Another crazy question.

Is this a countable set; $$A = \{-4, -2, -1, 0, 1, 2, 3, 4\}$$

I think it is. But my teacher says it isn’t due to the concept , he says, is of one to one mapping with a some subset of natural number. Since some elements there contain non - $\Bbb N$ elements.

From what I know, a countable set is either a finite set, or a set with the same cardinality with some subset of $\mathbb N$, which make me think it is.