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Home , Metamathematics, Model theory, Thoralf Skolem

W!orldof Mxodel T heory

This relatively new , ripe for potential research and application, may be a powerful new mathematical tool for analyzing many kinds of scientific problems by Lynn Arthur Steen

Georg Cantor's theory of infinite for example, satisfy this sentence as sets introduced into early 20th-century well, while still others, such as binary certain devastating para- , do not. doxes that posed serious challenges to , the logical bedrock of the logical foundations of mathematics. mathematics, are nothing but particu- One consequence of this upheaval was lar sentences in a particular . the development of a mathematical A central question in all mathematics, theory of mathematics itself-a new one which assumes special urgency in discipline called , Cantor's paradoxical theory of infinite akin to but far more rig- sets, is whether in fact there exist mod- orous and technical in its methodology. els that satisfy particular collections Now, nearly half a century after its of axioms. Collections of axioms that creation, aspects of metamathematics are logically contradictory or that en- are beginning to move beyond the tail logical falsehoods surely cannot analysis of existing mathematics to the be satisfied by any model, for such a creation of surprising new would have to contain an in- models. These models give promise of ternal . Collections of well representing certain large-scale axioms with this failing are called in- scientific problems and may even yield I - 01' consistent and are purged from mathe- a new perspective on the nature of matics whenever they are discovered. time. ,.A rfa"N S Mathematics deals only with consistent Details of this relatively new theory, collections of axioms, in two essentially known as "," were out- W E~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~tdifferent but closely related ways: a lined by H. Jerome Keisler of the Uni- syntactical process in which certain versity of Wisconsin in four Colloquium sentences (called ) are de- Lectures at the annual meeting of the rived from consistent collections of American Mathematical Society in axioms, and a semantical process in Washington in January. According to which models for consistent collec- society President Lipman Bers of Co- tions of axioms are constructed, ana- lumbia University, an invitation to de- standard models for many current sci- lyzed and utilized. liver the Colloquium Lectures is one entific problems. The first major result of model theo- of the highest honors in mathematics. The models provided by model ry a result which provided the pri- Rarely has the subject of these lectures theory come, of course, from within mary tool for 40 years of subsequent been so rich in history and so ripe for mathematics itself. Mathematics is research-is the so-called Compactness potential research and application. comprised of a langucage of symbols in first proved by the German Model theory has developed rapidly which sentences may be formed that logician Kurt Godel in 1930. (Godel is during the past two decades as a tool describe properties of certain abstract now a fellow of the Institute for Ad- for classifying and constructing models objects called models. For instance, vanced Study in Princeton.) This theo- that arise in diverse parts of matlhe- by using symbols such as +, =, 0, 1, rem establishes certain very general matics. The subject of its study is 2, we may express sentences ( e.g., " 1 grounds under which the existence of mathematics itself, and the tools are + 1 = 2") about a particular model. a model can be guaranteed even with- highly exotic methods of mathematical What a sentence says about a particu- out knowing what it looks like or how , particularly various techniques lar model may be true, or it may be to construct it. Specifically, it says that for constructing, enlarging or modify- false, or it may be meaningless. A mod- if every finite subcollection of an in- ing mathematical models to serve new el is said to satisfy a particular sen- finite collection of axioms has a model, requirements. As 19th-century models tence if the sentence is true in that par- then so must the whole infinite collec- for non-Euclidean and non- ticular model. For example, because tion. commutative arithmetic later proved "1-I 1 = 2" is a true about The power of this theorem lies in useful (indeed, indispensable) in rela- integers, we say that the integers form its ability to leap logically from evi- tivity theory and in atomic , so a model which satisfies the sentence dence based on finite collections to a advocates of model theory foresee "I + 1 = 2." Of course other models, conclusion that holds for an infinite model theory providing valuable non- the model of ordinary real numbers, collection: It is a mathematical bridge

