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March 2007 On finite approximations of topological algebraic systems
L. Yu. Glebsky, E. I. Gordon, C. Ward Hensen
J. Symbolic Logic 72(1): 1-25 (March 2007). DOI: 10.2178/jsl/1174668381


We introduce and discuss a concept of approximation of a topological algebraic system A by finite algebraic systems from a given class 𝔎. If A is discrete, this concept agrees with the familiar notion of a local embedding of A in a class 𝔎 of algebraic systems. One characterization of this concept states that A is locally embedded in 𝔎 iff it is a subsystem of an ultraproduct of systems from 𝔎. In this paper we obtain a similar characterization of approximability of a locally compact system A by systems from 𝔎 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.


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L. Yu. Glebsky. E. I. Gordon. C. Ward Hensen. "On finite approximations of topological algebraic systems." J. Symbolic Logic 72 (1) 1 - 25, March 2007.


Published: March 2007
First available in Project Euclid: 23 March 2007

zbMATH: 1115.03096
MathSciNet: MR2298468
Digital Object Identifier: 10.2178/jsl/1174668381

Rights: Copyright © 2007 Association for Symbolic Logic


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Vol.72 • No. 1 • March 2007
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