. If A ⊂ X then either A ∈F or X \A ∈F 5. If A ∈F and A ⊂ B then B ∈F. 3.2 The Equivalence Relation ≡ The equivalence relation we want to take is on the set of all innite real sequences, RN, will use an ultralter to grab sets that are equivalent. We will say that two hyper real numbers are equal if their real number sequences dier by at most a nite number of terms.

2This constuction is taken from Goldblatt Chapter 3.[3]

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Dention 3.4. Dene a relation, ≡, on RN, by saying that (rn) ≡ (sn) if and only if {n ∈N|rn = sn}∈F. This equivalence relation captures the idea of ’agreeing almost everywhere.’ Since the hyperreals are constructed using real numbers we should use real number equality since we know how that works. We say x = y in the hyperreals if the parts of the real number sequences dening x and y dier at only nitly many terms. For example let x = (1,2,3,4,...) and y = (1,2,2,4,...), if these sequences progressed in the natural way then we could say that x = y because each sequence only diers by nite terms, namely one term. The less than relation is dened similarly we want the terms for which the xj 6 yj to be only nite to say that yj < xj. The conite equality of the real number sequences is captured by the ultralter.

Dention 3.5. Hyperreal Arithmetic is dened componentwise.

[r] + [s] = [(rn + sn)] [r]·[s] = [(rn ·sn)] Dention 3.6. The set of equivalence classes of an ultralter is called an ultraproduct and r ∈RN under ≡ denoted by [r] will be [r] = {s ∈RN|r ≡ s}. This leads to the denition of the hyperreals from RN/ ≡. ∗R = {[r]|r ∈RN}

3.3 Transfer Principle

Statements about hyperreals require the use of rst-order logic. First-order logic discuss quantication over objects, whereas second-order logic allows quantication over predicates or relations.

Dention 3.7. First Order Formula is a formula involving quantication over objects in the domain of discourse.

Example 3.1. Let X be the set of students in 336, and let φ be the formula so that φ(x) says that ’x does analysis.’ Then we can say that ∀xφ(x) is a rst order forumla about students.

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Theorem 3.1. Transfer Principle: Any appropriately formulated statement φ about R holds i *φ holds for *R.

Proof. The rigorous proof uses sophisticated model theory or axiomatic deductions and is beyond the scope of this paper. The idea of why this holds is straightforward. Consider the ultralter U on a cartesian product of sets, Mi: Y i∈N Mi/U Then if a rst-order formula, φ, holds for each Mi and is captured by the ultralter, then φ holds in the ultraproduct.

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Here appropriately formulated means statements in rst-order logic. The transfer principle allows statements about the reals to be equivalent statements of the hyperreals and vice versa. For example commutativity is an appropriate statement about the reals and so it is an appropriate statement about the hyperreals. The details of all the properties of the reals that transfer to the hyperreals are too tedious for this paper.

3.4 Ordered Field Dention 3.8. An Ordered Field, F, is an algebraic stucture with a set, F, two operations, {+,·}, and an ordering relation, {6} that satisfy the following properties:

1. F is associative in both operations 2. F is invertable in both operations 3. F has an identity element in both operations 4. F is closed under both operations 5. F has a well dened notion of order

The Transfer Principle justies this. For our purpose it is enough to say that since the reals are a ordered eld then the hyperreals are an ordered eld.

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3.5 Innitesimals and Unlimiteds

The reals are a subeld of the hyperreals, in a similar sense that the rationals are a subeld of the reals. The reals in the hyperreals are identied with innte sequences of themselves, so in the hyperreals 0 is identied with the innite sequence of 0’s, (0,0,0...) and π with (π,π,π,...). Innitesimals are dened by seqences of real numbers approaching 0, for instance we may take = (1 2, 1 4, 1 8,...). This number is less than any real number in the hyperreals: Theorem 3.2. There exists  ∈*R, called an innitesimal, such that  > 0 and for all x ∈R, < x. Proof. Let  = (1 2, 1 4, 1 8,...), with each term called n and 0 = (0,0,0,...), with each term called 0n, then n > 0n for all n, so > 0. Take x = (x,x,x,...) with each term called xn, then since each xn is equal there exists n0 such that for all n > n0 then n < xn, so there are innitely many terms such that n < xn, therefore  < x.

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 Theorem 3.3. There exists ω ∈*R, called unlimited, such that ω > x for all x ∈R. Proof. Let ω = (1,2,3,...) with each term ωn and take x = (x,x,x,...) with each term called xn, then there exists n0 such that for all n > n0 then ωn > xn.

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Let’s recap what just happened. First we considered the set of innite sequences of real numbers. Then we dened an ultralter that picked out elements of those sequences that were dened to be equivalent if they diered by at most nite number of terms. This dened the hyperreals as an object of study. Next the arithmetic of how the equivlence classes would be worked with in terms of addition and multiplication was dened in terms of innite sequences. The transfer principal said that any well formulated statement about the reals would hold for the hyperreals, this happened because of model theory. This gave the arithmetic and ordering properties of the hyperreals as objects themselves. Finally we showed that this set had the innitesimals and unlimited numbers that were desired.

