哥德尔预言无穷小微积分是未来的数学分析

Posted 2023-01-27 yuanmeng001

tags:

篇首语：本文由小常识网(cha138.com)小编为大家整理，主要介绍了哥德尔预言无穷小微积分是未来的数学分析相关的知识，希望对你有一定的参考价值。

哥德尔预言无穷小微积分是未来的数学分析

二十世纪世界伟大的数学家哥德尔预言非标准分析是未来的数学分析。

哥德尔1974年预言的原文如下：

“There are good reasons to believe that non-standard analysis, in some version or other, will be the analysis of the future” [33]. Kurt G¨odel, 1974.请见本文附件1。

注：本文附件2是发表于2013年2月8日的非标准分析论文，此文附有44篇珍贵的非标准分析论文。

袁萌陈启清 9月17日

附件1

[33] T. Runge. Hyperﬁnite probability theory and stochastic analysis within Edward Nelsons internal set theory. 2011. URL http://www10.informatik. uni-erlangen.de/Publications/Theses/2010/Runge_DA10.pdf.

附件2：

Eoghan Staunton

ID Number: 09370803

Final Year Project

National University of Ireland, Galway

Supervisor: Dr. Ray Ryan

February 8, 2013

I hereby certify that this material, which I now submit for assessment on the programme of study leading to the award of degree is entirely my own work and has not been taken from the work of others save and to the extent that such work has been cited and acknowledged within the text of my work.

Author:

Eoghan Staunton

ID No:

09370803

Contents

1 Introduction 1

2 Construction of the Hyperreals 2

2.1 Our aim . . . . . . 2

2.2 Z, Q and R from N . . . . . . . . 2

.3 Free Ultraﬁlters . . . . .. . 4

2.4 Generating elements of ∗R . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 Arithmetic operations and inequalities in ∗R . . . . . . . . . . . . . 7 2.6 Some Notation & Deﬁnitions . . . . . . . . . . . . . . . . . . . . . 9 2.7 Other Ultrapower Constructions . . . . . . . . . . . . . . . . . . . 9 2.8 The ∗-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.9 Internal vs. External constants . . . . . . . . . . . . . . . . . . . . 11 2.10 Inﬁnitesimals and Hyperlarge numbers in ∗R . . . . . . . . . . . . 11

3 The Transfer Principle 14 3.1 History and Importance . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Mathematical Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 L o´s’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.4 The Transfer Principle . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.5 Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.6 Nonstandard Analysis as a Tool in Classical Mathematics . . . . . 23

4 The History of Inﬁnitesimals 25 4.1 Use in Ancient Greek Mathematics . . . . . . . . . . . . . . . . . . 25 4.2 Geometers of the 17th century and Indivisibles . . . . . . . . . . . 26

4.3 The Development of Calculus . . . . . . . . . . . . . . . . . . . . . 26

4.4 Modern Nonstandard Analysis . . . . . . . . . . . . .. . 29

5 Applications of Nonstandard Analysis 29 5.1 Economics and Finance . . . . . . . . . . . . . . . . . . . . . . . . 30 5.2

Selected Other Applications . . . . . . . . . . . . . . . . . . . . . . 33

6 Appraisal and Conclusion 34

1 Introduction

“There are good reasons to believe that non-standard analysis, in some version or other, will be the analysis of the future” [33]. Kurt G¨odel, 1974.

An inﬁnitesimal is a number that is smaller in magnitude than every positive real number. The word inﬁnitesimal comes from the Latin word inﬁnitesimus and was coined by the German mathematician Gottfried Wilhelm Leibniz around 1710 [1]. We learn early on in our study of standard analysis that nonzero inﬁnitesimals cannot exist. It is also true however that many people use the intuitive notion when trying to understand basic concepts in analysis and calculus such as derivatives or integrals. For example a student may think of the derivative of a function f at a point x as the slope of the secant line between the point (x,f(x)) and a point an inﬁnitesimal distance away. Informal notions of inﬁnitesimals have been used throughout history, indeed the concept was used by Isaac Newton and Leibniz in their formulation of calculus [9]. The use of inﬁnitesimals in the formative years of calculus however lacked rigour and their use was criticised by the Irish philosopher George Berkeley among others [44]. Eﬀorts were made to come up with a system that would admit the existence of inﬁnitesimals in a consistent manner. Little progress was made however, and most eﬀorts were abandoned when in the 1870’s Karl Weierstrass came up with the formal ‘epsilon-delta’ theory of limits which became the rigorous foundation needed for calculus. Modern mathematicians however again made eﬀorts in the 20th century to formalise a theory of nonstandard numbers and in 1961 Abraham Robinson succeeded in producing a consistent nonstandard analysis. I was introduced to the area by my supervisor who directed me to read a piece by 2006 Fields Medal winner Terence Tao on the subject in his book Structure and Randomness: Pages from Year One of a Mathematical Blog [39]. In the piece Tao gives a nice introduction to the area explaining many of the ideas in a nice intuitive way which piqued my interest in the subject. In his book Tao attributes the reluctance of many mathematicians to use non-standard methods to the tendency to “gloss over the actual construction of non-standard number systems”. Perhaps this is one of the reasons why although nonstandard analysis may still be the “analysis of the future”, as predicted by G¨odel, in mainstream mathematics, and certainly in undergraduate mathematics, it has yet to become the analysis of the present. The main aim of my project therefore is to give a clear introduction to the construction of the hyperreal numbers and the transfer principle of nonstandard analysis, suitable for any undergraduate mathematics student without any background in the area. A simple introduction to nonstandard analysis is given by Jerome Keisler in his book Elementary Calculus: An Inﬁnitesimal Approach [23]. His construction of the hyperreals is based on introducing inﬁnitesimals in an axiomatic way. My introduction will be based on the so-called ultraproduct construction of nonstandard analysis and uses some of the ideas of Tao and Jaap Ponstein [39], [30]. I will give an overview of the fascinating history of inﬁnitesimals. I will also present some of the interesting applications of nonstandard analysis with a focus

on the applications to economics and ﬁnance.

2 Construction of the Hyperreals

2.1 Our aim

In the ﬁrst article I read on the subject of non-standard analysis Tao attributes the reluctance of many mathematicians to use non-standard methods to the tendency to “gloss over the actual construction of non-standard number systems”. This causes the transfer principle and the construction behind it to be viewed as “some sort of “black box” which mysteriously bestows some certiﬁcate of rigour on nonstandard arguments” [39]. In this section I will attempt to clearly explain just how the hyperreals are constructed and to begin to demystify this “black box”. First we must be clear on exactly what we are attempting to do. We wish to introduce non-zero inﬁnitesimals to our set of real numbers. It is clear that 0 is the only element of the real numbers that is an inﬁnitesimal. Recall, an inﬁnitesimal is a number that is smaller in magnitude than any non-zero real number and so no non-zero real number ε is an inﬁnitesimal since |ε 2| < |ε| for every ε 6= 0. Our aim therefore must be to introduce non-zero inﬁnitesimals to the set of real numbers and come up with an extension of the real numbers. We also wish to treat these inﬁnitesimals as we would classical numbers and so each non-zero inﬁnitesimal δ can be inverted to give γ = 1 δ. Now ∀n ∈ N : |γ| > n, we will call such numbers hyperlarge. These hyperlarge numbers are also clearly greater than any real number x since ∀x ∈R : ∃n ∈N : n > x. Our ultimate goal therefore is to introduce these numbers to the reals and to come up with a consistent system of hyperreals ∗R. So how will we do this? To motivate our method of constructing the hyperreals we’ll ﬁrst look at how we can introduce Z, Q and R from our starting point of the natural numbers N.

