Let's first think about the object that we are counting: dogs.
Dogs cannot be counted in fractions.
Since you cannot have 1/2 dog or 4/5 dogs, then we should forget about the rational and irrational numbers.
Also, we cannot have -1 dogs or any negative number of dogs (or at least if you owe someone a dog but this is not the case), therefore, the correct set would be the Natural set of numbers
