今天，高考结束了。上了大学，学习什么呢？
世界著名数理逻辑模型论专家Keisler为此事写了一段文字，很有意思。请见附件。
附：
PREFACE
In 1960 AbrahamRobinson (1918–1974) solved the three hundred year old problem of giving a rigorous development of the calculus based onin-finitesimals. Robinson’s achievement was one of the major mathematical
advances of the twentieth century. This is an exposition of Robinson’sin-finitesimal calculus at the advanced undergraduate level. It isentirely self contained but is keyed to the 2000 edition of my first year collegetext Elementary Calculus [Keisler 2000]. Elementary Calculus is available freeonline at www.math.wisc.edu/.Keisler. This monograph can be used as a quick introduction to the subject for mathematicians, as background materialfor 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 ofInfinitesimalCalculus [Keisler 1976], which was published as a companion to thefirst (1976) edition of Elementary Calculus, and has been out of print for overtwenty years.
A companion to the second (1986) edition of Elementary Calculus wasnever written. The biggest changes are: (1) A new chapter on differentialequations, keyed to the corresponding new chapter in Elementary Calculus. (2) Theax
ioms for the hyperreal number system are changed to match those in thelater editions of Elementary Calculus. (3) An account of the discovery ofKanovei and Shelah [KS 2004] that the hyperreal number system, like the realnumber system, can be built as an explicitly definable mathematical structure.Earlier constructions of the hyperreal number system depended on anarbitrarily 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 aninfinitesimal,culminating in the work of Gottfried Leibniz (1646-1716) and IsaacNewton(1643-1727). When the calculus was put on a rigorous basis in thenineteenth century, infinitesimals were rejected in favor of the "; .approach, because mathematicians had not yet discovered a correct treatment ofinfinitesimals.Since then generations of students have been taught that infinitesimalsdo not exist and should be avoided.
The actual situation, as suggested by Leibniz and carried out byRobinson, is that one can form the hyperreal number system by adding infinitesimalsto the real number system, and obtain a powerful new tool in analysis. Thereason Robinson’s discovery did not come sooner is that the axioms needed todescribe the hyperreal numbers are of a kind which were unfamiliar tomathematicians 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 anonstandard model of analysis. The older name infinitesimal analysis isperhaps 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 aseconomics and physics as a source of mathematical models. (See, for example,the books [AFHL 1986] and [ACH 1997]). 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 andstudents. Infinitesimals provided the intuition for the original development of thecalculusand should help students as they repeat this development. The book Elementary Calculus treats infinitesimal calculus at the simplest possiblelevel,and gives plausibility arguments instead of proofs of theorems wheneverit is appropriate. This monograph presents the subject from a more advancedview point and includes proofs of almost all of the theorems stated inElementary Calculus.
Chapters 1–14 in this monograph match the chapters in Elementary Calculus, and after each section heading the corresponding sections ofElementary Calculus are indicated in parentheses.
In Chapter 1 the hyperreal numbers are first introduced with a set ofaxioms and their algebraic structure is studied. Then in Section 1G thehyperreal numbers are built from the real numbers. This is an optional sectionwhich is more advanced than the rest of the chapter and is not used later. It isincluded for the reader who wants to see where the hyperreal numbers come from. Chapters 2 through 14 contain a rigorous development of infinitesimalcalculus based on the axioms in Chapter 1. The only prerequisites are thetraditional three semesters of calculus and a certain amount ofmathematical maturity. In particular, the material is presented without usingnotions from mathematical logic. We will use some elementary set-theoretic notationfamiliar to all mathematicians, for example the function concept and thesymbols;;A [ B; fx 2 A: P(x)g.
Frequently, standard results are given alternate proofs usinginfinitesimals. In some cases a standard result which is beyond the scope of beginningcalculus is rephrased as a simpler infinitesimal result and used effectively inElementary Calculus; some examples are the Infinite Sum Theorem, and thetwo-variable criterion for a global maximum.
The last chapter of this monograph, Chapter 15, is a bridge between the simple treatment of infinitesimal calculus given here and the moreadvancedsubject of infinitesimal analysis found in the research literature. Togo beyond infinitesimal calculus one should at least be familiar with some basicnotions from logic and model theory. Chapter 15 introduces the concept of a nonstandard universe, explains the use of mathematical logic,superstructures, and internal and external sets, uses ultrapowers to build anonstandard universe, and presents uniqueness theorems for the hyperreal numbersystems and nonstandard universes.
The simple set of axioms for the hyperreal number system given here(and in Elementary Calculus) make it possible to present infinitesimalcalculus at the college freshman level, avoiding concepts from mathematical logic.It is shown in Chapter 15 that these axioms are equivalent to Robinson’sapproach.For additional background in logic and model theory, the reader canconsult the book [CK 1990]. Section 4.4 of that book gives further results onnonstandard universes. Additional background in infinitesimal analysis can befound in the book [Goldblatt 1991].
