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 cap
tures 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 meas
urable similarity similar to which we consider as most important and w
hich are captured in the present modeling are: To similar to there cor
responds an assignment mu(x) of integral values to the objects of the
domain of discourse, and a corresponding distance d(x,y) = \mu(x) - mu
(y)\, such that: (i) If x similar to y and d(x,z) less than or equal t
o d(x,y), then x similar to z. (ii) For any two similar to -similar ob
jects x,y and for any two similar to -nonsimilar objects x',y',d(x,y)
< d(x',y'). (iii) For any x,y such that x similar to y, we can find a
z in the same class, i.e. x similar to y similar to z such that d(x,y)
< d(x,z). (iv) For any similar to -nonsimilar objects x,y such that m
u(x) < mu(y), there is a third object z, nonsimilar to both x,y, such
that mu(x) < mu(z) < mu(y).Further we discuss the issue of what a ''ri
ght'' nonstandard model of numbers would be and some related questions
. (C) 1998 Elsevier Science B.V.