# Books by H Jerome Keisler

We present extracts from some reviews and Prefaces of books authored by H Jerome Keisler. We present the books in chronological order, beginning with the earliest, except for different editions of the same work which we present immediately after the first edition.

**1. Continuous model theory (1966), by Chen-chung Chang and H Jerome Keisler.**

**1.1. Review by: P J M.**

*The Review of Metaphysics*

**20**(2) (1966), 364.

This monograph is the first really systematic study of the model theory (semantics) of many-valued logic. The authors develop model theory for systems of logic whose truth values lie in a compact topological space; the results are analogous to those for two-valued logic - they yield the two valued logics as special cases - but often the methods of proof are more complicated and tend to reveal some of the deep structure of these logics. There is pre supposed a fair knowledge of naive set theory and point-set topology, but no knowledge of classical logic is required although it will be of help in seeing the motivation behind various results. The first three chapters are concerned with preliminaries on topology, model theory, and continuous logic. The next chapter examines the relation of elementary equivalence among models, including the downward Skolem-Löwenheim theorem; the fifth chapter contains the generalizations of such classical results as the compactness and upward S-L theorems. The authors specialize their work in the sixth chapter to consider certain particular kinds of models: saturated models, universal models. The last chapter considers classes of models closed under various algebraic operations. There is a bibliography, historical notes, and indices of exercises, symbols, and definitions. This book is the prolegomenon to any future study of many-valued logic.

**1.2. Review by: G Fuhrken.**

*Mathematical Reviews*MR0231708

**(38 #36)**.

The monograph contains a generalization of classic model theory. It was started by the first author in a series of papers. An outline appeared in a joint article [Theory of models, Proc. 1963 Internat. Sympos. Berkeley (1965)]. ...

The aim of the authors follows from their words: "The analogy (between their generalization and classic model theory) is perfect as far as the statements of the theorems are concerned; however, the proofs are far from analogous to the corresponding two-valued proofs. In many cases the new proofs are considerably more subtle and delicate, and they give further insight into why the standard two-valued proofs work the way they do." The book contains many, partly illustrative, and some advanced exercises. An appendix gives historical and bibliographical information.

**2. Elementary Calculus - An Approach Using Infinitesimals (experimental version) (1971), by H Jerome Keisler.**

This was a version of a text used by students which eventually was published in 1976. See 7. below.

**3. Model theory for infinitary logic. Logic with countable conjunctions and finite quantifiers (1971), by H Jerome Keisler.**

**3.1. Review by: Andreas Blass.**

*Mathematical Reviews*MR0344115

**(49 #8855)**.

Despite its small size, this book, intended both as an advanced graduate text and as a reference work, contains a wealth of material, ranging from the well-known theorems that any text on the subject must include to results of the author that appear here for the first time. ...

The book is very clear and should serve well as a text-book. Problems that graduate students should find challenging but not impossible are provided after several of the chapters. Exercises of a routine nature can be found throughout the book, for the author often omits tedious details; for example, when applying the model existence theorem, he usually verifies the one or two interesting clauses of its hypothesis and leaves the rest to the reader. Students should also specialize as many of the theorems as possible to the finitary case and see where compactness can simplify or strengthen the results.

**4. Model theory (1973), by Chen-chung Chang and H Jerome Keisler.**

**4.1. Review by: Gebhard Fuhrken.**

*The Journal of Symbolic Logic*

**41**(3) (1976), 697-699.

Several years ago two masters of the subject decided to write a textbook on "classical" model theory. Through the years people have referred to the book, solved open problems from it, even mentioned publishing date and publisher (different from the ultimate one). Finally the book actually appeared.

The authors set out to cover "most of first-order model theory and many of its applications to algebra and set theory." They have rewritten parts of the book more recently in order to incorporate some new developments. It is divided into seven chapters, each chapter into three to five sections ...

The book is no longer the only text on the subject, as it would have been when its first draft was written, but it is still the most comprehensive one. There is one major area of first-order model theory which is not touched upon: the area around Robinson's forcing. Some may feel that the lion's share of the applications goes to set theory, and that putting a good part of the material into the exercises makes it a little harder to use the book as a reference work. The pedantic reader will find inconsistencies in notation. The serious student will find - the reviewer has testimony thereof - the text very readable.

