Not every counting number is an integer true or false?

False. All counting numbers {1, 2, 3, 4, 5, ... } are integers. The set of integers is { ..., - 3, - 2, - 1, 0, 1, 2, 3, ... } which is basically the set of positive whole numbers, negative whole numbers and 0.

Note: the set of counting numbers is a subset of the set of integers.