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
In the signature of A we introduce positive bounded
formulas and their approximations; these are similar to
those introduced by Henson  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
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