Counting the Infinite
In some of the earlier chapters we encountered numbers so large that it was just not feasible to write them down in decimal form, because they would contain more digits than would fill the entire book. Nevertheless, no matter how large a finite number may be, it is always possible to to imagine one still larger, and this process can be continued indefinitely. Colloquially we say “and so on to infinity”. But is it possible to get any precise grasp of this ‘infinity’? We know that an infinite number of items is certainly larger than any finite number, but is there any more which can be said about it? Does it make any sense, for example, to ask whether there are more fractions than integers, or more irrational numbers than rational ones? Clearly there are an infinite number of each, so that we are really asking whether it is possible to find a method for comparing two different infinities to see which is the larger.