David Peat

On Incompleteness 31How Do We Count?It required the extremes of brainwashing to convince Winston Smiththat two plus two equals five. But when you take a second look, iscounting all that simple after all? We know how to count, but are wereally sure what it means to count? How, for example, do you count thenumber of all the numbers or the number of all the fractions? Betweenany two integers, say 3 and 4, can be found a series of fractions—3 1 , 23 3 , 3 5 , 3 11 , 3 99 , and so on. If you think about it, it becomes clear that4 8 12 100between 3 and 4 can be placed an infinite number of fractions. Likewisethere are an infinite number of fractions between 0 and 1, aninfinite number between 1 and 2, 2 and 3, and so on. Common sensetells us that, since you can put an infinite number of fractions betweenany two integers, the number of fractions must be vastly greater thanthe number of integers.But here, mathematicians are happy to tell us, common sense iswrong. The number of all possible fractions is exactly the same as thenumber of all possible integers! How can this be true? To find out we’llhave to further explore the world of counting. John and Jill each have abag of candies and, as children will, they argue about who has more.However, they are so young that once they count past five candies theyget confused as to what comes next. They decide to solve the problemin another way. John takes a candy out of his bag and Jill lays one ofhers beside it. Then John takes another candy and Jill matches it. Theycontinue in this way until one of the bags is empty. It turns out that,when John’s bag is empty, Jill still has some candies left in her bag.Then, even though Jill cannot count, she knows that she must havemore candies than John. On the other hand, if Jill’s bag had emptiedfirst, then she must have had fewer candies than John. And if both bagsempty at the same time then they know they have an equal number ofsweets—even though they do not know what the value of that numberhappens to be.The same thing happens with the number of fractions and thenumber of numbers. Take a fraction and put it down on the table. Nowmatch this with the number 1. The next fraction is matched up with 2,3, 4, 5, and so on. Because the number of integers is infinite they will