ABSTRACT
We now turn to the second group of paradoxes. Consider Figure 0.2. Are there as many apples here as bananas? Or, if you like, does the set of apples have as many members-is it the same size-as the set of bananas? We can see that the answer is yes, because we can see that there are seven of each. But to see this we must count; and counting is itself an operation that presupposes such comparisons of size. To say that there are seven apples is to say that there are as many apples as there are positive whole numbers up to and including seven. (So to count the apples and the bananas is simply to bring a third set into the reckoning.)
We could, however, have answered the question from scratch, without recourse to counting-by pairing the apples and bananas off with one another, in such a way that each apple corresponds to a unique banana and each banana to a unique apple, as shown in Figure 0.3. For it to be possible to pair off the members of two sets with one another in this way seems to be what it is for the two sets to have as many members as each other. Applying this principle to the infinite, however, yields further paradoxes.
