Combinatorial Games (Part II): Different Moves for Left and Right
4. Counting and sets
Counting finite collections of objects is something we teach children, but the transition to counting infinite collections of objects requires a surprising amount of sophistication. The German mathematician Georg Cantor developed ideas about counting infinite collections into a theory.
What is unexpected here is that the usual "rules of arithmetic" do not work! For example 2 + ω is not ω + 2. To see this, if we have to count beans where first there are two beans, and then ω beans (think of the situation visually as b, b, (0, 1, 2, ....) beans, then we would count 0 for the first b, 1 for the second, 2 for the next bean, 3 for the next, etc. The result would be ω beans. However, for the beans lined up (0, 1, 2, ....), b, b we would count 0, 1, ..., ω, ω +1 and report that we had counted ω + 2 beans. Remember that when we count four beans starting with 1 we count 1, 2, 3, and 4 and the number of beans is the last number we state. However, when we count 4 beans starting at 0 we get 0, 1, 2, and 3, so we report the number of beans as 1 more than the last number we count. Thus, above, we would have ω + 2 beans because the last bean we called out was ω + 1. Suffice it to say that in the ordinal number world Cantor pioneered, one is able to give meaning to the product and exponentiation of ordinals as well as to "limits" of such ordered counting of beans.
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