Summary: An affinely recursive set \(S\) consists of integers defined by these rules: \(1 \in S\), and if \(x \in S\), then \(f(x)\in S\) for every \(f\) in a prescribed set \(F\) of affine functions with integer coefficients. If these functions are indexed by a set \(A\), then each \(x\) in \(S\) except 1 corresponds to a word in the language \(L(A)\) of nonempty words over the alphabet \(A\). Conditions are found for the correspondence to be bijective, so that the ordering of numbers in \(S\) induces an ordering of \(L(A)\). Moreover, since each \(x\) except 1 in \(S\) is of the form \(f(y)\) for some \(y\) in \(S\) and \(f\) in \(F\), the set \(S\) is partitioned by \(F\). The partition extends to \(L(A)\); viz., a component consists of words that end in the same letter. The distribution and limiting density within \(S\) of the numbers in each component is considered.