Ambiguous functions are not really functions. For each prescription of values to the arguments the ambiguous function has a collection of possible values. An example of an ambiguous function is less(n) defined for all positive integer values of n. Every non-negative integer less than n is a possible value of less(n). First we define a basic ambiguity operator amb(x,y) whose possible values are x and y when both are defined: otherwise, whichever is defined. Now we can define less(n) by
less(n) has the property that if we define
then
There are a number of important kinds of mathematical arguments whose convenient formalization may involve ambiguous functions. In order to give an example, we need two definitions.
If f and g are two ambiguous functions, we shall say that f is a descendant of g if for each x every possible value of f(x) is also a possible value of g(x).
Secondly, we shall say that a property of ambiguous functions
is hereditary if whenever it is possessed by a function g it is
also possessed by all descendants of g. The property that iteration
of an integer valued function eventually gives 0 is hereditary, and
the function less has this property. So, therefore, do all its
descendants. Therefore any integer-function g satisfying g(0) = 0
and 
has the property that 
is identically 0 since g is a
descendant of less. Thus any function, however complicated, which
always reduces a number will if iterated sufficiently always give 0.
This example is one of our reasons for hoping that ambiguous functions will turn out to be useful.
With just the operation amb defined above adjoined to those
used to generate 
, we can extend 
to the class
which may be called the computably ambiguous
functions. A wider class of ambiguous functions is formed using the
operator 
whose values are all x's satisfying 