2.2 Z, Q and R from N Just as when we are attempting to construct the system of hyperreals we are trying to construct is an extension of the set of real numbers, the set of integers is an extension of the set of natural numbers. The set of rationals is in turn an extension of the integers and the set of reals is an extension of the rationals. The elements of each of these sets can be generated by the elements of the set we are trying to extend, for example the elements of Z can be generated by elements of N. Leopold Kronecker is famously quoted as saying “God created the natural numbers; all else is the work of man” [42]. When we are generating these new elements we must always follow two rules:

1. We must deﬁne the equality of elements by using an equivalence relation.

2. If the element we have generated is not new we must identify it explicitly with the element we already knew.

2.2.1 The Integers

We can generate each integer using an ordered pair of natural numbers. If (x,y) is a pair of natural numbers then let Z(x,y) be the integer generated by it. So the set of all integers Z = Z(x,y) : x ∈N,y ∈N. We now must deﬁne when two integers Z(x,y) and Z(w,z) are equal. To do this we ﬁrst deﬁne an equivalence relation ∼z on pairs of natural numbers: (x,y) ∼z (w,z) ⇔ x + z = y + w and then set Z(x,y) = Z(w,z) precisely when (x,y) ∼z (w,z). Finally we must identify when a pair of natural numbers generates one of our original set of natural numbers. We use the following rule: ∀n ∈N : Z(n,0) = n.

2.2.2 The Rationals

We can generate each rational number by using an ordered pair of integers. If (p,q) is a pair of integers with q 6= 0 then let Q(p,q) be the rational generated by it. So the set of all rationals Q = Q(p,q) : p ∈Z,q ∈Z,q 6= 0. We now must deﬁne when two rationals Q(p,q) and Q(r,s) are equal. To do this we ﬁrst deﬁne an equivalence relation ∼q on pairs of integers: (p,q) ∼q (r,s) ⇔ ps = qr and then set Q(p,q) = Q(r,s) precisely when (p,q) ∼q (r,s). Finally we must identify when a pair of integers generates one of our original set of integers. We use the following rule: ∀x ∈Z : Q(z,1) = z.

2.2.3 The Reals

We can generate each real number by using a Cauchy sequence of rationals. If (q1,q2,...) is a Cauchy sequence of rationals then let R(q1,q2,...) be the real generated by it. So the set of all reals R = R(q1,q2,...) : ∀i : qi ∈Q. As in the two cases above we now must deﬁne when two reals R(q1,q2,...) and R(r1,r2,...) are equal. To do this we ﬁrst deﬁne an equivalence relation ∼r on Cauchy sequences of rationals:

(q1,q2,...) ∼r (r1,r2,...) ⇔∀m ∈N : ∃k ∈N : ∀n ∈N,n > k : |qn −rn| <

1 m and then set R(q1,q2,...) = R(r1,r2,...) precisely when (q1,q2,...) ∼r (r1,r2,...). Finally, we must identify when a Cauchy sequence of rationals generates one of our original set of rationals. We use the following rule: ∀q ∈Q : R(q,q ...) = q. We have seen that we can extend the natural numbers N to the integers Z by using two natural numbers to generate each integer. The integers Z can again be

extended to the rationals Q by using two integers to generate each rational. Finally, the rationals Q can be extended to the reals R by using inﬁnite Cauchy sequences of rationals to generate each real number. Now we wish to extend the reals R to the hyperreals ∗R by using elements of the reals to generate each hyperreal. If man can construct Z, Q and R by simply using the natural numbers given to us by God why not go one step further and construct ∗R?

2.3 Free Ultraﬁlters To extend the real numbers to the hyperreals ∗R we are going to use inﬁnite sequences of real numbers to generate each hyperreal number. To do this we also need to have rules for equality and identiﬁcation, just as we had for Z, Q and R so we can come up with a mathematically consistent and sensible system of hyperreals. To help us do this we are going to use a free utraﬁlter. Ultraﬁlters were not originally introduced for the purpose of constructing the hyperreal numbers and have other applications outside non-standard analysis. The notion of an ultraﬁlter was ﬁrst introduced by the French mathematician Henry Cartan in two short notes in The Proceedings of The Academy of Sciences Paris in 1937, for use in the area of general topology [12]. Ultraﬁlters are particularly important when dealing with Hausdorﬀ spaces. They are used in the construction of the Stone-ˇCech and Wallman Compatiﬁcations. First we will explore the more general idea of a ﬁlter.

Deﬁnition A non-empty collection F of subsets of N is called a ﬁlter (over N) if: • ∅6∈ F • if A ∈ F and N⊇ B ⊇ A then B ∈ U, • if A ∈ U and B ∈ U, then A∩B ∈ U, A ﬁlter U is called an ultraﬁlter if for any A ⊆N, either A or Ac is an element of U, but not both. (Here Ac is the complement of A. Ac = N\\A.) A ﬁlter is called free (or non-principal) if all of its elements are inﬁnite sets. Combining our above deﬁnition of a ﬁlter and the two conditions above we come up with the following deﬁnition of a free ultraﬁlter:

Deﬁnition A non-empty collection U of subsets of N is called a free ultraﬁlter (over N) if: • if A ∈ U and N⊇ B ⊇ A then B ∈ U, • if A ∈ U and B ∈ U, then A∩B ∈ U, • if A ∈ U, then A is inﬁnite, and, • if A ⊆N, then either A ∈ U or Ac ∈ U, but not both. Now we have given the deﬁnition of a free ultraﬁlter we must ask ourselves if such a ﬁlter exists. The answer is that free ultraﬁlters over any inﬁnite set do exist. A proof of their existence, relying on the axiom of choice, was given by

Tarski in 1930. In our proof below we will invoke the axiom of choice through the use of Zorn’s Lemma. This reliance on the axiom of choice leads to the area of non-standard analysis being criticised by constructivist mathematicians due to the extremely nonconstructive nature of the axiom. We will discuss this criticism further in the ﬁnal section of this paper. Theorem 2.3.1. Free ultraﬁlters over N exist. Proof. Let F0 be the ﬁlter consisting of all coﬁnite subsets of N, sometimes known as the coﬁnite or Fr´echet ﬁlter, (Q ⊆ N is coﬁnite if and only if Qc is ﬁnite). Let E be the set of all ﬁlters F s.t. F ⊇ F0. E is nonempty (since F0 ∈ E) and can be partially ordered by set inclusion. Let G be any totally ordered subset of E. Then B =SF : F ∈ G∈ E andB is an upper bound for G. B ⊇ F0 since ∀F ∈ G : F ⊇ F0. B is also a ﬁlter: 1. ∅6∈ B since B is the union of ﬁlters. 2. If Q ∈ B and N ⊇ R ⊇ Q then Q ∈ F for some F ∈ G. Since F is a ﬁlter R ∈ F also ⇒ R ∈ B. 3. Let Q,R ∈ B, then Q ∈ F1 and R ∈ F2 for some F1,F2 ∈ G. G is totally ordered so F1 ⊆ F2 or F2 ⊆ F1. Suppose F1 ⊆ F2, then Q,R ∈ F2. But F2 is a ﬁlter so Q∩R ∈ F2 ⊆ B. Now Zorn’s lemma tells us that E contains a maximal element U. We wish to show that U is a free ultraﬁlter. U ⊇ F0 and so U is a free ﬁlter. We must now show that U is also an ultraﬁlter. Let Q0 ⊆N and consider the following cases: Case 1: Suppose ∀Q ∈ U : Q0∩Q is inﬁnite. Let, V = T : N⊇ T ⊇ Q0∩Q : Q ∈ U. where Q is some arbitrary element of U. Then V is a ﬁlter. Also V ∈ E, we can see this by taking Q = n,n+1,...1. We also have that U ⊆ V but since V ∈ E and U is the maximal element of E U ⊇ V . So V = U and since N⊇ Q0 ⊇ Q0∩Q we have that Q0 ∈ V ⇒ Q0 ∈ U. Case 2: Suppose on the other hand that ∃Q0 ∈ U : Q0 ∩ Q0 is ﬁnite then Q0 6∈ U. Now take any Q ∈ U. Q0 ∈ U also ⇒ Q0 ∩ Q ∈ U and Q0 ∩ Q is inﬁnite. Since Q0 ∩Q0 ∩Q is ﬁnite (Q0 ∩Q)\\(Q0 ∩Q0 ∩Q) = Q0c ∩Q0 ∩Q is inﬁnite. So Q0c∩Q must be inﬁnite. Applying Case 1, replacing Q0 with Q0c, gives Q0c ∈ U. Theorem 2.3.2. Suppose that N = A1 ∪A2 ∪...∪An for mutually disjoint Ai and n ∈N. Then Ai ∈ U for exactly one i. Proof. Let Bi = Ac i. Suppose that there is no i ∈1,2,...,n such that Ai ∈ U. Then ∀i ∈1,2,...,n : Bi ∈ U. So we now have that ∅ = B1∩B2∩...∩Bn−1∩ Bn ∈ U which is a contradiction. So ∃i ∈1,2,...,n : Ai ∈ U. But we also have that the Ai are mutually disjoint and so there can only be one i such that Ai ∈ U since if Ai ∈ U and Aj ∈ U for i 6= j we would have that ∅ = Ai ∩Aj ∈ U. 1U ∈ E ⇒ U ⊇ F0. Now Q ∈ F ⇔∃n ∈N : n,n + 1,...⊆ Q (since Q must be coﬁnite). So Q0∩Q is inﬁnite iﬀ ∃m ∈N : m,m + 1,...⊆ Q0∩Q.

