In 1980 Wattenberg constructed
the Dedekind completion of a nonstandard model of the real numbers and applied the
construction to obtain certain kinds of special measures on the set of integers.
We feel that the Dedekind completion is a structure of interest for its own
sake and we establish further properties here. Of particular interest is the
connection with [1]. Specifically, the main concept we introduce is that of the
absorption number of an element a which, roughly speaking, measures the
degree to which the cancellation law a + b = a + c → b = c fails for a. The
absorption number may be regarded as an element in the Dedekind completion
of the value group of the valuation ring of finite numbers as discussed in
[1].