**4.2. Review by: B Weglorz.**

*Mathematical reviews*MR0409165

**(53 #12927)**.

This book is an excellent monograph on first order finitary model theory. In fact, it contains almost everything known about this subject up to the beginning of the 1970's. ... The authors have also made a great effort to collect and systematize the large amount of "folklore" in model theory, as well as some almost inaccessible proofs of well-known theorems ...

The reviewer thinks that the last sentence summarizes the principal value of this book. On the other hand, he is afraid that its size would discourage anyone who is searching here only for the basic facts from model theory (e.g., algebraists) from reading it. It might also be too difficult as a book for self-instruction. Another objection to this book is that there has been a great deal of progress in model theory outside the areas explained in this book (e.g., forcing in model theory) - but it is rather difficult to write a book containing all the facts that may turn out to be necessary in the future. Nevertheless, the reviewer thinks that this book is a basic source for anyone working in any area of the foundations of mathematics.

**5. Model theory 2nd ed (1977), by Chen-chung Chang and H Jerome Keisler.**

**5.1. Review by: Mathematical Reviews Editors.**

*Mathematical Reviews*MR0532927

**(58 #27177)**.

The authors state that there are no major changes from the first edition [1973] errors have been corrected and there is a new progress report on the 24 open problems in Appendix B.

**6. Model theory 3rd ed (1990), by Chen-chung Chang and H Jerome Keisler.**

**6.1. Preface.**

It has been thirteen years since the Second Edition of this book was written, and as one would expect, the subject of model theory has changed radically. Model theory is now dominated by new areas which were in their infancy in 1976 and have blossomed into thriving fields in their own right. Among these fields are classification (or stability) theory, nonstandard analysis, model-theoretic algebra, recursive model theory, abstract model theory, and model theories for a host of nonfirst order logics. Model-theoretic methods have also had a major impact on set theory, recursion theory, and proof theory.

In spite of the changes in the field, this book still serves well as a beginning graduate textbook and reference work. Classical first order model theory as developed here remains a prerequisite for all of the newer branches of model theory, and many newer books have relied on this book for the necessary background.

In preparing this Third Edition, we have been careful to preserve the usefulness of the book as a first textbook in model theory. We have made no attempt to cover the whole field, but have added new topics which now belong in a first graduate course. Four new sections have been added. These sections have been placed at the end of the original chapters to minimize changes in the numbering of results. Throughout the book, new exercises have been added, usually at the end of the original exercise lists. We made a number of updates, improvements, and corrections in the main text, have updated the appendix on the current status of the open problems, and have added a list of additional references.

The new Section 2.4 introduces recursively saturated models, which have led to the simplification of many arguments in model theory by replacing large saturated or special models by countable models. As an illustration, we have replaced the proof of the Vaught two-cardinal theorem in Section 3.2 by a simpler proof using recursively saturated models.

The new Section 2.5 presents Lindstrøm's celebrated characterization of first order logic. This result has gained importance as the launching point for the subject of abstract model theory. "Because of the growing importance of model-theoretic algebra, our treatment of model completeness which had been in Section 3.1 was greatly expanded and moved to the new Section 3.5.

The new Section 4.4 on nonstandard universes was added to provide an interface which is needed to apply results from model theory to nonstandard analysis.

**6.2. Review by: Mathematical Reviews Editors.**

*Mathematical reviews*MR1059055

**(91c:03026)**.

The revisions in the third edition were made by Keisler and include (i) a new section on recursively saturated models, (ii) an exposition of Lindstrøm's theorem, (iii) an expanded treatment of model completeness, (iv) a model-theoretic discussion of superstructures as a foundation for nonstandard analysis (which Keisler favours calling "Robinsonian analysis"), and (v) corrections, additional exercises, 61 /2 pages of additional references, and an updated discussion of the current status of the open problems in Appendix B. This book remains the canonical account of classical first-order model theory.

