Journal of Symbolic Logic

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.

Article information

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

Dates
First available: 23 March 2007

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. Journal of Symbolic Logic 72 (2007), no. 1, 1--25. doi:10.2178/jsl/1174668381. http://projecteuclid.org/euclid.jsl/1174668381.