2.4 Generating elements of ∗R We now ﬁx an ultraﬁlter U on N. We will use inﬁnite sequences of real numbers, in conjunction with U, to generate the hyperreals. Just as in the case of Z, Q and R we will need to establish rules for the equality and identiﬁcation of these inﬁnite sequences of real numbers. This is where we will use our free ultraﬁlter. A nice way to think about how this works is to think of the inﬁnite sequence real numbers as the votes of an inﬁnite electorate. The ultraﬁlter helps us decide which votes matter when deciding the winner of the election. The fact that the ultraﬁlter is free means that if one number gets a coﬁnite number of “votes” it will always win against a number that gets only a ﬁnite number of “votes”.

2.4.1 Rules for Equality and Identiﬁcation

We can generate each hyperreal number by using an inﬁnite sequence of real numbers. If (x1,x2,...) is an inﬁnite sequence of real numbers let H(x1,x2,...) be the hyperreal number generated by it. So the set of all hyperreals ∗R = H(x1,x2,...) : ∀i : xi ∈R.

• Equality Again we must deﬁne when two hyperreals H(x1,x2,...) and H(y1,y2,...) are equal. To do this we ﬁrst deﬁne an equivalence relation ∼u on inﬁnite sequences of reals: (x1,x2,...) ∼u (y1,y2,...) ⇔i : xi = yi∈ U and then set H(x1,x2,...) = H(y1,y2,...) precisely when (x1,x2,...) ∼u (y1,y2,...). Theorem 2.4.1. ∼u is an equivalence relation. Proof. We must show that ∼u is reﬂexive, symmetric and transitive. 1. ∼u is reﬂexive. ∀(x1,x2,...) : (x1,x2,...) ∼u (x1,x2,...) since i : xi = xi = N∈ U for every free ultraﬁlter U. 2. ∼u is symmetric. (x1,x2,...) ∼u (y1,y2,...) ⇔i : xi = yi = i : yi = xi∈ U ⇔ (y1,y2,...) ∼u (x1,x2,...) 3. ∼u is transitive. (x1,x2,...) ∼u (y1,y2,...) and (y1,y2,...) ∼u (z1,z2,...) ⇔ A = i : xi = yi∈ U and B = i : yi = zi∈ U. Since U is a free ultraﬁlter we have that if A ∈ U and B ∈ U, then A∩B = i : xi = zi∈ U. i : xi = zi∈ U ⇔ (x1,x2,...) ∼u (z1,z2,...) So ∼u satisﬁes all three properties and is therefore an equivalence relation.

• Identiﬁcation We now must identify when an inﬁnite sequence of reals generates one of our original sets of reals. We use the following rule: ∀x ∈R : H(x1,x2,...) = x ⇔i : xi = x∈ U.

2.4.2 Some Examples 1. H(√2,√2,√2,...) = √2. This is true since i : xi = x = N and N∈ U for all free ultraﬁlters U. 2. What about the sequence (√2,π,√2,π,√2,...)? What number does it generate? Is it possible that our deﬁnition could lead us to the obvious contradiction that H(√2,π,√2,π,√2,...) = √2 and H(√2,π,√2,π,√2,...) = π? To answer this we must recall the deﬁnition of an ultraﬁlter given above. If U is an ultraﬁlter and A ⊆ N, then either A ∈ U or Ac ∈ U, but not both. So either the set of odd numbers 1,3,5,... or the set of even numbers 2,4,6,... is in our free ultraﬁlter U but not both. In this case if 1,3,5,...∈ U (we can think of this as the case where only the odd voters votes are taken into consideration) then H(√2,π,√2,π,√2,...) = √2 but if 2,4,6,...∈ U (we can think of this as the case where only the even voters votes are taken into consideration) then H(√2,π,√2,π,√2,...) = π.

3. What number does the sequence (1, 1 2, 1 3,...) generate? Recall that if U is a free ultraﬁlter and A ∈ U, then A is inﬁnite. Since no real number appears more than once the set Ax = i : xi = x is ﬁnite ∀x ∈ R. So ∀x : Ax 6∈ U ⇒ H(1, 1 2, 1 3,...) is an entirely new number that is not equal to any of our classical real numbers. Later we will show that H(1, 1 2, 1 3,...) is a non-zero inﬁnitesimal. Similarly the sequence(1,2,3,...) does not generate any of our classical real numbers. Later we will show that H(1,2,3,...) is a positive hyperlarge.

2.5 Arithmetic operations and inequalities in ∗R Now that we have constructed the hyperreals we must be able to carry out simple operations and deﬁne when we consider when one hyperreal number to be larger than another. For example for two hyperreal numbes x = H(x1,x2,...) and y = H(y1,y2,...) how do we deﬁne x + y or x/y? When can we say that x < y? The deﬁnitions and rules we use follow very intuitively from our construction of the hyperreals. If & is an operation such as addition, subtraction, taking the absolute value, multiplication or division we introduce our version of & for the hyperreal numbers ∗& = H(&1,&2,...) by simply taking &i = & for all i. So for example H(x1,x2,...) ∗ + H(y1,y2,...) = H(x1 + y1,x2 + y2,...) for any H(xi) and H(yi) ∈ R. Since the context makes it clear whether we mean the classical version of the operation or our version for the hyperreals we will drop the ∗. This is a list of simple deﬁnitions and operations in ∗R:

(i) Deﬁnition of ∗R ∗R = H(x1,x2,...) : ∀i : xi ∈R. (ii) Equality H(x1,x2,...) = H(y1,y2,...) precisely when (x1,x2,...) ∼u (y1,y2,...). (Recall that: (x1,x2,...) ∼u (y1,y2,...) ⇔i : xi = yi∈ U)

