We are concerned here with a mathematical modeling of vague predicates and vague partitions (or equivalences or similarities) by the help of nonstandard sets of integers. The modeling is faithful in that it captures all basic features of the notion of vagueness. Nevertheless, it is not applicable to concrete phenomena, since nonstandard numbers are too big to be used in actual counting. The properties of a vague measurable similarity ∼ which we consider as most important and which are captured in the present modeling are: To ∼ there corresponds an assignment μ( x) of integral values to the objects of the domain of discourse, and a corresponding distance d( x, y) = | μ( x) − μ( y)|, such that: • (i) If x∼ y and d( x, z)⩽ d( x, y), then x∼ z. • (ii) For any two ∼-similar objects x, y and for any two ∼-nonsimilar objects x′, y′, d( x, y) < d( x′, y′). • (iii) For any x, y such that x∼ y, we can find a z in the same class, i.e. x∼ y∼ z such that d( x, y) < d( x, z). • (iv) For any ∼-nonsimilar objects x, y such that μ( x) < μ( y), there is a third object z, nonsimilar to both x, y, such that μ( x) < μ( z) < μ( y). Further we discuss the issue of what a “right” nonstandard model of numbers would be and some related questions.