**6.3. Review by: Jan WoleDski.**

*Studia Logica: An International Journal for Symbolic Logic*

**51**(1) (1992), 154-155.

If someone will ask you about the most successful book in logical (classical) model theory, your answer may be only one: that is C C Chang and H J Keisler, Model Theory. This book was published for the first time in 1973. Second revised and enlarged edition appeared in 1977. Now we welcome the third edition of this classic book in classical model theory.

The book covers a wide variety of topics from model theory for sentential logic to large cardinals and the constructible universe. ...

Basically, the book is written from the standpoint of so called Western model theory originated with Tarski. Its ideology consists in proceeding from more general to more particular topics. Thus authors begin with abstract relation between languages and structures and then consider special algebraic constructions. So called Eastern model theory (A Robinson and his school) consists in generalizing concrete examples to abstract concepts. The book shows that the difference in these two ways of doing model theory is now of a secondary importance. This is documented in the third edition by the expansion the treatment of model-completeness, an idea introduced by Robinson.

The book is self-contained and shows the actual place of model theory in modern logic. Although Historical notes give a compact picture of the development of model theory but, in my opinion, might be more extended to give a general perspective of the field.

**7. Elementary Calculus, An Approach Using Infinitesimals (1976), by H Jerome Keisler.**

**7.1. Preface.**

The calculus was originally developed using the intuitive concept of an infinitesimal, or an infinitely small number. But for the past one hundred years infinitesimals have been banished from the calculus course for reasons of mathematical rigour. Students have had to learn the subject without the original intuition. This calculus book is based on the work of Abraham Robinson, who in 1960 found a way to make infinitesimals rigorous. While the traditional course begins with the difficult limit concept, this course begins with the more easily understood infinitesimals. It is aimed at the average beginning calculus student and covers the usual three or four semester sequence.

The infinitesimal approach has three important advantages for the student. First, it is closer to the intuition which originally led, to the calculus. Second, the central concepts of derivative and integral become easier for the student to understand and use. Third, it teaches both the infinitesimal and traditional approaches, giving the student an extra tool which may become increasingly important in the future.

Before describing this book, I would like to put Robinson's work in historical perspective. In the 1670's, Leibniz and Newton developed the calculus based on the intuitive notion of infinitesimals. Infinitesimals were used for another two hundred years, until the first rigorous treatment of the calculus was perfected by Weierstrass in the 1870's. The standard calculus course of today is still based on the "ε, δ definition" of limit given by Weierstrass. In 1960 Robinson solved a three hundred year old problem by giving a precise treatment of the calculus using infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century.

Recently, infinitesimals have had exciting applications outside mathematics, notably in the fields of economics and physics. Since it is quite natural to use infinitesimals in modelling physical and social processes, such applications seem certain to grow in variety and importance. This is a unique opportunity to find new uses for mathematics, but at present few people are prepared by training to take advantage of this opportunity.

Because the approach to calculus is new, some instructors may need additional background material. An instructor's volume, "Foundations of Infinitesimal Calculus," gives the necessary background and develops the theory in detail. The instructor's volume is keyed to this book but is self-contained and is intended for the general mathematical public.

This book contains all the ordinary calculus topics, including the traditional limit definition, plus one extra tool - the infinitesimals. Thus the student will be prepared for more advanced courses as they are now taught. In Chapters 1 through 4 the basic concepts of derivative, continuity, and integral are developed quickly using infinitesimals. The traditional limit concept is put off until Chapter 5, where it is motivated by approximation problems. The later chapters develop transcendental functions, series, vectors, partial derivatives, and multiple .integrals. The theory differs from the traditional course, but the notation and methods for solving practical problems are the same. There is a variety of applications to both natural and social sciences.

I have included the following innovation for instructors who wish to introduce the transcendental functions early. At the end of Chapter 2 on derivatives, there is a section beginning an alternate track on transcendental functions. and each of Chapters 3 through 6 have alternate track problem sets on transcendental functions. This alternate track can be used to provide greater variety in the early problems, or can be skipped in order to reach the integral as soon as possible. In Chapters 7 and 8 the transcendental functions are developed anew at a more leisurely pace. The book is written for average students. The problems preceded by a square box go somewhat beyond the examples worked out in the text and are intended for the more adventuresome.

