ABSTRACT
This chapter explains the theory of transfinite or infinite cardinal numbers as it results from a combination of his discoveries with those of Frege on the logical theory of numbers. The assumption that there are is what we call the “axiom of infinity.” In the first place, a cardinal number is a set of classes which are all similar to each other and are not similar to anything except each other. The most noteworthy and astonishing difference between an inductive number and this new number is that this new number is unchanged by adding 1 or subtracting 1 or doubling or halving or any of a number of other operations which we think of as necessarily making a number larger or smaller. One of the most striking instances of a “reflexion” is Royce’s illustration of the map: he imagines it decided to make a map of England upon a part of the surface of England.
