 We can go further and define the set \(\mathbf{N}\) of natural numbers. 0 is a natural number and if \(A\) is a natural number, so is \(A + 1\). If a set \(S\) contains 0 and is closed under successor, it will contain all natural numbers (this is one form of the principle of mathematical induction). Define \(\mathbf{N}\) as the intersection of all sets \(I\) which contain 0 and contain \(A + 1\) whenever \(A\) is in \(I\) and \(A + 1\) exists. One might doubt that there is any inductive set, but consider the set \(V\) of all \(x\) such that \(x = x\) (the universe). There is a formal possibility that \(V\) itself is finite, in which case there would be a last natural number \(\{V\}\); one usually assumes an Axiom of Infinity to rule out such possibilities.