A Generalisation of Productive Set
enumeration of the set in increasing order. Alternatively, we may give some effective test for membership in the set. Again, the cardinal number of the set and/or of its complement may be given in addition. Each of these methods of specification can be made precise by using the theory of partial recursive functions. However, we shall here give only two explicit examples and shall point out later why we consider them to be sufficient for our purposes. DEFINITIONS.