What does it mean for sets to have "the same size"? For finite sets, it's obvious. Even young children who aren't yet experts in arithmetic can quickly tell the difference between a bag containing three candies and another containing as many as there are fingers on one hand. In short, it's enough to count the number of elements (the cardinality) of each set.
But what happens with infinite sets? Relying on counting alone is a method that falls short. By contrast, relating the elements of one set to those of another sheds useful light on the matter.
As Cantor wrote at the head of his 1867 thesis (written in Latin), in mathematics the art of posing questions is more important than that of solving them. Identifying the right questions among those whose answers are trivial was the first stroke of genius in this adventure.
Another way to "count"
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