On approximation of topological algebraic systems by finite ones

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 [8] for Banach spaces. We prove that a positive bounded formula φ holds in A if all precise enough approximations of φ 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 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.