I was originally led to write this book when it became clear that Robinson's infinitesimal calculus could be made available to college freshmen. The theory is simply presented: for example, Robinson's work used mathematical logic, but this book does not. I first used an early draft of this book in a one-semester course at the University of Wisconsin in 1969. In 1971 a two-semester experimental version was published. It has been used at several colleges and at Nicolet High School near Milwaukee, and was tested at five schools in a controlled experiment by Sister Kathleen Sullivan in 1972-1974. The results (in her 1974 Ph.D. thesis at the University of Wisconsin) show the viability of the infinitesimal approach and will be summarized in an article in the American Mathematical Monthly.

**7.2. Review by: Errett Bishop.**

*Bull. Amer. Math. Soc.*

**83**(2) (1977), 205-208.

From a primary concern with numbers and geometrical objects, the pre-college curriculum has moved on to open sentences, sets of sets, distinctions between numbers and numerals, and the like. The students quickly get the idea that they are not supposed to take it seriously: the teachers do not, do they?

Now the colleges have been more conservative, but a new book, Elementary calculus by H Jerome Keisler, could change all that. To quote from the book: "In 1960 Robinson solved a three hundred year old problem by giving a precise treatment of infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century." Again: "Recently, infinitesimals have had exciting applications outside mathematics, notably in the fields of economics and physics. Since it is quite natural to use infinitesimals in modelling physical and social processes, such applications seem certain to grow in variety and importance. This is a unique opportunity to find new uses for mathematics, but at present few people are prepared by training to take advantage of this opportunity."

No evidence of these claims is given in Keisler's book, but the students will not notice that. Those students who think that mathematics is about something will be disabused. To quote Keisler: "Do not be fooled by the name 'real number'. The real number system is a purely mathematical creation which may or may not give an accurate picture of a straight line in physical space." Again: "In discussing the real line we remarked that we have no way of knowing what a line in physical space is really like. It might be like the hyperreal line, the real line, or neither. However, in applications of the calculus it is helpful to imagine a line in physical space as a hyperreal line." What are we to make of these statements? Is Keisler describing mathematics as we know it, and the world as we have come to perceive it? The answer would appear to be "no", but perhaps we have not kept pace. If not, and his statements are true, the evidence should be somewhere in the book. ...

The technical complications introduced by Keisler's approach are of minor importance. The real damage lies in his obfuscation and devitalization of those wonderful ideas. ... Although it seems to be futile, I always tell my calculus students that mathematics is not esoteric: It is common sense. (Even the notorious µ, ¥ definition of limit is common sense, and moreover is central to the important practical problems of approximation and estimation.) They do not believe me. In fact the idea makes them uncomfortable because it contradicts their previous experience. Now we have a calculus text that can be used to confirm their experience of mathematics as an esoteric and meaningless exercise in technique.

**7.3. Review by: E W Madison and K D Stroyan.**

*Amer. Math. Monthly*

**84**(6) (1977), 496-500.

We believe that the main importance of calculus is what the name implies - calculation. Calculation of areas, lengths, volumes, tangents, maxima, rates of change, sums of series, and so forth, especially by use of the derivative and integral. In this regard, Leibniz' approach to calculus has long held a recognized advantage at least in guiding correct calculation; his notation persists even in the most pedantic epsilon-delta courses, as it should. The infinitesimal approach permits the most direct interpretation of the calculations. A student who refuses to draw little disks or shells when computing volumes of revolution ultimately encounters difficulty and usually dislikes the problems. Keisler's approach makes the 'sum of infinitesimal disks' heuristic nearly true (within an infinitesimal). One can think of the small disks as part of a limiting process, but the infinite sum of infinitesimals Leibniz had in mind (in the form provided by Robinson-Keisler) is just a very good approximation very far along the process, it is more direct but not at odds with finite limits.

