Predicativity

Picking an object from several already existing objects is a very different process than building an object to specifications.

The key point is this. If you want an object to have a certain property with respect to all objects, then finding it doesn't change what properties it has. But building it changes the class of all objects and hence may not preserve a property with respect to all objects.

Example: (von Neumann's) a number is inductive if it has all the properties that 0 has that are "hereditary" (that is, such that if has the property, then so does .

The notion of an inductive number does not immediately require that the number have a property with respect to all numbers, but it does require that the number have a property with respect to all properties of numbers, and what numbers there are will change what properties of numbers there are.

There is no problem about determining whether a number is inductive if the properties of numbers already exist. If they don't, the definition is circular.

We use definitions in (at least) two ways: to pick something out among known things and to say what it is we are talking about in the first place.

Properties like inductive are called impredicative properties. If we use axioms to identify already existing systems, they are fine. However, if we use axioms in the manner in which they have been conceived in the Euclidean tradition to say what our subject matter is in the first place, they pose a real problem.

This is the problem Russell encountered: the axiom of reducibility is used precisely to turn impredicative definitions into predicative ones.

-- ProfessorShaughanLavine? - 02 Feb 2005