Lecture Notes ni Mathematics Edited yb .A Dold, Heidelberg dna .B Z0rich Eckmann, 369 Victoria Symposium no Nonstandard Analysis University of Victoria 1972 Edited yb Albert Hurd, University of Victoria, Victoria/Canada Peter Loeb, University of Illinois, Urbana, II/USA galreV-regnirpS Berlin. Heidelberg- New York 1974 AMS Subject Classifications (1970): 02 H 25, 26-02, 26A98 ISBN 3-540-06656-X Springer-Verlag Berlin • Heidelberg • New York ISBN 0-387-06656-X Springer-Verlag New York • Heidelberg. Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photo- copying machine or similar and means, storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is to payable the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1974. Library of Congress Catalog Card Number 73-22552. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr. FOREWORD This volume is a record of the Symposium on Nonstandard Analysis held at the University of Victoria in Victoria, Canada during the period May 8-11, 1972. The symposium followed in spirit the previous symposia in nonstandard analysis held at the California Institute of Technology (1968), and Oberwolfach, Germany (1970). The principal invited speakers were Abraham Robinson, W.A.J. Luxemburg, H. Jerome Keisler , and Elias Zakon. It was our intention, hopefully almost completely realized, to have everyone working in nonstandard analysis in North America attend the conference. Most of the papers included here are based on lectures presented at the symposium, but in several instances we have included papers submitted in addition to those read at the conference. The following is a list of the papers which were presented at the conference. Andrew Alder Model theoretic ideas in topology Michael Behrens Analytic and meromorphic functions in the open unit disk Steven F. Bellenot A nonstandard theory of topological vector spaces Harry Gonshor Enlargements of Boolean algebras C. Ward Henson The nonstandard hulls of a uniform space Joram Hirschfeld Existentially complete models for arithemtic Albert E. Hurd Nonstandard dynamical systems H. Jerome Keisler Freshman calculus and measureable cardinals Peter J. Keleman Applications of nonstandard analysis to Quantum Mechanics and Quantum Field Theory Anders Kock Elementary Topoi Nonstandard extensions and the theory of Topoi Lawrence D. Kugler Weak almost periodicity Peter A. Loeb Some results in nonstandard measure theory W.A.J. Luxemburg On a theorem of Helly and a theorem about liftings L. J. Moore The nonstandard theory of topological vector spaces IV Louis Narens Homeomorphisms of generalized metric spaces Rohit Rarikh Conditional probability can be defined for all pairs of sets of reals Robert Phillips Omitting types in arithmetic David Pincus The power of the Hahn-Banach theorem Abraham Robinson Nonstandard exchange economies Keith D. Stroyan Infinitesimal relations on the space of bounded holomorphic functions Janes K. Thurber Applications of fractional powers of delta functions Frank Wattenberg Two different topologies with the same monads Elias Zakon A new variant of nonstandard analysis Metrization and completeness of the hyperreals In cases where the work presented at the conference is being published elsewhere the abstract included in the table of contents is the only record of the symposiu~ lecture. Also participating in the symposium were Gary L. Bender .U( of Colorado), D. .S Carter (Oregon State U.), David Cozart (Duke U.), Martin Davis (New York U.), Don Easton (Brandon U.), Bill Glassmire (Oregon State U.), Melvin Hausner (New York U.), James .P Jones .U( of Calgary), Steve Kloster (Simon Fraser U.), Susan Lenker .U( of Montana), George McRae .U( of Montana), Stephan Sperling (U.C.L.A.), Arthur L. Stone (Simon Fraser U.), Doug Super (Simon Fraser U.). Almost all of the papers included here were typed at the institutions of the respective authors, but we have proofread the manuscripts and have had them refereed. ACKNOWLEDGMENTS The symposium was co-organized by one of us (A.E.H.) together with Professor C. Robert Miers. It would not have been possible without the enthusiastic support of Professor Phoebe Noble and the generous financial aid provided by the University of Victoria. That the conference flowed as smoothly as it did is a tribute to the staff of the Department of Mathematics and especially to Mrs. Ruth McRae. Helping us with the refereeing were David Berg, Earl Berkson, John Gray, Lester Helms, Carl Jockusch, Thomas McLaughlin, Lee Rubel, Gaisi Takeuti. To everyone mentioned above our sincere thanks. Albert E. Hurd Peter Loeb October l, 1973 CONTENTS AND ABSTRACTS Date following abstract is date of receipt. Andrew