(iii) Identiﬁcation ∀x ∈R : H(x1,x2,...) = x ⇔i : xi = x∈ U. (iv) Addition H(x1,x2,...) + H(y1,y2,...) = H(x1 + y1,x2 + y2,...), xi,yi ∈R. (v) Subtraction H(x1,x2,...)−H(y1,y2,...) = H(x1 −y1,x2 −y2,...), xi,yi ∈R. (vi) Multiplication H(x1,x2,...)×H(y1,y2,...) = H(x1 ×y1,x2 ×y2,...), xi,yi ∈R. (vii) Absolute Value |H(x1,x2,...)| = H(|x1|,|x2|,...), xi ∈R. (viii) Division When we are deﬁning division we use the same method as usual however we must be careful since just like in the case of classical mathematics our divisor cannot be zero. We can deal with this extra condition easily. For H(x1,x2,...) 6= 0 we deﬁne: 1/H(x1,x2,...) = H(r1,r2,...), xi ∈R with ri = 1/xi if x 6= 0 and ri arbitrary if xi = 0. Note that our arbitrary choice of ri when xi = 0 has no eﬀect on the value of 1/H(x1,x2,...) since we can see from part (iii) that H(x1,x2,...) 6= 0 ⇔i : xi 6= 0∈ U. (viii) Inequalities H(x1,x2,...) < H(y1,y2,...) ⇔i : xi < yi∈ U, xi,yi ∈R. (We have similar deﬁnitions for ≤,>,≥.) Theorem 2.5.1. Let x,y ∈ ∗R then exactly one of x < y, x = y and x > y is true. Proof. Let x = H(x1,x2,...) and y = H(y1,y2,...). Then A1 = i ∈N : xi < yi, A2 = i ∈ N : xi = yi and A1 = i ∈ N : xi > yi are mutually disjoint sets with A1∪A2∪A3 = N. Now by Theorem 2.3.2 exactly one of A1, A2 and A3 is in U.

2.5.1 Some examples

1. From (iii) above we have that H(1,1,...) = 1 and H(2,2,...) = 2 and so H(1,1,...) + H(2,2,...) = 1 + 2 = 3. This is consistent with the deﬁnition of addition for ∗R given in (iv): H(1,1,...) + H(2,2,...) = H(1 + 2,1 + 2,...) = H(3,3,...) = 3

2. From (ii) and (iii) above we have that H(0,2,2,...) = H(2,2,2,...) = 2 and so 1/H(0,2,2,...) = 1/2. This is consistent with our deﬁnition of division for ∗R given in (viii): 1/H(0,2,2,...) = H(r,1/2,1/2,1/2,...), for arbitrary r ∈R. Now from (ii) and (iii) we have that: H(r,1/2,1/2,1/2,...) = H(1/2,1/2,1/2,...) = 1/2.

3. From (iii) above we have that H(1,1,...) = 1 and H(2,2,...) = 2. We have that 1 < 2 and so H(1,1,...) < H(2,2,...). This is consistent with the deﬁnition of < for ∗R given in (ix): H(1,1,...) < H(2,2,...) ⇔i : 1 < 2 = N∈ U. But from the deﬁnition of a free ultraﬁlter N∈ U for all ultraﬁlters U. So H(1,1,...) < H(2,2,...).

2.6 Some Notation & Deﬁnitions

Before we go much further we will quickly introduce some notation that we will use to deal with the new elements we can now introduce. Hyperlarge Numbers Let x ∈ ∗R be a positive hyperlarge i.e. ∀n ∈N : x > n then we write x ∼∞. For x ∈ ∗R, x a negative hyperlarge i.e. ∀n ∈N : x < −n we write x ∼−∞. Inﬁnitesimals Let δ ∈ ∗R be an inﬁnitesimal i.e. ∀n ∈N : x < 1/n then we write δ ' 0. If δ is non-zero we write δ ∼ 0. Limited Numbers Let x ∈∗R be a number that is not hyperlarge. Then we call x a limited number. Appreciable Numbers Let x ∈ ∗R be a limited number that is not an inﬁnitesimal. Then we call x an appreciable number. Standard Part of a Limited Number Let x ∈∗R be a limited number, then the unique real number r that is inﬁnitesimally close to x is called the standard part of x and we write st(x) = r.

2.7 Other Ultrapower Constructions

We have shown that it is possible to use an ultraﬁlter to generate hyperreal numbers, but is there anything special about real numbers or can we generate new hyperconstants from other types of mathematical constants such as functions, sequences, sets and n-tuples in a similar manner? For example, is it possible to generate hypersets or hyperfunctions? Unsurprisingly perhaps, the answer is yes. Although our focus in this paper will be on hyperreals we will brieﬂy deal with hyperfunctions and hypersets. Our ﬁrst task is to give a generalised deﬁnition of when two hyperconstants are equal, a deﬁnition that will hold for any type of mathematical constant. These constants could be sets, functions, n-tuples or sequences for example.

2.7.1 Equality We have already used the equivalence relation ∼u to deﬁne when two hyperreals are equal. Recall that for two inﬁnite sequences, this relation is given by: (x1,x2,...) ∼u (y1,y2,...) ⇔i : xi = yi∈ U. Now, to remain consistent, and to allow us to have a generalised deﬁnition of when any two hyperconstants are equal we shall use it again, setting H(x1,x2,...) = H(y1,y2,...) precisely when (x1,x2,...) ∼u (y1,y2,...).

2.7.2 Identiﬁcation of Hypersets

We now wish to give a deﬁnition of a hyperset that will be consistent with this deﬁnition of equality. To motivate this we consider the following theorem.

Theorem 2.7.1. Let (X1,X2,...) and (Y1,Y2,...) be inﬁnite sequences of sets. Then

H(xi) : xi ∈ Xi = H(yi) : yi ∈ Yi⇔ H(X1,X2,...) = H(Y1,Y2,...). Proof. Suppose H(X1,X2,...) = H(Y1,Y2,...) and let Q = i : Xi = Yi then Q ∈ U. Let X = H(xi) : xi ∈ Xi and let Y = H(yi) : yi ∈ Yi. Now for any H(xi) ∈ X for all i we have that xi ∈ Xi and for i ∈ Q, xi ∈ Yi. Now by the deﬁnition of equality we have that H(xi) ∈ Y . So X ⊆ Y , similarly we have that Y ⊆ X ⇒ X = Y . Conversely suppose H(X1,X2,...) 6= H(Y1,Y2,...), then Q 6∈ U and so by the deﬁnition of a free ultraﬁlter Qc = i : Xi 6= Yi ∈ U. Now we have that either i : xi 6∈ Yi for some xi ∈ Xi∈ U or, i : yi 6∈ Xi for some yi ∈ Yi∈ U, or both. Suppose R = i : xi 6∈ Yi for some xi ∈ Xi ∈ U. If i ∈ R take xi ∈ Xi s.t xi 6∈ Yi, and if i 6∈ R take any xi ∈ Xi. Now i : xi 6∈ Yi⊇ R so i : xi 6∈ Yi∈ U. Suppose now that H(xi) ∈ Y , then i : xi ∈ Yi ∈ U. But since U is a free ultraﬁlter we have that it is closed under intersection and so: i : xi 6∈ Yi∩i : xi ∈ Yi = ∅∈ U. which is a contradiction. So H(xi) 6∈ Y and X 6= Y . The proof in the other case is similar.

Our deﬁnition for identiﬁcation of hypersets follows intuitively from the result of the previous theorem. We use the following deﬁnition: If all Si are sets then H(S1,S2,...) = H(si) : si ∈ Si.

2.7.3 Identiﬁcation of Hyperfunctions

Again when giving the deﬁnition for a hyperfunction we wish for it to be consistent with the deﬁnition of equality that we have given above. Although we have omitted the proof, the following deﬁnition of a hyperfunction is consistent with that deﬁnition of equality. Given an inﬁnite sequence of functions (f1,f2,...) with fi: X → Yi, we let H(f1,f2,...): H(X1,X2,...) → H(Y1,Y2,...) be the function deﬁned by H(fi)(H(xi)) = H(fi(xi)).

Our focus will be on hyperfunctions of the form ∗f = H(f,f,f,...) where f : R→R. This is known as the ∗-transform of f. We note that since ∗f : ∗R → ∗R we have that ∗f 6= f. However, ∗f is an extension of f. We should also note that since a sequence (xn) of real numbers is simply a special type of function x: N→R where x(n) = xn we use the same rules as we use for functions to generate hypersequences. Again we will be most interested in the ∗-transform

of a sequence which is a function ∗x: ∗N→∗R which we will denote ∗(xn). When referring to the mth term of the ∗-transform of a sequence (xn) we will simply write xm. Our meaning will be clear from the context but in any case of possible ambiguity a note will be included for clarity.