Moreover, Keisler does a very careful job of showing the student the epsilon-delta or limit formulations of the concepts of infinitesimal calculus when he discusses numerical approximation, for example, the trapezoidal rule. We believe that a student who has gone carefully thru Keisler's chapter 5 will know as much about "epsilon-delta" as the students we have taught using Apostol's Calculus (2nd edition) while the Keisler student should have a stronger intuitive feeling for the integral and derivative because of the infinitesimals. On the other hand, error estimates are not easy. Even in our computer lab offered in conjunction with calculus they meet with limited success. Therefore, good heuristic arguments which infinitesimals can now provide rigorously are important in any approach. We think it is desirable to have intuition and rigor more closely linked than is possible in conventional approaches at the usual freshman level. ...

We highly recommend Keisler's book to interested instructors for "Engineering Calculus" through "Honors Calculus," with the change in emphasis coming from the instructor. Keisler's book gives correct intuitive proofs, once the instructor has helped the student over the axiom hurdle in chapter one, that is, once the student understands computationally and intuitively what infinite and infinitesimal numbers are. More or less emphasis on theoretical details and choice of topics make the book suitable for the full spectrum of calculus courses. Since his proofs are intuitive, a sketch provides the background for theorems in an intuitive course. Keisler's book has a large number of population and economics problems as well, so it may be interesting for "Liberal Arts Calculus."

We qualify our recommendation of Keisler's book. There are pitfalls, most notably the axioms in chapter one (comparable to Apostol's axiomatics). The book is revolutionary and we feel that instructors should learn the meaning of the axioms well before they attempt to teach them. The axioms need to be played down, explained intuitively in terms of pictures and examples, and dispensed with within a few days (unless students really need the excellent review of algebra also contained in chapter one). Not all the proofs of the basic properties need to be given in class. There is a great temptation for the instructor to apologize for, explain in excess, and philosophize about something which we find the students readily accept - infinitesimals. (They don't know "the reals are categorical.")

**7.4. Review by: Peter A Loeb.**

*The Journal of Symbolic Logic*

**46**(3) (1981), 673-676.

Infinitesimals have been used as a source of mathematical inspiration and heuristic justification for over two thousand years. The Greek Archimedes used infinitesimals to obtain his far-reaching results, as he showed in his treatise The method, which was discovered in 1906. Seventeenth- century mathematicians developed and used infinitesimal methods without being aware of their previous use by the Greeks or of the passionate debate over their validity. As Otto Toeplitz (The calculus, a genetic approach, 1963) notes, this trend in seventeenth-century research came to focus in the work of Leibniz, whose successors - Euler, the Bernoullis, Taylor, and others - then created, "with uninhibited zeal of discovery, the edifice of the new mathematics on the foundation of such non-rigorous, heuristic methods". Of course, infinitesimals were subsequently removed from the foundations of mathematics, but they remained part of the calculus training until the latter part of this century. Physicists and engineers have, for the most part, never relinquished infinitesimals as a source of discovery and a method of instruction. Consequently, a gap has developed between the language and concepts in courses taught by many physicists and engineers, and the language and concepts ($\epsilon$'s and $\delta$'s) in mathematics courses.

In 1960 Abraham Robinson used model theory to obtain a rich ordered field extension of the real numbers that allowed the use of infinitely small and infinitely large numbers in calculus). Although his discovery, non-standard analysis, went far beyond the calculus in its applications, Robinson's original purpose was to justify the use of infinitesimals in analysis, and he considered a number of classical arguments from this now rigorous viewpoint. For an instructor, however, even one not prejudiced by the previous lack of rigor in using infinitesimals, there still remained the problem of communicating Robinson's results to an undergraduate class.

In his text, Elementary calculus, Keisler has formulated Robinson's discovery so that it is easily accessible to beginning calculus students. While infinitesimals are used in developing the basic concepts of calculus, the $\epsilon-\delta$-method is also discussed as a way of obtaining numerical approximations of the limit, derivative, and integral. The reviewer has used this text in taking one honours class through three semesters of calculus and another through the first semester. The book has been widely used in teaching non-honours students and even bright high school classes. Experience shows that students who have used this text for one or more semesters are well prepared for subsequent mathematics courses and also for mathematics taught in other departments.