Adler An application of elementary model theory to topological Boolean algebras Every topological Boolean algebra is isomorphic to a topological subalgebra of the ordered space on a non-standard model of the rationals. So in particular a countable topological boolean algebra is isomorphic to a topological subalgebra of the ordered space on an ~i set. An atomic countable 3 T dense-in-itself algebra is isomorphic to a topological subalgebra of the space of rationals. (January i0, 1973) Michael Behrens Analytic sets inK(D) Nonstandard techniques are used to investigate the maximal ideal space ~(D) of the Banach algebra H~(D) of bounded analytic functions on a planar domain D, and, more especially, to study the analytic structure inK(D) - D. A relatively complete discussion is presented for the unit disk, and a few results for infinitely connected domains are discussed. (August ,i 1973) Michael Behrens Boundary values for meromorphic functions defined in the 23 open unit disk Several classical boundary value theorems including the Gross principal value theorem are proved using nonstandard methods. (August 20, 1973) Michael Behrens 34 A local inverse function theorem An inverse function theorem is proved under a local assumption which is weaker than differentiability in a neighborhood. (August 20, 1973) VIII Steven F. Bellenot Nonstandard topological Vector spaces 37 A simple short nonstandard proof of the uniqueness of finite dimensional Hausdorff topological vector spaces. (October 17, 1972) Allen R. Bernstein and Peter A. Loeb A nonstandard integration theory for unbounded functions 4O This paper applies the nonstandard partition method of measure theory to the problem of integrating unbounded functions in a linear fashion, i.e., without truncation. As an application, one obtains a projection T of the extension *LI(X,~) of an arbitrary 1 L space onto a * finite dimensional subset of *LI(X,~,~) so that if h ~ LI(X,~,~) then II~*h-*hll ~ o . (March 5, 1973) Allen R. Bernstein and Frank Wattenberg Cardinality-dependent properties of topological spaces 50 In this paper we characterize certain topological properties whose definitions depend on a particular infinite cardinal by using ultrapowers over sets of that cardinality. In particular we obtain a nonstandard characterization of e-Baire spaces in terms of certain s-indexed ultrapowers. As a corollary, Baire spaces have a nonstandard characterization valid in any non-trivial countably-indexed ultrapower. This characterization may be used to give nonstandard proofs of results which depend on the Baire category theorem. This provides, at least in part, a solution to problem number 9 asked by Abraham Robinson in his retiring presidential address delivered to the Association for Symbolic Logic, January 1973 in Dallas, Texas. The paper also uses countably-indexed ultrapowers to examine certain countable equivalence conditions between topological spaces. (April 20, 1973) Donald J. Brown and Abraham Robinson Nonstandard exchange economies An exchange economy consists of a set of traders each of whom is characterized by an initial endowment and a preference relation. In addition, one usually assumes that the set of traders is finite. But IX in order to model perfectly competitive markets, i.e., markets where each trader's economic influence is negligible, we assume that the economy has ~ traders, where ~ is an infinite integer, and the average endowment of each trader is infinitesimal. In these nonstandard exchange economies, we examine the relationship between outcomes obtained through bargaining, called core allocations, and the allocations arising out of the competitive price system. We show that Edgeworth's conjecture, that every core allocation is a competi- tive allocation, is true in nonstandard exchange economies. AS a consequence of this theorem we also show that core allocations in large finite standard economies are approximately competitive allocations. References: Brown, D. J. and A. Robinson, "Nonstandard Exchange Economies", Econometrica (to appear). - - , "A Limit Theorem on the Cores of Large Standard Exchange Economies," Proc.Nat.Acad.Sc., U.S.A., Voi.69, No. 5, 1258-1260. Harry Gonshor Enlargements contain various kinds of completions 60 In this paper we show that various types of completions in different senses may be obtained as subquotients of enlargements. Among the examples considered are the following: The rationals as a "completion" of the integers, the Stone-Cech compactification of a completely regular space, the second conjugate space of a Banach space, rings of quotients of rings of continuous functions, the projective cover of a compact Hausdorff space, and the completions of a Boolean algebra. The last example is studied in detail. (March 22, 1973) C. Ward Henson and L. C. Moore, Jr. Semi-reflexivity of the nonstandard hulls of a locally convex 71 space Let E,F be vector spaces over R or C paired by a nonsingular bilinear form <...,...