Abstract

In spite of the fact that the Z.F. universe is not well-ordered, it behaves in some respects like the ordinals. It is possible to define on it the usual operations of addition, multiplication and exponentiation, which enjoy similar properties to those on the ordinals. Further when restricted to the ordinals, the operations coincide, so that ordinal arithmetic can be regarded as a restriction of the universe arithmetic. But more than that, rank which retracts the universe of sets onto the ordinals is a homomorphism between the universe arithmetical structure 〈V,+,·,exp〉 and the ordinal arithmetical structure 〈Ord, +, ·,exp〉