**8. Elementary Calculus: An Infinitesimal Approach (internet version) (2007), by H Jerome Keisler.**

**8.1. Preface.**

This book contains all the usual topics in an elementary calculus course, plus one extra concept that makes the basic ideas more easily understood by beginning students. This concept is the infinitesimal, or infinitely small number. The calculus was originally developed in the 1670's by Leibniz and Newton using infinitesimals in an informal way, and infinitesimals were given a rigorous foundation by Abraham Robinson in 1960.

The work is designed as the main textbook for a three or four semester calculus course. It has been used at both the college and high school levels. Because of its unique approach with infinitesimals, it can also serve as a supplementary book for courses based on other textbooks, and as an interesting refresher for those who have had calculus courses in the past. In addition. it can be a star ting point for mathematicians, scientists. and engineers who want to learn about the recent use of infinitesimals beyond calculus.

The First Edition of this book was published in 1976, and a considerably revised Second Edition was published in 1986, both by Prindle, Weber, and Schmidt. When the Second Edition went out of print in 1992, the copyright was returned to me as the author. Since 2002, the Second Edition with minor revisions has been available free in digital form under a Creative Commons license. It is listed by the California Digital Textbook Initiative as meeting all content standards for calculus.

Many people find it easier and more convenient to learn mathematics from a printed book. With this Third Edition, Dover Publications has given people the option of purchasing a high-quality affordable printed version of the free digital book.

I have written a higher-level companion to this book, Foundations of Infinitesimal Calculus (2007). This companion is intended for instructors and advanced students. It is a self-contained treatment of the mathematical background for this book, and can be a bridge to more advanced topics.

**9. Foundations of Infinitesimal Calculus (1976), by H Jerome Keisler.**

**9.1. Review by: Peter A Loeb.**

*The Journal of Symbolic Logic*

**46**(3) (1981), 673-676.

Experience shows that students are relatively comfortable with the beginning of Keisler's book Elementary Calculus, An Approach Using Infinitesimals (1976). They have already taken a number of courses that started with a set of basic assumptions or axioms and Keisler's book is just more of the same. The whole of Keisler's text, in fact, is given an uncommonly positive evaluation by most students. It has one of the highest levels of clear exposition of any currently available calculus text. Most of a course taught from this book is like any other, and even where infinitesimals are not used, this text makes the job easier. Keisler has included in this first edition more details about the hyperreal numbers than should be used in a class. These details provide a background for better students and provide background and reassurance for the instructor. A complete background for each chapter is given in Keisler's Foundations of infinitesimal calculus. Here one also finds an excellent introduction to Robinson's non-standard analysis, including a discussion of internal and external sets for higher-order structures, an ultrapower construction of $\mathbb{R}^{*}$, and a characterization of enlargements via limit ultrapowers. Most of this background does not find its way into the classroom. ... It is sometimes said that the broad acceptance of calculus taught with infinitesimals will have to wait for a further accumulation of successes in mathematical research in which non-standard analysis plays a role. If this is true, it is not because the usefulness of infinitesimals in calculus depends on new research. That usefulness was established long ago. Neither is it true that students have much interest in the frontiers of research when they are struggling with calculus. They do welcome whatever makes their task easier; in particular, they welcome Keisler's approach. What is true is that the majority of calculus instructors have been trained to avoid infinitesimal arguments. They know little or nothing about non-standard analysis and its successes, but they are strongly committed to carrying on the decades-long experiment of removing infinitesimals from the undergraduate curriculum. The goal of that experiment has been more or less met, but the resulting courses have been somewhat less than successful. The re-establishment of a more balanced approach to teaching calculus should not have to wait for the next generation of instructors.

**10. Foundations of Infinitesimal Calculus, Online edition (2007), by H Jerome Keisler.**

**10.1. Preface.**

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' and the second printed edition [1986]. 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' [1976], which was published as a companion to the first (1976) edition of 'Elementary Calculus', and has been out of print for over twenty years. 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 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 favour of the µ, ¥ 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.

