 Cantor noted that for any two well-ordered sets, the initial positions in one ordering (the first, the second, the third, etc.) correspond to the initial positions in the other, the way that they do for finite sets. In fact, he showed that all of the positions of one well-ordering must correspond to initial positions in the other. (If this weren’t true, then the set of positions in one that don’t correspond to positions in the other would be non-empty for each set, and the first elements of these sets would correspond, which would contradict the claim that these positions don’t correspond.) Thus, there is a single list of all the possible positions in well-ordered sets, beginning with the first, second, third, and so on, and these positions are called the ordinal numbers. A well-ordered set can be said to have its own ordinal number, which is the first ordinal number that does not correspond to a position in that set.