How many distinct positive integers can be represented as the difference of two numbers in the set $\{1, 3, 5, 7, 9, 11, 13\}$?
Since all of the integers are odd, the differences between any pair of them is always even. So, $13 - 1 = 12$ is the largest even integer that could be one of the differences. The smallest positive (even) integer that can be a difference is 2. So, the integers include 2, 4, 6, 8, 10 and 12, for a total of $\boxed{6}$ integers.