On finite approximations of topological algebraic systems

Abstract

We introduce and discuss a definition of approximation of a topological algebraic system $A$ by finite algebraic systems of some class $\K$. For the case of a discrete algebraic system this definition is equivalent to the well-known definition of a local embedding of an algebraic system $A$ in a class $\K$ of algebraic systems. According to this definition $A$ is locally embedded in $K$ iff it is a subsystem of an ultraproduct of some systems in $\K$. We obtain a similar characterization of approximation of a locally compact system $A$ by systems in $\K$. We inroduce the bounded formulas of the signature of $A$ and their approximations similar to those introduced by C.W.Henson \cite{he} for Banach spaces. We prove that a positive bounded formula $\f$ holds in $A$ if all precise enough approximations of $\f$ hold in all precise enough approximations of $A$. We prove that a locally compact field cannot be approximated by finite associative rings (not necessary commutative). Finite approximations of the field $\mathbb{R}$ can be concedered as computer systems for reals. Thus, it is impossible to construct a computer arithmetic for reals that is an associative ring.

20 pages, sent to Journal of Symbolic Logic