  Instead, one should introduce a ramified hierarchy of properties and numbers. At the beginning, one has only first-order inductive properties, which do not refer in their definitions to a totality of properties, and one defines the numbers of order 1 to be the elements satisfying all first-order inductive properties. One can next consider the second-order inductive properties, that can refer to the collection of first-order properties, and the numbers of order 2, that are the elements satisfying the inductive properties of order 2. One can then similarly consider numbers of order 3, and so on. Poincaré emphasizes the fact that a number of order 2 is a fortiori a number of order 1, and more generally, a number of order \(n+1\) is a fortiori a number of order \(n\). We have thus a sequence of more and more restricted properties: inductive properties of order 1, 2, … and a sequence of more and more restricted collections of objects: numbers of order 1, 2, … Also, the property “to be a number of order \(n\)” is itself an inductive property of order \(n+1\). 