COUNTING THE RATIONAL NUMBERS

To show that a set is countable, you need to set up some kind of a scheme so that anyone can see which member of the set is the first, the second, the third, etc.. This puts the elements of the set in a one-to-one correspondence with the counting numbers, and thereby proves that the cardinality of the set is equal to aleph null (À₀).

1) Here's a possible scheme for counting all the positive rational numbers:

1 1 2 1 2 3 1 2 3 4 ...
1 2 1 3 2 1 4 3 2 1

I'm not attempting to put them in order of size. Figure out what I am doing to list them, and use the same method to produce the next ten numbers in the list.

2) You can see that two-thirds is the eighth term in the list. When will seven-ninths occur?

3) Try to generalize and predict when any rational number a/b will occur.

4) Assume the list can be continued indefinitely in the same fashion. Will it be comprehensive? Will it eventually capture every positive rational number without forgetting any?

5) You may have noticed by now that the list contains some duplication. For instance, it has both two-thirds and also four-sixths, which really are the same rational number. Call me over to discuss if we think that's a problem or not in establishing the countability of an infinite set.

6) Of course, the true set of rational numbers also includes negative values (and zero, too). Can the scheme above be modified to accommodate all of the rational numbers?

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