2.8 The ∗-transform For any classical constant w the∗-transform of w is the hyperconstant H(w,w,w,...) generated by the inﬁnite sequence (w,w,w,...). We denote this hyperconstant ∗w. Note that as shown above in the case of a function from R to R, ∗w is not necessarily equal to w. Another example is the ∗-transform of the set of natural numbers ∗N, since the hyperlarge number H(1,2,3,...) ∈ ∗N we have that ∗N6= N . However it is also true that the two can be equal. Consider for example a real number x or a ﬁnite set of real numbers A, then ∗x = x and ∗A = A.

2.9 Internal vs. External constants

Later when dealing with the transfer principle it will be important to distinguish between two types of constants in our nonstandard system. We deﬁne an internal constant to be any constant that is an element of a some set ∗X where ∗X is the ∗-transform of the classical set X. We will see that these internal constants will behave “well” i.e. in a manner very similar to classical constants. We will ﬁnd on the other hand that external constants i.e. constants which are not internal, can behave very unpredictably. Our focus will be on results for internal constants and we will be careful to avoid the problems introduced by external constants.

2.9.1 Some Important Examples

1. Every hyperreal number is internal. This is immediately clear since ∗R is the ∗-transform of R. 2. P(∗A)\\∗P(A) are the external subsets of ∗A. Clearly every element of ∗P(A) is internal since P(A) is a classical set. Now it remains to show that P(∗A) ⊇ ∗P(A). Let X ∈ ∗P(A) then X = H(Si) for Si ⊆ A, so X = H(Si) = H(si) : si ∈ Si⊆∗A, and so X ∈P(∗A) as required. This inclusion is strict if and only if the set A is inﬁnite [30] and so ∗A has external subsets if and only if A is inﬁnite. In fact if A is an inﬁnite set then A is an external subset of ∗A.

2.10 Inﬁnitesimals and Hyperlarge numbers in ∗R Theorem 2.10.1. Hyperlarge numbers and non-zero inﬁnitesimals exist in ∗R.

Proof. First we will prove the existence of non-zero inﬁnitesimals. Consider the hyperreal number H(1, 1 2, 1 3,...). Clearly 0 < H(1, 1 2, 1 3,...) since i : 0 < 1 i = N. Now let x ∈ R be positive. ∃M ∈ N : ∀n > M : 1 n < x. Since 1,2,3,...,M is

ﬁnite it is not an element of U but its complement M + 1,M + 2,... is. This implies that i : 1 i < x∈ U ⇒ H(1, 1 2, 1 3,...) < H(x,x,...) = x. So H(1, 1 2, 1 3,...) is a non-zero inﬁnitesimal. Recall we denote this H(1, 1 2, 1 3,...) ∼ 0. Similarly, H(1,2,3,...) is hyperlarge since ∀x ∈ R : ∃M ∈ N : ∀n > M : x < n ⇒i : x < i = M + 1,M + 2,...∈ U. Recall we denote this H(1,2,3,...) ∼ ∞. Theorem 2.10.2. Every inﬁnite sequence of positive real numbers converging to zero generates a positive inﬁnitesimal.

Proof. Let (xi) be a sequence of positive real numbers converging to zero. For any ε > 0, ε ∈ R there exists M ∈ N s.t. ∀n > M : 0 < xn < ε. Now since 1,2,3,...,M is ﬁnite it is not an element of U but its complement M +1,M + 2,... is. This implies that i : xi < ε∈ U ⇒ 0 < H(x1,x2,...) < H(ε,ε,...) = ε. So H(x1,x2,...) is a positive inﬁnitesimal.

Corollary 2.10.3. Every inﬁnite sequence of negative real numbers converging to zero generates a negative inﬁnitesimal.

Corollary 2.10.4. Every inﬁnite sequence of real numbers with a ﬁnite number of nonpositive terms which converges to zero generates a positive inﬁnitesimal.

Corollary 2.10.5. Every inﬁnite sequence of real numbers with a ﬁnite number of nonnegative terms which converges to zero generates a negative inﬁnitesimal.

Theorem 2.10.6. An inﬁnite sequence of real numbers generates an inﬁnitesimal for some ultraﬁlter U0 if and only if it has a inﬁnite subsequence converging to zero.

Proof. Let (xi) be a sequence of real numbers with an inﬁnite subsequence (xin) converging to zero. Now since A = i1,12,... is an inﬁnite set, there is some ultraﬁlter U0 such that A ∈ U0. (For proof that such an ultraﬁlter exists we refer to Theorem 1.16.1 of [30].) Now for any ε ∈R,ε > 0, we have |xin| < ε for all but ﬁnitely many in ∈ A, and so B = in ∈N : |xn| < ε∈ U0. This implies that H(x1,x2,...) < H(ε,ε,...) = ε, for our chosen ultraﬁlter U0 and since ε was an arbitrary positive real number we have that H(x1,x2,...) ' 0. Conversely suppose (xi) be a sequence of real numbers with no inﬁnite subsequence (xin) converging to zero. Then ∃ε ∈R,ε > 0 such that the set C = i ∈N : xi < ε is ﬁnite. Since C is ﬁnite it cannot be in any ultraﬁlter U. This implies that H(x1,x2,...) ≥ H(ε,ε,...) = ε for any ultraﬁlter U.

2.10.1 Some examples

1. Addition and hyperlarge numbers What happens when we add two positive hyperlarge numbers? Can we say that the resulting number is also hyperlarge? What about when we add a hyperlarge number to a positive classical real number? Consider ﬁrst an example of the second case: H(1,2,3,...) + 1 = H(1,2,3,...) + H(1,1,1,...) = H(2,3,4,...). Which is a hyperlarge number by the same argument as used in our theorem above. In fact it is true in general for any H(xi) ∼ ∞ and positive real number y since ∀i : xi + y > xi. Now consider an example of the ﬁrst case: H(1,2,3,...)+H(2,4,6,...) = H(3,6,9,...). Which is a hyperlarge number by the same argument as used in our theorem above. Clearly it is also true in general for any H(xi),H(yi) ∼∞ since if A,B ∈ U then A∩B ∈ U. 2. Subtraction and hyperlarge numbers What happens if we subtract one positive hyperlarge number from another? What can we say about subtracting a positive classical real number from a positive hyperlarge number? Again we will look at an example of the second case ﬁrst: H(1,2,3,...)−1 = H(1,2,3,...)−H(1,1,1,...) = H(0,1,2,...). Which is a hyperlarge number by the same argument as used in our theorem above. In fact it is true in general for any H(xi) ∼ ∞ and positive real number y since if ∀xi : x < xi∈ U then i : x + y < xi∈ U. Now consider an example of the ﬁrst case: H(1,2,3,...) − H(0,1,2,...) = H(1,1,1,...) = 1. So by subtracting one hyperlarge number from another we get a ﬁnite real number, but we can’t say this is true in general since: H(2,4,6,...)−H(1,2,3,...) = H(1,2,3,...) which is again a hyperlarge number and H(2,5/2,10/3,...)−H(1,2,3,...) = H(1,1/2,1/3,...) which is an inﬁnitesimal.

3. Addition and Subtraction of non-zero inﬁnitesimals What happens when we add two positive non-zero inﬁnitesimals? What happens when we subtract one non-zero inﬁnitesimal from another? First, consider an example of the ﬁrst case: H(1,1/2,1/3,...)+H(2,1/4,1/6,...) = H(3,3/4,3/6,...) which is again an inﬁnitesimal. This is again true in general. Now consider an example of the second case: H(2,1/4,1/6,...)−H(1,1/2,1/3,...) = H(1,1/2,1/3,...) which is again an inﬁnitesimal. This is again true in general since if xi,yi > 0 then |xi −yi| < maxxi,yi. 4. The ∗-transform of a function evaluated at an inﬁnitesimal What happens when we evaluate the ∗-transform of a function at a non-zero inﬁnitesimal?