The actual situation, as suggested by Leibniz and carried out by Robinson, is that one can form the hyperreal number system by adding infinitesimals to the real number system, and obtain a powerful new tool in analysis. The reason Robinson's discovery did not come sooner is that the axioms needed to describe the hyperreal numbers are of a kind which were unfamiliar to mathematicians until the mid-twentieth century. Robinson used methods from the branch of mathematical logic called model theory which developed in the 1950's.

Robinson called his method nonstandard analysis because it uses a nonstandard model of analysis. The older name infinitesimal analysis is perhaps more appropriate.

The method is surprisingly adaptable and has been applied to many areas of pure and applied mathematics. It is also used in such fields as economics and physics as a source of mathematical models. However, the method is still seen as controversial, and is unfamiliar to most mathematicians.

The purpose of this monograph, and of the book 'Elementary Calculus', is to make infinitesimals more readily available to mathematicians and students. Infinitesimals provided the intuition for the original development of the calculus and should help students as they repeat this development. The book 'Elementary Calculus' treats infinitesimal calculus at the simplest possible level, and gives plausibility arguments instead of proofs of theorems whenever it is appropriate. This monograph presents the subject from a more advanced viewpoint and includes proofs of almost all of the theorems stated in 'Elementary Calculus'.

**11. An infinitesimal approach to stochastic analysis (1984), by H Jerome Keisler.**

**11.1. Review by: Jens Erik Fenstad.**

*The Journal of Symbolic Logic*

**51**(3) (1986), 822-824.

A preliminary version of the memoir was written in 1978. It was enthusiastically received, and it stimulated much further research. Thus at the final publication of the book in 1984 many of the results have been extended, and some proofs have been simplified, sometimes by the author himself. The bibliography gives a reasonably complete set of references to current results.

Keisler's memoir is a contribution to stochastic analysis. In a certain sense it does not belong to the field of mathematical logic. Of course, concepts and tools from logic are important (transfer, saturation). But these are tools that by now should belong to the kit of every soldier in the field.

We should note that there is a model-theoretic side to the topic, viz. the study of probability quantifiers and their associated logic; a basic reference is Keisler's Hyperfinite model theory. The model theory and the stochastic theory are not independent enterprises; one would hope for a fruitful interaction as in the well-established relationship between algebra and model theory.

Keisler's memoir is a landmark; it deserves to be carefully studied.

**12. Mathematical Logic and Computability (1996), by J Keisler and J Robbin.**

**12.1. From the Introduction.**

In this book we shall study certain formal languages each of which abstracts from ordinary mathematical language (and to a lesser extent, everyday English) some aspects of its logical structure. Formal languages differ from natural languages such as English in that the syntax of a formal language is precisely given. This is not the case with English: authorities often disagree as to whether a given English sentence is grammatically correct. Mathematical logic may be defined as that branch of mathematics which studies formal languages.

**12.2. Review by Leon Harkleroad.**

*Modern Logic*

**8**(1-2) (1998), 138-141.

In 1976 Prindle, Weber & Schmidt published a book that stood out in the traditionally homogenized realm of calculus texts. That book was H Jerome Keisler's Elementary Calculus. Adapting Abraham Robinson's nonstandard analysis for first-year students, Keisler presented an infinitesimal-based development of the material. Whatever the other pros and cons of the book, its distinctive viewpoint set it apart as a genuine alternative to the usual clones.

Keisler's Mathematical Logic and Computability, however, lacks that flair. Perhaps the authorship by committee of this undergraduate text helps account for its hint of blandness. The cover credits Keisler as the author, but the list of titles in the flyleaf also names Joel Robbin. And the title page adds the names of Arnold Miller, Kenneth Kunen, Terrence Millar, and Paul Corazza as contributors. Given so many authors, one is not surprised that this book does not possess the individuality of Elementary Calculus.

Not that a text needs to be path-breaking to be a valuable addition to the literature. Books can certainly distinguish themselves by doing things well rather than differently. But Keisler and company have produced a work of uneven quality, with some nice features but also with some very definite drawbacks. This applies both to the text proper and to the accompanying computer software package.

