Cantor asks us to think about the following. Suppose we have any infinite list L of numbers a1, a2,…thought of as being given in decimal form. Then it is possible to write down another number,a, that does not appear anywhere in the list L:we simply take a to be different from a1 in the first place after the decimal point,different from a2 in the second decimal place, different from a3 in the third decimal place, and so on–in this way, we maybuild our number a making sure it is not equal to any number in the list.This observation looks innocuous but it has the immediate consequence that it is absolutely impossible for the list L to contain all numbers, because the number a will be missing from L. It follows that the set of all real numbers, that is all decimal expansions, cannot be written in a list, or in other words cannotbe put into a one-to-one correspondence with the natural counting numbers, the way we saw in Figure 8. This line of reasoning is known as Cantor s DiagonalArgument, as the number a that lies outside the set L is constructed by imagining a list of the decimal displays of L as in Figure 9 and defining a in terms of the diagonal of the array.