AFAIK 0 is not a counting number. BUT, it is an Integer.

Because the set of Integers are { ... -2, -1, 0, 1 , 2 ...}
and counting numbers only include { 1, 2, 3, ...}

BUT then again, standards do change and I see in Wolfram that some mathematicians do include Zero as counting numbers.
http://mathworld.wolfram.com/CountingNumber.html