 The system of PM differed significantly from Frege’s system, in a large part because of the introduction of the theory of types whose purpose was to avoid the paradox that had affected Frege in a principled fashion. A second important difference from Frege’s system is that PM is based on a logic of relations of various numbers of arguments, whereas Frege’s system was based on the notions of function and object, with even his distinctively logical concepts being seen as functions (from a number of objects to the truth values T and F, which are also objects in Frege’s system.) So, it might be said that PM is based on a theory of ramified types of relations, in contrast to Frege’s second order predicate calculus with concepts. The most important step is to define set expressions in terms of higher-order functions. Thus the paradoxical “Russell set”, the set of all sets which are not members of themselves, \(\{ x \mid x \notin x\}\), is defined by an expression involving functions that will violate the theory of types. The expression for the offending class is ruled out on the basis of the theory of types, as is its seeming innocuous complement, the set of all sets that are members of themselves, \(\{ x \mid x \in x\}\). In contemporary set theory \(\{ x \mid x \notin x\}\) is the universe of sets, which is not itself a set, and because no set is an element of itself, \(\{ x \mid x \in x\}\) is just the empty set. An additional cost of this method is that while for Frege sets are objects of the lowest types, there will be sets in the PM theory in a simple theory of types, which distinguishes individuals and sets of individuals and sets of sets of individuals, etc. Even to derive a hierarchy of sets in the simple theory the axiom of reducibility is needed to guarantee that more complex “impredicative” definitions pick out sets of the same simple type. Thus the “least upper bound” of a closed interval of real numbers will identify a member of that set of a higher order in the ramified theory. That this least upper bound will be of the same simple type requires the axiom of reducibily. The cost of adopting the theory of types to avoid the paradox extends to difficulties in constructing the natural numbers. While Russell follows Frege in many important details, in particular in using Frege’s notion of the ancestral of the successor relation to define the natural numbers, other parts of the construction are importantly different. Frege was able to define the successor of a number by using the set of its predecessors.  The number 2 is the set containing 0 and 1, and thus it has two members. They will, however, be of different types in the hierarchy of simple types, and so the whole set of natural numbers cannot be defined within the theory of simple types. Since each step from 0 to 1, to 2, etc, raises the simple types from 0 to 1 to 2, there will be no simple type of all the natural numbers, so defined. Instead PM adopts the axiom of infinity which assures the existence of an infinite number of individuals, allowing for the construction of the natural numbers for each type above a lower bound of 3 or so (as numbers will be sets of equinumerous sets of individuals…).