 Define 0 as the set whose only element is the empty set. Let \(A\) be any set; define \(A + 1\) as the collection of all sets \(a \cup \{x\}\) where \(a \in A\) and \(x \not\in a\) (all sets obtained by adding a new element to an element of \(A)\). Then \(0 + 1\) is clearly the set we want to understand as \(1, 1 + 1\) is the set we want to understand as \(2, 2 + 1\) is the set we want to understand as 3, and so forth.