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4 Working With *R 4.1 Arithmetic in *R Let x ∈*R, x is called nite if a < x < b for a,b ∈R, and x is not innitesimal. A number  is called innitesimal if  < x for all x ∈ R+ and  > x for all x ∈ R−. Here  can be positive or negative, but it is not necessary to worry about for the purposes of this paper. One way to think about an innitesimal is that if the limit of the terms of the real number sequence dening the hyperreal number goes to zero then it will dene an inntesimal. Suppose ,δ are innitesimal, x,y are nite and ω,α are unlimited then the following is true:

1.  + δ is innitesimal

2. x +  is nite

3. x + y is appreciable possible innitesimal 4. ·δ is inntesimal 5. ·x is innitesimal 6. ω + x is unlimited

7. 1  is unlimited 8. 1 ω is innitesimal

Several indeterminate forms arise when working with operations of hyperreal numbers:

1.  δ 2. ω α 3. ω·4. ω + α

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4.2 The ’Arbitrarily Close’ Equivalence Relation Dention 4.1. Let ', be an equivalence relation on *R such that x ' y means that x−y is innitesimal or 0. Checking this is a well dened equivalence relation is straightforward and left to the reader. It immediately follows that  ' 0, this will be important because it captures of idea of ”arbitrary close” that lies at the heart of usual ,δ type proofs in analysis.

4.3 Set Enlargement Dention 4.2. Set Enlargement: A set I ⊂ R can be extended to a set *I ⊂*R if for each r ∈RN then [r] ∈∗A ↔{n ∈N|rn ∈ A}∈F The ultralter gives us the concept of almost all, by the conite denition of equality. So we can extend an interval to be a hyperreal interval by saying that a hyperreal number is in the interval if the real valued sequence dening it is almost all in the real interval.

4.4 Least Upper Bound Property

The least upper bound (LUB) property for the real number states that every set of real numbers with an upper bound has a least upper bound. This is also called the dedekind completeness property or Cauchy completeness. This has problems in the language of the transfer princple, since here we are not quantifying over objects, but sets of objects. This means that the LUB property is not expressable in rst-order language. But the properties of the hyperreals give us an equivalent statement.

Theorem 4.1. Because Every limited hypereral is innitely close to a real number implies the completeness of R.3 Proof. Let s : N→R be a Cauchy sequence, so there exists k ∈N so that ∀m∀n ∈N(m,n > k →|sm −sn| < 1). 3[3] pg 55

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This is a rst-order statement(only quantied over numbers, not sets), so the transfer principle applies in a sequence with an unlimited hyperreal N. Take N unlimited so k,N > k then |sk −sN < 1|, and so sN is limited. By assumption that every limited hyperreal is innitely close to a real number then say that sN ' L for L ∈R. Now we show that the the original sequence s → L. Let  > 0, since s is Cauchy then there exists j0 ∈ N so that for j > j0then: ∀m∀n ∈N(m,n > j →|sm −sn| < ). We can show that beyond sj that all the terms are within  of L. This is because all such terms are within  of sN, which is itself with in  of L. Let m ∈N be such that m > j then m,N > j so by transfer principle then |sm −sN| < . Pushing the inequalities this becomes: |sm −L| < |sm −sN|+|sN −L| <<span style="margin: 0px;">  + δ. Where here δ is another innitesimal. And since Cauchy sequence convergence implies completeness, this completes the proof.

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The last result is interesting in that it shows how transfer can be used in ingenous ways to give us properties that we might have lost. In this case we might lose the LUB property since it is a second-order formula, but the rstorder formulation of convergent sequences gave use that property. Though the work of showing that Cauchy Completeness is equivalent to LUB and to Dedekind completeness is taken as given here.

5 Non-Standard Calculus

5.1 Continuity

Dention 5.1. Let f(x) be a real-valued function on [a,b], then say that f(x) is continuous if when x ' c then f(x) ' f(c) for all x ∈ [a,b]. 4 Example 5.1. All lines of the form y = mx + b are continuous. Suppose x ' c then x = c + and: f(x)−f(c) = m(c + ) + b−(mc + b) = m ' 0. 4These dentions are taken from Keisler[4]

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This simple example shows the power of innitesimal caluclus. In regular continuity proofs, much inequality pushing was necessary and nding appropriate bounding inequalities could be very dicult. But here we appeal to the intuition that Leibniz and many great mathematicians had, but that intuition now has a very rigorous foundation that could be laid out at any time. No longer is it necessary in many cases to nd a δ because that δ exists.

5.2 Derivative Dention 5.2. Let f(x) be real valued, the derivative of f(x) denoted f0(x) is given by f0(x) = f(x + )−f(x)  provided that the function is dened at x and f0(x) ' f0(x) + .

