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- W2115145127 abstract "Abstract We introduce and discuss a concept of approximation of a topological algebraic system A by finite algebraic systems from a given class . If A is discrete, this concept agrees with the familiar notion of a local embedding of A in a class of algebraic systems. One characterization of this concept states that A is locally embedded in iff it is a subsystem of an ultraproduct of systems from . In this paper we obtain a similar characterization of approximability of a locally compact system A by systems from using the language of nonstandard analysis. In the signature of A we introduce positive bounded formulas and their approximations; these are similar to those introduced by Henson [14] for Banach space structures (see also [15, 16]). We prove that a positive bounded formula φ holds in A if and only if all precise enough approximations of φ hold in all precise enough approximations of A . 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 ℝ 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." @default.
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- W2115145127 date "2007-03-01" @default.
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- W2115145127 title "On finite approximations of topological algebraic systems" @default.
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- W2115145127 doi "https://doi.org/10.2178/jsl/1174668381" @default.
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