Let’s take for example the ∗-transform of sin(x), ∗sin(x), evaluated at an inﬁnitesimal. Since sin(x) is continuous and is positive on (0,π) with sin(0) = 0 we would expect ∗sin(δ) where δ is a positive inﬁnitesimal to also be a positive inﬁnitesimal. Let δ = H(1,1/2,1/4,...) then ∗sin(δ) = H(sin(1),sin(1/2),...). Now ∀x ∈ (0,1) : sin(x) > 0, so (sin(1),sin(1/2),...) is a sequence of positive real numbers. We also have that sin(x) is continuous with sin(0) = 0 and so (sin(1),sin(1/2),...) converges to zero. Now by Theorem 1.8.2 above we have that ∗sin(δ) is a positive inﬁnitesimal. In general we can apply Theorem 2.10.6 to prove that ∗sin(δ) ' 0 for any δ ' 0. We have only to notice that if any inﬁnite subsequence (xin) converges to zero so too does the sequence (sin[xin]).

3 The Transfer Principle

3.1 History and Importance

This section brings us ﬁnally to the transfer principle. The transfer principle is the powerful “black box” that allows us to use the methods of non-standard analysis to prove results in standard analysis. Essentially it provides a ‘bridge’ between nonstandard analysis and classical mathematics. We will interpret the term classical to mean ‘not involving any nonstandard mathematical ideas’. It is therefore, in my opinion, the most important result pertaining to nonstandard analysis. It gives us one of our greatest motivations to study the area. It is as a result of the transfer principle that non-standard methods are powerful tools, tools that we can use to help our understanding of other areas of mathematics. Being able to transfer reasoning from a system of numbers that included inﬁnitely large and small numbers to a system which does not such as the real numbers was naturally of great interest to the founders of calculus. Since Leibniz and Newton both used such inﬁnitely small and large numbers when developing calculus the validity of these results depended on such a principle. The idea was described by Leibniz and given the name the “Law of Continuity”. In a 1702 letter to the French Mathematician Pierre Varignon, Leibniz formulated the Law of Continuity as follows: “...et il se trouve que les r`egles du ﬁni r´eussissent dans l’inﬁni comme sil y avait des atomes (c’est `a dire des ´el´ements assignables de la nature) quoiquil ny en ait point la mati`ere ´etant actuellement sousdivis´ee sans ﬁn; et que vice versa les r´egles de l’inﬁni r´eussissent dans le ﬁni, comme s’il y’avait des inﬁniment petits m´etaphysiques, quoiqu’on n’en n’ait point besoin; et que la division de la mati`ere ne parvienne jamais `a des parcelles inﬁniment petites: c’est parce que tout se gouverne par raison, et qu’autrement il n’aurait point de science ni r`egle, ce qui ne serait point conforme avec la nature du souverain principe” [24]. Many academics including Robinson identify this passage as a formulation of the law of continuity, which can be summarized as follows: “the rules of the ﬁnite succeed in the inﬁnite, and conversely” [21]. This principle was a forerunner to the transfer principle that we will discuss in this section. It is a consequence of

a theorem proved by the Polish mathematician Jerzy L o´s in 1955 [29]. Before we tackle L o´s’ Theorem we will ﬁrst give a quick outline of some ideas in mathematical logic.

3.2 Mathematical Logic

3.2.1 Atomic Statements

Atomic relations are simple mathematical relations that don’t contain either logical connectives or quantiﬁers such as =,<,> and ∈. A relation on n arguments is called n-ary. An n-ary relation can be thought of as a function R : X1 ×X2 × ...×Xn → B. Where B is the set of Boolean constants B = TRUE,FALSE. For example (−1 ∈ N) ≡ FALSE and (1 < 2) ≡ TRUE. Since = is one of our atomic relations for clarity we will always use ≡ to denote equivalence. For example (0 = 1) ≡ FALSE and (1 = 1) ≡ TRUE. An atomic statement is a statement given by applying atomic relations to suitable arguments. We deﬁne ∗TRUE ≡ TRUE and ∗FALSE ≡ FALSE. Note that since B is ﬁnite ∗B ≡∗TRUE,∗FALSE≡TRUE,FALSE≡ B. Now since we have a correspondence between these relations R and functions it follows intuitively that our corresponding relation ∗R in non-standard analysis is given by: H(xi)∗RH(yi) ≡ H(xiRyi) where by deﬁnition H(xiRyi) ≡ (i : xiRyi∈ U), so H(xi)∗RH(yi) ≡ TRUE ⇔i : xiRyi ≡ TRUE∈ U. Lemma 3.2.1. (A Special Case of L o´s’ Theorem) Let R be a binary relation R : X×Y → B. Then ∗[xRy] ≡ xRy for x ∈ X, y ∈ Y . Proof. ∗[xRy] ≡ ∗x∗R∗y ≡ (i : xRy ∈ U) Note that xRy does not depend on i so either i : xRy = N ∈ U if xRy ≡ TRUE or, i : xRy = ∅ / ∈ U if xRy ≡ FALSE. So we have that ∗[xRy] ≡∗x∗R∗y ≡ xRy as required. This is an example of a transfer principle for simple atomic statements. Although not very powerful it is an interesting result that lets us know that if the atomic statement ∗x∗R∗y is equivalent to the classical statement xRy. For example the statement (∗f(∗x∗y) < ∗x∗y∗z) ≡ TRUE ⇔ (f(xy) < xyz) ≡ TRUE.

3.2.2 Arbitrary Statements

Building on the notion of atomic statements, an arbitrary statement is one which is made up of a ﬁnite number of atomic relations, logical connectives, quantiﬁers, constants, free variables and bound variables. using these we can construct more complex mathematic statements.

Logical connectives Given two basic statements P and Q we can combine them with logical connectives to construct a more complex statement. The basic logical connectives are:

1. Negation (“not”), denoted q. Not P has the opposite Boolean value to P. 2. Conjunction (“and”), denoted ∧. P and Q is true only when both P and Q are both true. 3. Disjunction (“or”), denoted ∨. P or Q is true only when either one or both of P and Q are true. 4. Conditional (“if-then” or “implication”), denoted ⇒. P implies Q is true unless P is true and Q is false. 5. Biconditional (“if and only if” or “double implication”), denoted ⇔. P ⇔ Q is true when P and Q are both true or both false but is false otherwise. Quantiﬁers Quantiﬁers are used in statements containing variables. There are two quantiﬁers: 1. The universal quantiﬁer (“for all”), denoted ∀. 2. The existential quantiﬁer (“there exists”), denoted ∃. Constants, free variables and bound variables Apart from relations, logical connectives and quantiﬁers, each statement also contains a number of variables and constants.

1. Constants Speciﬁed or unspeciﬁed ﬁxed numbers, n-tuples, sets and functions.

2. Free variables If replacing any variable occurring in a statement by some constant leads to another meaningful statement that variable is called a free variable with respect to the statement.

