The purpose of this text is to give a quick presentation of Non-Standard Analysis. Non-Standard Analysis is a mathematical theory invented in the 60's by Robinson, from the Model Theory. It was then developped by Nelson and others. This new analysis (by contrast to the classical one) allows to compute with actual infinite numbers and infinitesimals, the way Physicists have always been doing.

The notion of infinitesimals is very old : it appeared in Europe at the end of the 17th century, with mathematicians such as Leibniz, Newton, De l'Hospital, etc. This notion lead to the concept of limits and differenciation, which are the basis of the whole Analysis. But the idea of infinitesimal numbers was rejected by mathematicians, because they could not give a proper definition to them. Rigor found its way through the formal definition of limit given by Cauchy, but it was at the expense of intuition and direct understanding. Indeed, reasonnings such as: « For all epsilon > 0, there is n great enough such as... » are rigorous but very complicated to manipulate !

However, Mathematics are full of infinitely great and infinitely small numbers. Indeed, the intuitive definition of an inifinitely great number is 'a number greater than every common (i.e. finite) number. In order to compute with infinitely great numbers, we just need to find an algebra with an appropriate relation of order, so that there are some elements which are greater than every element of a sub-set that can be identified with the natural integer set N. For instance, in the algebra of rationnal fractions, all constant polynoms (equivalent to real numbers) are less than the X polynom. X is therefore an infinite number, and X^{2 as well}, etc, whereas 1/X, 1/X² are infinitesimals. Fantastic ! It means we can eventually compute with infinite numbers and infinitesimals ! Yes, but these calculus are limited to ordinary operations such as +, -, multiplicate and division. What is, for example, the image of an infinite number by a function that cannot be decomposed into elementary operations ? We would like to say that it is the limit of the function in the infinity. And what if the function has no limit ?

We see that the problem is not really the existence of infinitesimals and inifitely great numbers, but rather the difficulty of defining their properties. We would like to transfer naturally the properties of common numbers to infinite numbers and infinitesimals. This is just what Non-Standard Analysis managed to do, thus giving us an efficient means of calculating with infinitesimals

Thus we have two kinds of objects : on the one hand, there are internal (to classical mathematics) objects, the definition of which does not require the word 'standard', and which can be either standard, either non-standard ; and on the other hand, there are external objects.

We now have to specify what the word 'standard' means, otherwise it would not be very useful. The IST (Internal Set Theory) has three axioms in order to tell what 'standard' means ; or more precisely, what are the rules of the use of the word 'standard'. These three axioms are to be added to those of the ZFC theory (classical formal Set Theory on which today's mathematics are based). These three axioms are : Idealisation, Transfer, and Standardisation. Without giving too much details, we can say that these three axioms lead to three principles which are the foundation of Non-Standard Analysis.

The fisrt principle says that there is a kind of transmission of 'standardness' among internal objects. (Thus, it is possible to demonstrate that Ø is standard, and that all objects build from Ø (1, 2, 3, ..., N, R, ...) are standard as well.)
The second principle tells us where to find standard and non-standard objects : the sets which have only standard elements are the finite sets. It also means that infinite sets are a kind of «containers» for non-standard elements.
The transfer principle lets us manipulate non-standard objects the way we do with standard objects.

Non-standard integers

According to the second principle, the set N of natural integers does have non-standard elements, because it is infinite.
Let n be an standard integer of N. The set I_n of all integers q where q < n is an internal set. This set is finite, so according to the second principle, all its elements are standard, so that if we take a non-standard integer, it is necessarily greater than n (otherwise it would belong to I_n and it would be standard). As n is any integer, we deduce that a non-standard integer is greater than any standard integer.

The non-standard integers are therefore called unlimited integers, and the standard integers are called limited integers. Nevertheless, whatever limited or unlimited they are, integers are all finite.

Indeed, integers are (from the point of view of formal Set Theory) equivalent to finite sets (called finite ordinals). Thus, 0 = Ø, 1 = {Ø}, 2 = {Ø,{Ø}}, 3 = {Ø,{Ø},{Ø,{Ø}}}, ... and so on, because all these sets are ordinals, and obviously finite. Yet, nothing tells us that there are no finite ordinals which are not build from Ø. The ZFC formal Set Theory tells us that 1, 2, 3 ...are integers, but does not say wether there are others. The IST theory was created to give an answer to this very question, and it states that there are other finite ordinals than 1, 2, 3, 4, ... . In fact, 1, 2, 3, ... are the standard finite ordinals (as they are defined from Ø which is standard) and the other ones (which exist because of the second principle) are the non-standard finite ordinals. These non-standard finite ordinals do correspond to idea of “infinitely great numbers”, (that's why we call them 'unlimited'); still, they are finite ordinals, according to the mathematical definition of finity (note : a set is finite if, and only if, there is no possible bijection between one of its sub-sets and itself).

This is a good example : it shows that we must be very wary of the meaning that we associate with mathematical objects, which are only meaningless formal structures. The traditionnal mathematical definition of an infinite set is very conventionnal : a set is infinite if, and only if, there is a bijection between itself and one of its parts. It is not very intuitive, but we choose this definition because it is the one that generates properties of the infinite sets which are the closest to what we think infinite sets are. Yet, it is only a convention. In order to define an infinitely great number (which also makes part of our intuitive conception of infinity, though), we prefered to use the formal concept of non-standardness. So, a general and intuitive idea such as infinity may actually lead to distinct formalisms, depending on the chosen point of view. It is a bit like in Physics, where we use different (even contradictory !) models in order to describe the same phenomenon. Alhough 'unlimited' and 'infinite' are totally different notions, it is worth noticing that they are linked thanks to the second principle : 'unlimited' numbers wouldn't exist without 'infinite' sets.

Non-standard real numbers

The foundation of Non-Standard Analysis are somewhat subtle, but it then enables us to write very natural and intuitive definitions, such as :

Unlike integers, the limited real numbers are not the standard real numbers. Nevertheless, it is possible to demonstrate that every limited real is infinitely close to a unique standard real number, which is called its standard part. From this, it is possible to define S-properties that are simlilar to those used in Classical Analysis. For example, a function is S-continuous if the images of two infinitely close real numbers are infinitely close to one another. For the standard functions, S-continuity and (classical) continuity are the same. But, for instance, the non-standard function which has the value 1 for all infinitesimal real numbers, and the value 0 for all non-infinitesimal real numbers, is S-continuous (it's easy to check it). Yet, its standard part (0 everywhere, except in 0 where it takes the value 1) is a discontinuous function !