Rose,

The problem with using I to stand for the set of integers is that the letter is used in so many other places, particularly to stand for the identity. A capital Z has become the standard among mathematicians -- it comes from the German word Zahl, which means "number." The history page http://jeff560.tripod.com/nth.html suggests that the notation was first used in 1930 both by van der Waerden and by Landau, then was adopted by Bourbaki, after which it became accepted universally.

It has been so long since I last saw "J" used for the set of integers, I wonder if anybody still uses it. My guess: it was used because it is close to I in the alphabet and was available.

Chris