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The Metamathematical World of Model Theory

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The Metamathematical World of Model Theory W!orldof Mxodel T heory This relatively new theory, 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 mathematics certain devastating para- arithmetic, do not. doxes that posed serious challenges to Axioms, 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 language. 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 metamathematics, Cantor's paradoxical theory of infinite akin to metaphysics 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 conceptual model would have to contain an in- models. These models give promise of ternal contradiction. 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 "model theory," 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 theorems) 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 Theorem 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 logic, 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 geometry and non- ticular model. For example, because tion. commutative arithmetic later proved "1-I 1 = 2" is a true statement 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 physics, 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 108 Science News, Vol. 107 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 infinitesimal 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 geometries." 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 Compactness Theorem 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 language of mathematics as presence of the existing model makes infinitesimals 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 truth 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. Thoralf Skolem showed that the theory Part of the reason 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 arguments 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 Abraham Robinson 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 type whose existence is guaran- new model for calculus-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 ultraproduct. 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 name 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 ultraproducts 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 universe 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. February 15, 1975 109 This content downloaded on Thu, 14 Mar 2013 16:59:48 PM All use subject to JSTOR Terms and Conditions interest in model theory.
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