In 1960 Abraham Robinson (1918–1974) solved the three hundred year old problem of giving a rigorous development of the calculus based on infinitesimals. Robinson’s achievement was one of
the major mathematical advances of the twentieth century. This is an exposition of Robinson’s infinitesimal calculus at the advanced undergraduate level. It is entirely self-contained but is
keyed to the 2000 digital edition of my first year college text Elementary Calculus: An Infinitesimal Approach [Keisler 2000] and the second
printed edition [Keisler 1986]. Elementary Calculus: An Infinitesimal Approach is available free online at www.math.wisc.edu/∼Keisler/calc. This monograph can be used as a quick introduction to the subject for mathematicians, as background material for instructors
using the book Elementary Calculus, or as a text for an undergraduate seminar.
This is a major revision of the first edition of Foundations of Infinitesimal Calculus [Keisler 1976], which was published as a companion
to the first (1976) edition of Elementary Calculus, and has been out of print for over twent yyears.
A companion to the second (1986) edition of Elementary Calculus was never written. The biggest changes are: (1) A new chapter on differential
equations, keyed to the corresponding new chapter in Elementary Calculus. (2) The axioms for the hyperreal number system are changed to match those in the later editions of Elementary Calculus. (3) An account of the discovery of Kanovei and Shelah [KS 2004]
that the hyperreal number system, like the real number system, can be built as an explicitly definable mathematical structure. Earlier constructions of the hyperreal number system depended on an arbitrarily chosen parameter such as an ultrafilter.
The basic concepts of the calculus were originally developed in the seventeenth and eighteenth centuries using the intuitive notion of an
infinitesimal, culminating in the work of Gottfried Leibniz(1646-1716) and Isaac Newton (1643-1727). When the calculus was put on a rigorous basis in the nineteenth century, infinitesimals were rejected in favor of the ε, δ approach, because mathematicians
had not yet discovered a correct treatment of infinitesimals. Since then generations of students have been taught that infinitesimals do not exist and should be avoided.
Theactual situation, as suggested by Leibniz and carried out byRobinson, is that one can form the hyperreal number system byadding infinitesimals
to the real number system, and obtain apowerful new tool in analysis. The reason Robinson’s discovery didnot come sooner is that the axioms needed to describe the hyperrealnumbers are of a kind which were unfamiliar to mathematicians untilthe mid-twentieth
century. Robinson used methods from the branchof
mathematical logic called model theory which developed inthe 1950’s.
Robinson called his method nonstandard analysis because ituses a nonstandard model of analysis. The older name infinitesimalanalysis is perhaps
more appropriate. The method is surprisinglyadaptable and has been applied to many areas of pure and appliedmathematics. It is also used in such fields as economics andphysics as a source of mathematical models. (See, for example, thebooks [AFHL 1986] and
[ACH 1997]). However, the method is stillseen as
controversial, and is unfamiliar to mostmathematicians.
Thepurpose of this monograph, and of the book Elementary Calculus, isto make infinitesimals more readily available to mathematicians andstudents.
Infinitesimals provided the intuition for the originaldevelopment of the calculus and should help students as they repeatthis development. The book Elementary Calculus treats infinitesimalcalculus at the simplest possible level, and gives plausibilityarguments
instead of proofs of theorems whenever it is appropriate.This monograph presents the subject from a more advanced viewpointand includes proofs of almost all of the theorems stated inElementary Calculus. Chapters 1–14 in this monograph match thechapters in
Elementary Calculus, and after each section heading thecorresponding sections of Elementary
Calculus are indicated inparentheses.
InChapter 1 the hyperreal numbers are first introduced with a set ofaxioms
andtheir algebraic structure is studied. Then in Section 1G thehyperreal numbers are built from the real numbers. This is anoptional section
which is more advanced than the rest of thechapter and is not used later. It is included for the reader whowants to see where the hyperreal numbers comefrom.
Chapters 2 through 14 contain a rigorous development ofinfinitesimal calculus based on the axioms in Chapter 1. The onlyprerequisites are the
traditional three semesters of calculus and acertain amount of mathematical maturity. In particular, thematerial is presented without using notions from mathematicallogic. We will use some elementary set-theoretic notation familiarto all mathematicians, for
example the function concept and thesymbols
∅, A ∪B, {x ∈ A : P (x)}.
Frequently, standard results are given alternate proofsusing infinitesimals. In some cases a standard result which isbeyond the scope of beginning
calculus is rephrased as a simplerinfinitesimal result and used effectively in Elementary Calculus;some examples are the Infinite Sum Theorem, and the two-variablecriterion for a global maximum.
Thelast chapter of this monograph, Chapter 15, is a bridge between thesimple treatment of infinitesimal calculus given here and the moreadvanced
subject of infinitesimal analysis found in the researchliterature. To go beyond infinitesimal calculus one should at leastbe familiar with some basic notions from logic and model theory.Chapter 15 introduces the concept of a nonstandard universe,explains the
use of mathematical logic, superstructures, andinternal and external sets, uses ultrapowers to build anonstandard
universe, and presents uniqueness theorems for thehyperreal number systems and nonstandarduniverses.
Thesimple set of axioms for the hyperreal number system given here(and in Elementary Calculus) make it possible to presentinfinitesimal calculus
at the college freshman level, avoidingconcepts from mathematical logic. It is shown in Chapter 15 thatthese axioms are equivalent to Robinson’s approach. For additionalbackground in logic and model theory, the reader canconsult
thebook [CK 1990]. Section 4.4 of that book gives further results onnonstandard universes. Additional background in infinitesimalanalysis can
be found in the book [Goldblatt1991].
Ithank my late colleague Jon Barwise, and Keith Stroyan of theUniversity of Iowa, for valuable advice in preparing the FirstEdition of this
monograph. In the thirty years between the firstand the present edition, I have benefited from equally valuable andmuch appreciated advice from frie rased as a simpler infinitesimalresult and used effectively in Elementary Calculus; some examplesare the Infinite
Sum Theorem, and the two-variable criterion for aglobal maximum.
Thelast chapter of this monograph, Chapter 15, is a bridge between thesimple treatment of infinitesimal calculus given here and the moreadvanced
subject of infinitesimal analysis found in the researchliterature. To go beyond infinitesimal calculus one should at leastbe familiar with some basic notions from logic and model theory.Chapter 15 introduces the concept of a nonstandard universe,explains the
use of mathematical logic, superstructures, andinternal and external sets, uses ultrapowers to build anonstandard
universe, and presents uniqueness theorems for thehyperreal number systems and nonstandarduniverses.
Thesimple set of axioms for the hyperreal number system given here(and in Elementary Calculus) make it possible to presentinfinitesimal calculus
at the college freshman level, avoidingconcepts from mathematical logic. It is shown in Chapter 15 thatthese axioms are equivalent to Robinson’sapproach.
Foradditional background in logic and model theory, the reader canconsult the book [CK 1990]. Section 4.4 of that book gives furtherresults
on nonstan-
darduniverses. Additional background in infinitesimal analysis can befound in the book [Goldblatt 1991].
Ithank my late colleague Jon Barwise, and Keith Stroyan of theUniversity of Iowa, for valuable advice in preparing the FirstEdition of this
monograph.
In thethirty years between the first and the present edition, I havebenefited from equally valuable and much appreciated advice fromfriends
and colleagues too numerous to recounthere.