3. Bounded variables A variable that is not free is called a bounded or dummy variable.

Notation and conventions for arbitrary statements While it was useful to write an atomic statement derived from the binary relation R as xRy above we now write this statement as R(x,y) and a general atomic statement as (P(x,x0,...) where P is an atomic relation and x,x0,... is an expression of constants and free variables. We assume that arbitrary statements are in their prenex normal form2 are free variables with all logical connectives to the right of the quantiﬁers. Every statement of ﬁrst-order logic can be converted to an equivalent statement in prenex normal form [34]. It is also assumed that each bound variable occurs to the left of the ∈ relation, helping us to ensure that each bound variable is internal. Now instead of regarding a statement R as a function of substatements P(s,s0,...), Q(t,t0,...),S(u,u0,...),..., and of sets X,X0,..., required in the quantiﬁcations we

2For example the following statement where P(a,b,...,q,r,...,xyz) is a statement containing no quantiﬁers, a,b,... are constants and q,r,... are free variables is in its prenex normal form: ∀x : ∃y : ∃z : P(a,b,...,q,r,...,xyz).

can regard it simply as a function of the constants and free variables X,X0,X00,...; s,s0,s00,..., . We will write a general arbitrary statement with a ﬁnite number of constants or free variables X,X0,X00,...;s,s0,s00,..., and a ﬁnite number of logical connectives and quantiﬁers as, R(X,X0,X00,...;s,s0,s00,...),. Here X,X0,X00,... are the sets required to formulate the quantiﬁcations properly; that is to say that X must occur in ∃x ∈ X or in ∀x ∈ X, for some suitable bound variable x, and similarly for X0,X00,... and that conversely each quantiﬁcation is taken care of this way.

3.3 L o´s’ Theorem

In this section we will present L o´s’ Theorem which is also sometimes known as The Fundamental Theorem of Ultraproducts. This name reﬂects its signiﬁcance in our study of non-standard analysis. In essence the Theorem tells us that a ﬁrst-order statement is true in the ultraproduct if and only if the set of indices for which the formula is true is an element of our ultraﬁlter U. A proof consistent with our approach can be found in [30]. We give its formal statement below.

Theorem 3.3.1. (L o´s’ Theorem) Let any classical statement, R(X,X0,X00,...;s,s0,s00,...), with a ﬁnite number of constants or free variables X,X0,X00,...;s,s0,s00,..., and a ﬁnite number of logical connectives and quantiﬁers be given. X,X0,X00,... are the sets required to formulate the quantiﬁcations properly; that is to say that X must occur in ∃x ∈ X or in ∀x ∈ X, for some suitable bound variable x, and similarly for X0,X00,... and that conversely each quantiﬁcation is taken care of this way. Then, H[R(Xi,X0 i,X00 i ,...;si,s0 i,s00 i ,...)] ≡ R(H(Xi),H(X0 i),H(X00 i ),...;H(si),H(s0 i),H(s00 i ),...)

3.4 The Transfer Principle

The transfer principle comes as a direct consequence of L o´s’ Theorem.

Theorem 3.4.1. (Transfer Principle) Let any classical statement, R(X,X0,X00,...;s,s0,s00,...), with a ﬁnite number of constants or free variables X,X0,X00,...;s,s0,s00,..., and a ﬁnite number of logical connectives and quantiﬁers be given. X,X0,X00,... are the sets required to formulate the quantiﬁcations properly; that is to say that X must occur in ∃x ∈ X or in ∀x ∈ X, for some suitable bound variable x, and similarly for X0,X00,... and that conversely each quantiﬁcation is taken care of this way. Then, R(X,X0,X00,...;s,s0,s00,...) ≡ R(∗X,∗X0,∗X00,...;∗s,∗s0,∗s00,...)

Proof. Taking Xi = X and si = s for every i and similarly for X0 i,s0 i,... etc. and applying L o´s’ Theorem we get that ∗[R(X,X0,X00,...;s,s0,s00,...)] ≡ H[R(X,X0,X00,...;s,s0,s00,...)] ≡ R(H(X),H(X0),H(X00),...;H(s),H(s0),H(s00),...) ≡ R(∗X,∗X0,∗X00,...;∗s,∗s0,∗s00,...).

But, we also have that ∗[R(X,X0,X00,...;s,s0,s00,...)] ≡ R(X,X0,X00,...;s,s0,s00,...). And so R(X,X0,X00,...;s,s0,s00,...) ≡ R(∗X,∗X0,∗X00,...;∗s,∗s0,∗s00,...).

The transfer principle in this formulation tells us that any classical statement is equivalent to the non-standard statement we get by replacing everything by its ∗-transform except the bound variables in the statement. This is so important, we do not think of real numbers as inﬁnite Cauchy sequences and now we no longer need to think of hyperreal numbers as inﬁnite sequences of real numbers. Instead we can treat them in a similar way as we treat the real numbers. It is this transfer principle, that acts like a “bridge” between analysis in R and analysis in ∗R, that makes our study of nonstandard analysis so useful. Consider the following examples.

Theorem 3.4.2. (The Archimedean Law) Let x be a real number. Then there exists a natural number n that is greater than x.

We can write this statement using the tools of mathematical logic as follows: ∀x ∈R : ∃n ∈N : n > x. Now applying the transfer principle given in Theorem 1.4.1 above we get that this is equivalent to: ∀x ∈∗R : ∃n ∈∗N : n > x. So for any hyperreal number x there exists a hypernatural number n that is greater than x. Obviously this is not true if we replace ∗N with N or the word hypernatural with the word natural in the statement above.

Theorem 3.4.3. Let n be a natural number that is greater than 1. Then n has at least one prime factor. Let P = P ∈N : p is prime. We can now write this statement using the tools of mathematical logic as follows: ∀n ∈N : n > 1 : ∃p ∈P : n/p ∈N.

Now applying the transfer principle given in Theorem 1.4.1 above we get that this is equivalent to: ∀n ∈∗N : n > 1 : ∃p ∈∗P : n/p ∈∗N. So every hypernatural number greater than 1 has at least one hyperprime factor. We will use this result in section 3.6 to give an elegant proof that P is an inﬁnite set.

Theorem 3.4.4. Let p and q be real numbers and let q be greater than p. Then there is a real number r that is greater than p but less than q. (i.e. The real numbers are dense.)

We can write this statement using the tools of mathematical logic as follows: ∀p,q ∈R : p < q : ∃r ∈R : p < r < q. Now applying the transfer principle given in Theorem 1.4.1 above we get that this is equivalent to: ∀p,q ∈∗R : p < q : ∃r ∈∗R : p < r < q. In other words the hyperreal numbers are also dense.

Theorem 3.4.5. (Dedekind Completeness of the Real Numbers) Let X be a non-empty subset of R that has an upper bound b ∈ R, then X has a least upper bound β ∈R. We can write this statement using the tools of mathematical logic as follows: ∀X ∈P(R) : x 6= ∅∧[∃b ∈R : ∀x ∈ X : x ≤ b]⇒ ∃β ∈R : [∀x ∈ X : x ≤ β]∧[∀ε ∈R,ε > 0 : ∃x ∈ X : x > β −ε], Now applying the transfer principle given in Theorem 1.4.1 above we get that this is equivalent to: ∀X ∈∗P(R) : x 6= ∅∧[∃b ∈∗R : ∀x ∈ X : x ≤ b]⇒ ∃β ∈∗R : [∀x ∈ X : x ≤ β]∧[∀ε ∈∗R,ε > 0 : ∃x ∈ X : x > β −ε]. In other words if X is an internal subset of ∗R that is bounded above by some hypperreal number b, which could be hyperlarge or indeed an inﬁnitesimal, then there is a hyperreal number β that is a least upper bound for X. (Again this could be hyperlarge or indeed an inﬁnitesimal). Note that it is important that X is an internal set, for example the statement above is not true for the set of real numbers R since R is external. Suppose β was a least upper bound for R in ∗R, then the β is a hyperlarge number but β −1 is also hyperlarge and so is also an upper bound for R which is a contradiction since β was our least upper bound for R. So ∗R is not Dedekind complete3. 3This it turns out is a major relief since every Dedekind complete ordered ﬁeld is isomorphic to R.

3.5 Deﬁnitions

Nonstandard analysis can be used to give simpliﬁed, elegant deﬁnitions of many concepts in classical mathematics. It is especially useful to give intuitive alternative deﬁnitions of things that are deﬁned using ε’s and δ’s in classical mathematics. Such deﬁnitions are often found to be very diﬃcult and unintuitive for many undergraduate mathematicians. I have tried to include some nonstandard deﬁnitions that are not easily found in the literature. Since Weierstrass developed the concept of a limit to eliminate the need to use inﬁnitesimals in calculus before a valid model of nonstandard analysis was developed, it makes sense to ﬁrst give a nonstandard version of the ε−δ deﬁnition of a limit. As an added bonus the deﬁnition we will give is a very intuitive deﬁnition of a concept so many undergraduates ﬁnd diﬃcult to grasp when starting their studies.

