{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,5,15]],"date-time":"2023-05-15T11:10:20Z","timestamp":1684149020418},"reference-count":27,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":2568,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2007,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We introduce and discuss a concept of approximation of a topological algebraic system<jats:italic>A<\/jats:italic>by finite algebraic systems from a given class<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200005454_inline1\" \/>. If<jats:italic>A<\/jats:italic>is discrete, this concept agrees with the familiar notion of a<jats:italic>local embedding<\/jats:italic>of<jats:italic>A<\/jats:italic>in a class<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200005454_inline1\" \/>of algebraic systems. One characterization of this concept states that<jats:italic>A<\/jats:italic>is locally embedded in<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200005454_inline1\" \/>iff it is a subsystem of an ultraproduct of systems from<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200005454_inline1\" \/>. In this paper we obtain a similar characterization of approximability of a locally compact system<jats:italic>A<\/jats:italic>by systems from<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200005454_inline1\" \/>using the language of nonstandard analysis.<\/jats:p><jats:p>In the signature of<jats:italic>A<\/jats:italic>we introduce<jats:italic>positive bounded<\/jats:italic>formulas and their<jats:italic>approximations;<\/jats:italic>these are similar to those introduced by Henson [14] for Banach space structures (see also [15, 16]). We prove that a positive bounded formula \u03c6 holds in<jats:italic>A<\/jats:italic>if and only if all precise enough approximations of \u03c6 hold in all precise enough approximations of<jats:italic>A<\/jats:italic>.<\/jats:p><jats:p>We also prove that a locally compact field cannot be approximated arbitrarily closely by finite (associative) rings (even if the rings are allowed to be non-commutative). Finite approximations of the field \u211d can be considered as possible computer systems for real arithmetic. Thus, our results show that there do not exist arbitrarily accurate computer arithmetics for the reals that are associative rings.<\/jats:p>","DOI":"10.2178\/jsl\/1174668381","type":"journal-article","created":{"date-parts":[[2007,12,19]],"date-time":"2007-12-19T16:24:33Z","timestamp":1198081473000},"page":"1-25","source":"Crossref","is-referenced-by-count":1,"title":["On finite approximations of topological algebraic systems"],"prefix":"10.1017","volume":"72","author":[{"given":"L. Yu.","family":"Glebsky","sequence":"first","affiliation":[]},{"given":"E. I.","family":"Gordon","sequence":"additional","affiliation":[]},{"given":"C. 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