## Journal of Symbolic Logic

- J. Symbolic Logic
- Volume 72, Issue 1 (2007), 1-25.

### On finite approximations of topological algebraic systems

L. Yu. Glebsky, E. I. Gordon, and C. Ward Hensen

#### 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.

#### Article information

**Source**

J. Symbolic Logic Volume 72, Issue 1 (2007), 1-25.

**Dates**

First available in Project Euclid: 23 March 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.jsl/1174668381

**Digital Object Identifier**

doi:10.2178/jsl/1174668381

**Mathematical Reviews number (MathSciNet)**

MR2298468

**Zentralblatt MATH identifier**

1115.03096

#### Citation

Glebsky, L. Yu.; Gordon, E. I.; Hensen, C. Ward. On finite approximations of topological algebraic systems. J. Symbolic Logic 72 (2007), no. 1, 1--25. doi:10.2178/jsl/1174668381. http://projecteuclid.org/euclid.jsl/1174668381.