ABSTRACT
9.1. Finite and enumerable classes. A short way of writing that a one-one relation is possible between any two classes x and y is ‘x≈y’ If x≈y, then by a definition of Cantor’s (1845-1918), x and y are said to have the same cardinal number or the same power, the sign for this fact being the equation‘https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780367854218/12ada8dd-5a4b-4fab-a7c8-92ab2453559a/content/pg77.tif"/>’. (This definition is clearly like Frege’s definition of ‘cardinal number’; but Cantor does not give a definition of ‘cardinal number’, only of ‘having the same cardinal number’. Cantor and Frege were working at the same time, but quite separately; one was not copying the other at all.) If Nm is the class of the ‘natural’ numbers from 1 through m, so the elements of it are 1, 2, 3,…, m—1, m, then a class x is finite if and only if it has no elements or, for some natural number m, x≈Nm. If N is the class of all the natural numbers, its elements being 1, 2, 3, … where ‘…’is used for marking that the list goes on for ever, then a class x is said to be enumerable if and only if x≈N. It is strange how great a number of quite different classes are enumerable. For example, Cantor gave demonstrations that the class Z of all the numbers 0, ±1, ±2, ±3, … is enumerable; the class of all the numbers p\q greater than 0 and less than 1, where p and q are elements of N, is enumerable; but what to the mind’s eye would be seen as a very much greater class, the class Q of all the numbers p/q, where p and q are elements of Z (q≠0), is again enumerable; and, further, the class of all numbers which are roots of an equation of the form a n+an-1wn-1+…+a1w+a0. where each a j is an element of Q, is enumerable.