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This content downloaded on Thu, 14 Mar 2013 16:59:48 PM All use subject to JSTOR Terms and Conditions "This is the essence of meta- " . Robinson's solution of mathematics-to study the lan- the problem of guage of mathematics as distinct numbers is as significant and from the objects of mathematics, revolutionary as the 19th- like studying linguistics as opposed century discovery of non- to writing a novel." Euclidian ." into the transfinite. Mathematicians chevski and Riemann. Moreover, his employ infinite collections of axioms method-an application of basic prin- quite frequently, even for such simple ciples of model theory-may be even things as a rigorous treatment of arith- more important than his result. In- metic. The pro- Cs stead of studying details of existing vides a powerful means for inferring models, he studied as a distinct mathe- the existence of a model for such an matical entity the language which one infinite collection of axioms. It is es- uses to describe these models. This is pecially useful in enlarging existing the essence of metamathematics-to models to meet new conditions: The "Newton's and Leibniz's intuition about study the as presence of the existing model makes was not only quite pro- distinct from the objects of mathe- it possible to verify the hypothesis of found but also, in the proper context, matics, like studying linguistics as op- the Compactness Theorem, thus en- capable of rigorous verification." M. C. posed to writing a novel. Model theory suring the of its conclusion. Escher's "Smaller and Smaller" ex- provides the translation from the lan- A second major event in the classical presses the mystery concerning what guage in which mathematics is ex- development of model theory occurred happens when things, or numbers, be- pressed to the models of which it is in 1935 when the Norwegian logician come infinitely small. comprised. showed that the theory Part of the that Robinson's of ordinary arithmetic must have mod- cians, especially Augustin Cauchy and work had greater impact than Skolem's els other than the intended one-in Karl Weierstrass, developed many dif- was that he had available more com- particular, models containing infinitely ferent designed to show that pelling techniques. In 1955 the Polish large numbers. This idea lay fallow, the entire concept of an infinitesimal mathematician Jerzy Los extended one however, until about 10 years ago was absurd, and they succeeded in of Skolem's ideas to actually construct when the late of banishing it from mathematics as a bit (in an abstract sort of way) models Yale University refined it to create a of clever but misguided intuition. Rob- of the whose existence is guaran- new model for -called "non- inson's nonstandard model, however, teed by the Compactness Theorem. standard" analysis-which incorporated has reaffirmed the early heroes. It Los's model is called an . both infinitely large and infinitely shows, in fact, that Newton's and It is formed by complex algebraic op- small (infinitesimal) numbers. Leibniz's intuition was not only quite erations (similar to simple products Isaac Newton and Gottfried Leibniz, profound but also, in the proper con- but of infinite extent; hence the the co-inventors of calculus, both em- text, capable of rigorous verification. ultraproduct) performed on the ele- ployed infinitesimals in their work. But Many mathematicians feel that Rob- ments of more basic models. subsequent generations of mathemati- inson's solution of the problem of in- The introduction of as cians, despite strenuous effort, were finitesimal numbers is as significant a relatively concrete realization of the unable to find a logically coherent ex- and revolutionary as, for instance, the heretofore totally existential models planation for the behavior of infinitesi- 19th-century discovery of non-Euclidean guaranteed by the Compactness Theo- mals. Nineteenth-century mathemati- geometries by Bolyai, Gauss, Loba- rem generated a major resurgence of

INFINITESIMAL MICKOSCOPE

MONAD OF - I MONAD OF I

NEGAIVE - -2- -_ 0 1 2 POSITIVE INFINITEG INFINITE GAL,AXIES NITVE. GALAXY LXE

INFINITF TELE-5C0PE hikk

NONSTANDARD MODFELOF THE REAL NOM6EP$l On a macroscopic level, the nonstandard of real which is the ordinary real line. On a microscopic level, ex- numbers contains, in addition to the ordinary real number amination of any point in a galaxy through an "infinitesimal line, many "galaxies" just like it which extend on both sides microscope" reveals that it is not just a point, but an in- of it. These galaxies appear, when viewed through an "in- finitely small copy of the entire real line called, after Leib- finite telescope," to be exact replicas of the finite galaxy niz, a monad.

