There are many examples of non-isomorphic pairs of finitely generated abstract groups that are elementarily equivalent. We show that the situation in the category of profinite groups is different: If two finitely generated profinite groups are elementarily equivalent (as abstract groups), then they are isomorphic. The proof applies a result of Nikolov and Segal which in turn relies on the class...

We study the relationship between the Loeb measure °(*fi) of a set £ and the /¿-measure of the set S(E) = [x\*x G E) of standard points in E. If E is in the a-algebra generated by the standard sets, then °(*f0(£) — piS(Ef). This is used to give a short nonstandard proof of Egoroffs Theorem. If £ is an internal, * measurable set, then in general there is no relationship between the measures of S...

Recent applications of decision procedures for nonlinear real arithmetic (the theory of real closed fields, or RCF) have presented a need for reasoning not only with polynomials but also with transcendental constants and infinitesimals. In full generality, the algebraic setting for this reasoning consists of real closed transcendental and infinitesimal extensions of the rational numbers. We pre...

We study classes of universal algebras in a variety that are closed under the formation of Cartesian products, homomorphic images and extensions. We prove a Birkhoff type theorem for such axiomatic classes. Let V be a variety of algebras of a fixed type τ and let S be a set of first order sentences in the language of τ of the form (∃x1) . . . (∃xn)((u1 = v1) ∧ · · · ∧ (um = vm)), where u1, v1, ...