Why does an intersection of all inductive sets comprise natural numbers?

Question

I suppose we cannot say "Let natural numbers be $1, 1+1, 1+1+1, \dots$", but I fail to put into words why we can't. What's the reason to define natural numbers as an intersection of inductive sets? If we could define it as I wrote above, we wouldn't be troubled with questions like "Is there a natural number between $1$ and $2$?"