I thank my late colleague Jon Barwise, and Keith Stroyan of theUniversity of Iowa, for valuable advice in preparing the First Edition of thismonograph. In the thirty years between the first and the present edition, I havebenefited from equally valuable and much appreciated advice from friends andcolleagues too numerous to recount here
世界著名数理逻辑模型论专家Keisler为此事写了一段文字，很有意思。请见附件。
袁萌 6月9日
附：
PREFACE
In 1960 AbrahamRobinson (1918–1974) solved the three hundred year old problem of giving a rigorous development of the calculus based onin-finitesimals. Robinson’s achievement was one of the major mathematical
advances of the twentieth century. This is an exposition of Robinson’sin-finitesimal calculus at the advanced undergraduate level. It isentirely self contained but is keyed to the 2000 edition of my first year collegetext Elementary Calculus [Keisler 2000]. Elementary Calculus is available freeonline at www.math.wisc.edu/.Keisler. This monograph can be used as a quick introduction to the subject for mathematicians, as background materialfor 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 ofInfinitesimalCalculus [Keisler 1976], which was published as a companion to thefirst (1976) edition of Elementary Calculus, and has been out of print for overtwenty years.
A companion to the second (1986) edition of Elementary Calculus wasnever written. The biggest changes are: (1) A new chapter on differentialequations, keyed to the corresponding new chapter in Elementary Calculus. (2) Theax
ioms for the hyperreal number system are changed to match those in thelater editions of Elementary Calculus. (3) An account of the discovery ofKanovei and Shelah [KS 2004] that the hyperreal number system, like the realnumber system, can be built as an explicitly definable mathematical structure.Earlier constructions of the hyperreal number system depended on anarbitrarily 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 aninfinitesimal,culminating in the work of Gottfried Leibniz (1646-1716) and IsaacNewton(1643-1727). When the calculus was put on a rigorous basis in thenineteenth century, infinitesimals were rejected in favor of the "; .approach, because mathematicians had not yet discovered a correct treatment ofinfinitesimals.Since then generations of students have been taught that infinitesimalsdo not exist and should be avoided.
The actual situation, as suggested by Leibniz and carried out byRobinson, is that one can form the hyperreal number system by adding infinitesimalsto the real number system, and obtain a powerful new tool in analysis. Thereason Robinson’s discovery did not come sooner is that the axioms needed todescribe the hyperreal numbers are of a kind which were unfamiliar tomathematicians 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 anonstandard model of analysis. The older name infinitesimal analysis isperhaps 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 aseconomics and physics as a source of mathematical models. (See, for example,the books [AFHL 1986] and [ACH 1997]). 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 andstudents. Infinitesimals provided the intuition for the original development of thecalculusand should help students as they repeat this development. The book Elementary Calculus treats infinitesimal calculus at the simplest possiblelevel,and gives plausibility arguments instead of proofs of theorems wheneverit is appropriate. This monograph presents the subject from a more advancedview point and includes proofs of almost all of the theorems stated inElementary Calculus.
Chapters 1–14 in this monograph match the chapters in Elementary Calculus, and after each section heading the corresponding sections ofElementary Calculus are indicated in parentheses.
In Chapter 1 the hyperreal numbers are first introduced with a set ofaxioms and their algebraic structure is studied. Then in Section 1G thehyperreal numbers are built from the real numbers. This is an optional sectionwhich is more advanced than the rest of the chapter and is not used later. It isincluded for the reader who wants to see where the hyperreal numbers come from. Chapters 2 through 14 contain a rigorous development of infinitesimalcalculus based on the axioms in Chapter 1. The only prerequisites are thetraditional three semesters of calculus and a certain amount ofmathematical maturity. In particular, the material is presented without usingnotions from mathematical logic. We will use some elementary set-theoretic notationfamiliar to all mathematicians, for example the function concept and thesymbols;;A [ B; fx 2 A: P(x)g.
Frequently, standard results are given alternate proofs usinginfinitesimals. In some cases a standard result which is beyond the scope of beginningcalculus is rephrased as a simpler infinitesimal result and used effectively inElementary Calculus; some examples are the Infinite Sum Theorem, and thetwo-variable criterion for a global maximum.
The last chapter of this monograph, Chapter 15, is a bridge between the simple treatment of infinitesimal calculus given here and the moreadvancedsubject of infinitesimal analysis found in the research literature. Togo beyond infinitesimal calculus one should at least be familiar with some basicnotions from logic and model theory. Chapter 15 introduces the concept of a nonstandard universe, explains the use of mathematical logic,superstructures, and internal and external sets, uses ultrapowers to build anonstandard universe, and presents uniqueness theorems for the hyperreal numbersystems and nonstandard universes.
The simple set of axioms for the hyperreal number system given here(and in Elementary Calculus) make it possible to present infinitesimalcalculus at the college freshman level, avoiding concepts from mathematical logic.It is shown in Chapter 15 that these axioms are equivalent to Robinson’sapproach.For additional background in logic and model theory, the reader canconsult the book [CK 1990]. Section 4.4 of that book gives further results onnonstandard universes. Additional background in infinitesimal analysis can befound in the book [Goldblatt 1991].
I thank my late colleague Jon Barwise, and Keith Stroyan of theUniversity of Iowa, for valuable advice in preparing the First Edition of thismonograph. In the thirty years between the first and the present edition, I havebenefited from equally valuable and much appreciated advice from friends andcolleagues too numerous to recount here