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This content downloaded on Thu, 14 Mar 2013 16:59:48 PM All use subject to JSTOR Terms and Conditions interest in model theory. Robinson contains certain sets, called *-finite THINGSof science used ultraproduct constructions to give (pronounced "star-finite") sets, which For Boys and Girls- 10 to 16 concrete form to his nonstandard mod- appear to be finite when viewed from Send TODAY for free information on how els of calculus: He could actually rep- within the nonstandard model, yet are you can receive a different experimental easily recognized as infinite when science kit once each month. Kits include resent infinitely small numbers by sim- materials and booklet of interesting experi- ple sequences of ordinary fractions. viewed from outside the model. The ments that can be done at home or in It is this possibility which has led schools. Write to: Keisler to develop an entire intuitive "Semantically, a *-finite THINGS of science introductory calculus course based on Department P27 1719 N St., N.W. Robinson's nonstandard model of anly- is infinite, but syntactically, Washington, D.C. 20036 sis. In this course the routine pattern it is finite. The formal of reasoning, quite similar to the origi- language of mathematics is nal intuition of Newton and Leibniz, simply not sufficiently rich Howto Argueand Win! is overlaid on a tightly woven logical to express all the Here is a clear simply writ- fabric of ultraproducts and applications ten basic guide to logical of the Compactness Theorem. bound up in its models." thinking, showing how to spot the fallacies, the prejudices Model theory offers a powerful and emotionalism, the inap- propriate analogies, etc., m mathematical tool for rigorously syn- reason for this ambiguity is that cer- the other fellow's thesizing common theoretical conflicts tain facts about a model (for example, and how to watch for and avoid the irrational in your between discrete and continuous in- that a particular set is infinite) may be own judgments. The author terpretations of scientific phenomena. recognized as true by an informed in- makes plain not only how but also why people resist facing This conflict occurs, for instance, in tuition, yet be incapable of expression the truth. our conceptualization of time. In the or A tool for clear thinking as within the grammar of the well as convincing others. ORDER NOW: continuous model, with time repre- mathematical language which describes THE ART OF ARGUMENT, By Giles St. Aubyn sented by a straight line, no moment of the model. Semantically, a *- 50, Money-Back Guar. $5.95 plus handling 10-Day time is followed by a next moment: is infinite, but syntactically, it is finite. EMERSON BOOKS, Inc., Dept. 157B Between each two moments there is al- The of mathematics is Buchanan, N.Y. 10511 ways an interval of other moments of simply not sufficiently rich to express time. In the discrete model, represented all the meaning bound up in its mod- THE NEXT INDUSTRIAL REVOLUTION by the counting of years or days or els. seconds, each moment is separated The part-finite, part-infinite charac- ROBOTICS from the next moment by an indivisi- ter of *-finite sets makes them particu- nEWSLETTERO ble unit of some fixed length. larly suita!ble for use as mathematical SAMPLE COPY $1.00 ROBOTICS NEWSLETTER In the nonstandard model of time, representations of very large finite sets 260 GODWIN AVE. each moment is followed by a next such as the voters in an election, trad- N.J. 07481 WYCKOFF, moment, yet two successive moments ers in an economy, or molecules in a are infinitely close together. In this fluid. Robinson and Yale economist model, it is possible to conceptualize Donald Brown have employed *-finite A SOURCESOOKONANCIENT MAN a physical or social process as taking sets to analyze such classical economic Over 100 articles on Stonehenge, Scottish Fused Forts, place one step at a time, each step problems as Edgeworth's conjecture Phoenicians in America, ancient Arizona canals, and other mysterious artifacts as originally printed in Nature, Archae- infinitely close to the next. Of course, that as the number of traders in an ologia and other publications. Thorough index. scientists have often this way exchange economy increases, the core 262 pages, hard covers, $6.95 postpaid of Publishedby: THE SOURCEBOOKPROJECT intuitively; the importance of this new the economy approaches competitive BOX1076, GLENARM, MARYLAND 21067 model is that, for the first time, such equilibrium. Earlier approaches to this intuitive reasoning can be made logical- type of problem relied either on a hypo- ly defensible. thetical sequence of economies growing It has uses. For without bound, or on a model of an Manuscriptsinvited for prompt review and practical example, terms of publication. All subjects. Professional in his lectures Keisler used this non- economy with an infinite number of editing, design, production and marketing since standard conceptualization of time to traders. 1920. Send or manuscript, or call (215) 563-3553. Ask for free Authors' Guide B-75. construct a model of an exchange Keisler's most recent work concerns DORRANCE & COMPANY economy in which prices would be de- the development of nonstandard prob- 1617 J.F. Kennedy Blvd. Philadelphia, Pa. 19103 termined by the aggregate effect of a ability theory. Traditional elementary large number of traders acting in a (used to analyze the flip succession of infinitesimally spaced of a coin or the throw of a die) em- moments. By means of this model he ploys a finite model; more sophisti- was able to set forth explicit upper cated and realistic problems require and lower bounds on the momentary infinite models. Unfortunately, many price fluctuations to make market sta- of the basic methods of analysis that bility possible. If prices change too work well in the finite models (e.g., slowly, they will not converge to counting, adding) become technically equilibrium in a finite amount of time, meaningless in the infinite models, even WINDWATCH TM "95 while if they change too rapidly, they though they often retain considerable ANEMOMETER-WINDVANE L will never have a chance to become intuitive value. Neon lights in indoor indicator show wind speed and direction. Decorative indicator with gold dial stabilized. Keisler showed in his lectures that it in dark mahogany case. Designed for desk use or wall mount. Outdoor instruments of durable The distinction between the lan- is possible to substitute for the tra- aluminum with electrical parts enclosed in rigid guage in which mathematics is ex- ditional infinite probabilistic model a vinyl housings. Easy to assemble and install. Attach directly to roof or pole. Comes with wire pressed and the models about which nonstandard one based on *-finite sets and materials for installation up to 60 feet. Uses regular 120V AC. Unique low amp circuit as- this language speaks leads to a sur- in which counting arguments may be sures absolute safety. A fascinating addition to prisingly subtle device for utilizing non- employed on sets which in any home. Five year guarantee. 15 day trial. Your are, reality, satisfaction or money back.Only $22.95 postpaid. standard models in interpreting scien- infinite. Although this investigation is WINDW1ATCHDept.sN-4,c/e Science News, 1719 tific phenomena. For example, every still young, some surprising results have W1 11A l NSt., N.W., Wash., O.c. 20036; (Make check payable to Science News) nonstandard model of the real numbers already emerged. One of the more in-

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