Journal of Symbolic Logic
- J. Symbolic Logic
- Volume 72, Issue 1 (2007), 1-25.
On finite approximations of topological algebraic systems
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  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.
J. Symbolic Logic Volume 72, Issue 1 (2007), 1-25.
First available in Project Euclid: 23 March 2007
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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. https://projecteuclid.org/euclid.jsl/1174668381.