Before describing these in more detail, let me first say what Mathematical Logic and Computability is and is not. The authors have clearly targeted an audience of upper-division mathematics majors, rather than aiming for a broader market including students in, say, computer science or philosophy. The treatment presupposes an appropriate level of mathematical maturity, while content prerequisites are minimal. An appendix covers the necessary rudiments of naive set theory, functions, cardinality, and so on. (As the authors rightly point out, this material gets short shrift in the curriculum at many institutions, with unfortunate results.) From the title one might expect a broad-based coverage of the various subfields of mathematical logic, as in Mendelson's standard text, for example. But such is not the case.

**13. Model theory of stochastic processes (2002), by Sergio Fajardo and H Jerome Keisler.**

**13.1. Review by: Alasdair Urquhart.**

*The Bulletin of Symbolic Logic*

**10**(1) (2004), 110-112.

The present monograph is a general study of stochastic processes on adapted spaces using the notion of adapted distribution; it combines ideas from stochastic analysis, model theory and nonstandard analysis. The authors describe one of their goals as follows: "From the viewpoint of nonstandard analysis, our aim is to understand why there is a collection of results about stochastic processes which can only be proved by means of nonstandard analysis." This remark may seem surprising to logicians with some acquaintance with nonstandard techniques, since it is a fundamental result that nonstandard analysis is a conservative extension of standard mathematics. The explanation for this apparent discrepancy is that the authors are working in adapted spaces derived from hyperfinite adapted spaces by the Loeb construction. These spaces are much richer than the spaces of conventional probability theory, and allow a much wider variety of constructions. For example, there are examples of stochastic differential equations that have no strong solution in the usual standard space of continuous functions from [0, 1] into $\mathbb{R}$ with the Wiener measure, but have strong solutions in Loeb adapted spaces, by the results of Keisler.

It is essential, of course, that these solutions should be considered as legitimate as solutions in the standard spaces. The fact that this is so reflects a general attitude of stochastic analysts; they are not fussy about the underlying probability space, provided that the process in question has the appropriate distribution. In the basic case of random variables, we can say that two random variables are "alike" if they induce the same probability distribution on the Borel sets. Similarly, two real valued stochastic processes are "alike" if they have the same finite dimensional distributions.

However, the main focus of the present monograph is adapted spaces. In these models, the single probability space of the basic theory of stochastic processes is replaced by a time- indexed increasing family of √-algebras (a filtration) representing the increase in information through time. This setting, basic to recent probabilistic work, allows the application of martingale techniques. Here, there is no generally agreed notion of what it means for two stochastic processes on adapted spaces to be "alike". The appropriate definition of this concept is one of the main themes addressed in the monograph.

**13.2. Review by: Constantinos A Drossos.**

*Mathematical reviews*MR1939107

**(2003k:60004)**.

This book studies stochastic processes using ideas from model theory and especially from nonstandard analysis. Sufficiently saturated models introduce a plethora of "ideal points" in such a way that almost any existence problem finds its solution! The power of this blending is exactly a kind of "imaginary" or "ideal points theory" that is behind the scenery. ...

The application of the methods of logic and model theory to ordinary mathematics, and in particular to probability theory, comes out of the necessity to study the formal linguistic structure which goes into the description of the concepts of probability. The main theme of the book is the generalization of probability spaces and finite-dimensional distributions to adapted spaces and adapted distributions, and one of the main questions is, "When are two stochastic processes alike?" The favourite answer of the authors is that "two processes are alike if they have the same adapted distribution".

One can say that based on the work of Keisler and his associates a species of "probabilistic logician", which J Łoś was speaking about in his fundamental paper "On the axiomatic treatment of probability", has been successfully formed!

The basic objective of this book is to import technology from model theory and blend it with probability, in order to get a more powerful theory. In particular the authors want to understand "why there is a collection of results about stochastic processes which can only be proved by means of nonstandard analysis". Specifically the program of the book is to define a language that is adequate for expressing the properties of stochastic processes on adapted spaces, define notions such as "elementary equivalence", etc., and build the model theory associated to the new language. In model theory the language is a vehicle that allows us to compare mathematical objects, possibly defined over different universes, that share a basic structure, and to explore its consequences in probability theory.

Last Updated April 2020