A nonnegative integer is called a number, a collection of numbers a set and a collection of sets a class. We write ε for the set of all numbers, o for the empty set, N(α) for the cardinality of α, ⊂ for inclusion and ⊂+ for proper inclusion. Let α, β 1,…, β k be subsets of some set υ. Then α′ stands for υ−α and β 1 … β k for β 1 ∩ … ∩ β k . For subsets α 1, …, α r of υ we write:

Note that α 0 = υ, hence s 0 = N(υ). If the set υ is finite, the classical inclusion-exclusion principle (abbreviated IEP) states

In this paper we generalize (a) and(b) to the case where α 1, …, α r are subsets of some countable but isolated set υ. Then the role of the cardinality N(α) of the set α is played by the RET (recursive equivalence type) Req α of α. These generalizations of (a) and (b) are proved in §3. Since they involve recursive distinctness, this notion is discussed in §2. In §4 we consider a natural extension of “the sum of the elements of a finite set σ” to the case where σ is countable. §5 deals with valuations, i.e., certain mappings μ from classes of isolated sets into the collection Λ of all isols which permit us to further generalize IEP by substituting μ(α) for Req α.