On finite approximations of topological algebraic systems

Abstract

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.