> and let @ be a locally convex topology on E which is admissible for the pairing. Given an enlargement * ~ of a set theoretical structure ~ which contains E and F , let (~,~) be the associated nonstandard hull of (E,@) . Also let ~ be A the space of linear functionals on E which are represented by those points q in *F for which <p,q> is finite for all @-finite p in A A *E • Then F is contained in the dual space E' of (~,~) and A A is an admissible topology relative to the pairing between E and F . A The principal result is that (E,~) is semi-reflexive if and only A if F is ~ ~( ',E)-dense A in E'A . (This extends a result for normed spaces proved earlier by the authors. ) Moreover, a geometric condition on (E,8) is given which is equivalent to the semi-reflexivity of A A (~,~) (and therefore this property of (E,@) does not depend on the particular enlargement *~). The main technical tool is the following result which seems of interest itself. Retraction Theorem: If *T~ is <-saturated and S is a subspace of ~ which has Hamel dimension less than < , then for each ~ ~ ~' there exists 9 c ~ such that #(x) = ~(x) for all x e S . (March 8, 1973) C. Ward Henson and L. C. Moore, Jr. Invariance of the nonstandard hulls of a uniform space 85 Let (X,~) be a uniform space and let *~ be an enlargement of some set-t/leoretical structure ~ which contain X . It is shown that if p { *X is not ~-pre-nearstandard and if *~ is <-saturated, then the filter monad ~(Fil(p)) : n{*YIp e *Y} intersects at least < distinct ~-monads ~ )q( . As a consequence, either )i( the nonstandard hulls of (X,~) (constructed using the ~-finite elements of *X ) are all equal to the completion of (X,%~) and are therefore independent of *~ or (ii) the nonstandard hulls of (X,~) can be made arbitrarily large by varying the choice of *~ . Similar results hold for the classes of nonstandard hulls defined by Luxemburg. In case )i( holds, we say (X,~) has invariant nonstandard hulls (extending the terminology introduced by the authors for locally convex topological vector spaces). A metric space (X,d) has invariant nonstandard hulls if and only if every subset of X on which every uniformly continuous function is bounded must be totally bounded. This equivalence is not true, however, for arbitrary uniform spaces. It is also shown that if (X,~) is a separable, topologically complete (metrizable) space, then ~ can be defined by a metric d such that (X,d) is complete and has invariant nonstandard hulls. (March 8, 1973) XI Joram Hirschfeld Models of arithmetic and the semi-ring of reeursive functions 99 S. Feferman, D. Scott, and S. Tennenbaum proved that every non-trivial homomorphic image of the semi-ring R of recursive functions fails to be a model of arithemtic. The aim of this paper is to show that every countable model of full arithmetic can be embedded in such a homormorphic image. To prove this we modify a theorem by H. Friedman to obtain a sufficient condition for a model of arithmetic to be embeddable in a model of a fragment of arithmetic. We then introduce recursive ultrapowers - homomorphic images of R which are models of that fragment. Finally, given a model, we show how to construct a recursive ultrapower which satisfies the condition of Friedman's theorem. (April 3, 1973.) Albert E. Hurd Nonstandard dynamical systems If D = (X,~,R) is a dynamical system then in an enlargement *D = (*X,*z,*R) we see that the flow can be followed for infinite time. This leads to simple characterizations of such standard notions as limit set, stability, prolongations, etc. as well as some notions in the enlargement which have no standard counterpart. At the same time X has been blown up to *X which is a sort of compactification. This leads to effective ways of studying flows on non-compact phase spaces. Using these ideas we can give necessary and sufficient conditions for topological equivalence of compact metric dynamical systems (Amer. J. Math. 93 (1971), 742-52). Results on almost periodicity have also been obtained. Albert E. Hurd Near periods and Bohr compactifications 106 A nonstandard construction is given for the Bohr compactification of a class of uniformly closed, translation invariant subalgebras of the almost periodic functions on a group G which includes the almost periodic functions themselves. The compactification is obtained by