5.3 Riemann Sum Dention 5.3. Let f(x) be dened on I = [a,b] and let π = {x0,x1,...,xω} be a partition of I, and let α = {α1,α2,...,αω}, where xj 6 αj 6 xj+1, then dene

S(π,α) =

ω X j=0

f(αj)(xj+1 −xj) 'Z b a

f(x)dx.5

Since there exists unlimited numbers in the hyperreals we can use those to capture the idea ”for suciently large” in a rigourous sense.

6 Non-Standard Analysis

We know present some theorems from introductory analysis from a nonstandard point of view. The goal here is to see simplications that come from not having to choose or ’suciently large n.’

5[5] Pg 71-72

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6.1 Uniform Convergence

Dention 6.1. Uniform Convergence of a Sequence of Functions A sequence of functions fn is said to converge uniformly to f on I if fn ' f for n unlimited.6

6.2 Adapted Analysis Proofs

Theorem 6.1. Cauchy Criterion for Uniform Convergence Let fn be a sequence of bounded funtions on I ⊂ R. Then this sequence converges uniformly on I to a bounded function f if and only if for unlimited n and m then fn −fm ' 0.7 Proof. Suppose fn converges uniformly to f on I, then for unlimited n and m, fn ' f and fm ' f. So by transitivity fm −fn ' 0. Now suppose that fm −fn ' 0, so for each x ∈ I then fn(x)−fm(x) ' 0. Because fn(x) adn fm(x) are cauchy they both converge to f(x). So fn(x) converges to f(x) for each x ∈ I then fn converges uniformly to f on I.

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 Theorem 6.2. Suppose fn is a sequence of continuous functions on and fn → f uniformly on I, then f is continous I.8 Proof. Take x ' c and n unlimited then fn(x) ' fn(c) for all n from the assumption of continuity of each fn. Because of uniform convergence fn(x) ' f(x) and fn(c) ' f(c). So then by transitivity f(x) ' f(c).

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 We know that for Z b a f = lim n→∞Z b a fn to hold then fn must converge uniformly to f on [a,b], we prove that now using non-standard analysis. Where n →∞ is replaced with the unlimited hyperreal number ω. Theorem 6.3. Let fn be a sequence of functions in R[a,b] and suppose fn converges uniformly on [a,b] to f. Then f ∈R[a,b].9 6[5] Pg 130-131 7[1] Pg 246 8[6] Pg 443-444 9[1] Pg250-251

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Proof. Since Riemann integrable functions are bounded then by the Cauchy Criterion for uniform convergence we know that for x ∈ [a,b] that fm−fn ' 0 for n,m > ω, so then fm converges and this implies that Z b a fm ' I. Now take m > ω so that fm(x)−f(x) ' 0 for all x ∈ [a,b]. And consider a signicant partition, π, of [a,b] then:

S(π,fm)−S(π,f) '

ω X i=1

(fm(xi)−f(xi))(xi −xi−1) ' 0(b−a) ' 0.

Considering the fact that for unlimited n that: Z b a fm −I ' 0. And we can say that because each fm is Riemann integrable Z b a fm −S(π,fm) ' 0. We nally get: S(π,f)−I ' S(π,f)−S(π,fm) + S(π,fm) +Z b a fm −Z b a

fm −I ' 0.

 

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7 Conclusions

The importance in non-standard analsis comes from its simplication of proofs. No triangle inequality, no adding and subtracting terms, and no ingenuity when trying to nd inequalties. You could say that inequalities are in the equivalence relation, ', but its function is almost that of equality. Non-Standard Analysis works great for heuristic arguments when it comes to analysis. If we want to show something, it could suce to show that it does work for the hyperreals and use the transfer principle to claim that it

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workds for the reals too, though this can be tricky. This paper left alot out because of the nature of the paper, and the deep topic that Non-Standard Analysis is. Goldblatt’s book is a very are lecture ntoes for a course that he taught on this subject. In 1976, Kielser, who was involved in the development of the some of the machinery that helped developed for the hyperreals, wrote a book called ”Elementary Calculus.” This book, is a textbook style calculus text, that uses the hyperreal number system for its proofs. Though it never caught on, this could be a great teaching method for those people learning elementary calculus, since it relies more on the intuitions and less on the seemily criptic analytical denitions of calculus topics.

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8 References

[1]R.G. Bartle. D.R Sherbert. Introduction To Real Analysis. John Wiley and Sons Inc. 2011. Chapter 8.

[2] H. B. Enderton Elements of Set Theory. Pg 90-127. Academic Press. 1977.

[3] R. Goldblatt. Lectures on the Hyperreals An Introductions to Nonstandard Analysis. Pg 23-33. Springer. 1998.

[4]J. Keisler. Elementary Calculus. Prindle, Wber, Schmidt. 1976.

[5] A. Robinson. Non-Standard Analysis. North-Holland Publishing Company. 1966. Pg 130-131

[6]C.L. Thompson. Nonstandard Continuity and Uniform Convergence. The American Mathematical Monthly, Vol. 96, No. 5(May 1989), pg 443-444.

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