Deﬁnition (Nonstandard Deﬁnition of a Limit) Let f : R→R and let a,l ∈R then we say that “the limit of f as x tends to a is l” and write lim x→a f(x) = l if and only if ∀δ ∈∗R,δ ∼ 0 : ∗f(a + δ)−l ' 0. In other words if and only if ∀δ ∼ 0 : l = st(∗f(a + δ)). This can be read as “The limit of the function f at a is l if and only if the value of the function when we are inﬁnitesimally close to a is inﬁnitesimally close to l.”, which is the intuitive way that many people think of a limit. Analogously we can give the following deﬁnition of one-sided limits.

Deﬁnition (Nonstandard Deﬁnition of One-sided Limits) Let f : R→R and let a,l+,l− ∈R. Then lim x→a+ f(x) = l+ if and only if ∀δ,δ > 0,δ ∼ 0 : ∗f(a + δ)−l+ ' 0, and lim x→a− f(x) = l− if and only if ∀δ,δ > 0,δ ∼ 0 : ∗f(a−δ)−l− ' 0. Theorem 3.5.1. The nonstandard deﬁnition of a limit given above is equivalent to the classic “ε−δ” deﬁnition of a limit: ∀ε ∈R,ε > 0 : ∃δ ∈R,δ > 0 : ∀x ∈R,0 < |x−a| < δ : |f(x)−l| < ε. Proof. By the transfer principle the statement above is equivalent to ∀ε ∈∗R,ε > 0 : ∃δ ∈∗R,δ > 0 : ∀x ∈∗R,0 < |x−a| < δ : |∗f(x)−l| < ε, and this can be simpliﬁed to ∀δ ∈∗R,δ ∼ 0 : ∗f(a + δ)−l ' 0.

From this nonstandard deﬁnition of a limit the following two intuitive nonstandard deﬁnitions of continuity and diﬀerentiability quickly follow.

Deﬁnition (Nonstandard Deﬁnition of Continuity) Let f : R→R and let a ∈R. Then f is continuous at a if and only if, ∀δ ∼ 0 : ∗f(a + δ)−f(a) ' 0. “A function f is continuous at a if and only if the value of the function inﬁnitesimally close to a is inﬁnitesimally close to f(a).”

Theorem 3.5.2. The nonstandard deﬁnition of continuity given above is equivalent to the classic deﬁnition of continuity: f : R→R is continuous at a ∈R if and only if lim x→a f(x) = f(a). Proof. By our nonstandard deﬁnition of a limit

lim x→a

f(x) = f(a) ⇔∀δ ∼ 0 : ∗f(a + δ)−f(a) ' 0.

Deﬁnition (Nonstandard Deﬁnition of Diﬀerentiability) Let f : R→R and let a,d ∈R. Then f is diﬀerentiable at a if and only if, ∀δ ∼ 0 : ∗f(a + δ)−f(a) δ ' d = f0(a). And so f0(a) = st[∗f(a+δ)−f(a) δ ].

“The derivative of the function f at a is the slope of the line between f(a) and f evaluated at a point inﬁnitesimally close to a.”

The fact that this is equivalent to the classic deﬁnition of diﬀerentiability again follows directly from our nonstandard deﬁnition of a limit.

Deﬁnition (Nonstandard Deﬁnition of a Convergent Sequence) Let (xn) be an inﬁnite sequence of real numbers then the sequence converges to l ∈R (xn → l) if and only if ∀N ∈∗N,N ∼∞ : xN −l ' 0. In other words st(xN) = l. (Here xN is the Nth element of ∗(xn).)

“The sequence (xn) converges to l if and only if for inﬁnitely large values of n, xn is inﬁnitesimally close to l.”

Deﬁnition (Nonstandard Deﬁnition of a Cauchy Sequence) Let (xn) be an inﬁnite sequence of real numbers then the sequence is a Cauchy sequence if and only if ∀N,M ∈∗N,N,M ∼∞ : xN −xM ' 0. (Here xN is the Nth and xM is the Mth element of ∗(xn).)

“The sequence (xn) is Cauchy if and only if for inﬁnitely large values of n the terms of the sequence are inﬁnitesimally close.”

Theorem 3.5.3. The nonstandard deﬁnition of a Cauchy sequence given above is equivalent to the standard deﬁnition of a Cauchy sequence: ∀n ∈N : ∃k ∈N : ∀N,M ∈N,N,M > k : |xN −xM| < 1/n. (∗) Proof. Fixing n ∈N and k ∈N the statement ∀N,M ∈N,N,M > k : |xN −xM| < 1/n by the transfer principle is equivalent to the statement ∀N,M ∈∗N,N,M > k : |xN −xM| < 1/n Now letting N,M ∼∞, then ∀k ∈N : N,M > k so the statement ∀n ∈N : ∃k ∈N : ∀N,M ∈∗N,N,M > k : |xN −xM| < 1/n can be simpliﬁed to the statement ∀N,M ∈∗N,N,M ∼∞ : ∀n ∈N : |xN −xM| < 1/n which is equivalent to ∀N,M ∈∗N,N,M ∼∞ : xN −xM ' 0. (∗∗) Conversely if we consider the negation of (∗) ∃n ∈N : ∀k ∈N : ∃N,M ∈N,N,M > k : |xN −xM|≥ 1/n Fixing n ∈N the statement ∀k ∈N : ∃N,M ∈N,N,M > k : |xN −xM|≥ 1/n by the transfer principle is equivalent to the statement ∀k ∈∗N : ∃N,M ∈∗N,N,M > k : |xN −xM|≥ 1/n now ﬁxing k ∼∞ we have that N,M ∼∞ and so the negation of (∗) implies that ∃N,M ∈∗N,N,M ∼∞ : ∃m ∈N : |xN −xM|≥ 1/n which is equivalent to the negation of (∗∗); ∃N,M ∈∗N,N,M ∼∞ :q[xN −xM ' 0].

Deﬁnition (Nonstandard Deﬁnition of Uniform Convergence) Let (fn) be a sequence of functions with fn : R → R. Then (fn) converges uniformly to the function f : R→R on R if and only if ∀x ∈∗R : ∀N ∈∗N,N ∼∞ : ∗f(x)−∗fN(x) ' 0.

“The sequence of functions (fn) converges uniformly to f on R if and only if for an inﬁnitely large N, ∗fN is inﬁnitesimally close to ∗f at all points of ∗R”

Theorem 3.5.4. The nonstandard deﬁnition of uniform convergence given above is equivalent to the classical deﬁnition given by: ∀ε ∈R,ε > 0 : ∃N ∈N : ∀n ∈N,n > N : ∀x ∈R : |fn(x)−f(x)| < ε. Proof. Suppose (fn) converges to the function f uniformly on R then by the transfer principle the statement ∀ε ∈R,ε > 0 : ∃N ∈N : ∀n ∈N,n > N : ∀x ∈R : |fn(x)−f(x)| < ε is equivalent to ∀ε ∈∗R,ε > 0 : ∃N ∈∗N : ∀n ∈∗N,n > N : ∀x ∈∗R : |∗fn(x)−∗f(x)| < ε Now letting N be hyperlarge we must have that ε is inﬁnitesimal and hence ∀n ∈∗N,n ∼∞ : ∀x ∈∗R : ∗f(x)−∗fn(x) ' 0. Conversely suppose that ∀n ∈∗N,n ∼∞

以上是关于哥德尔预言无穷小微积分是未来的数学分析的主要内容，如果未能解决你的问题，请参考以下文章