 But this line of thought can be resisted. First, suppose that the procedure just described completely divides the object into non-overlapping parts. (There is a problem with this supposition that we will see just below.) It involves doubling the number of pieces after every division and so after \(N\) divisions there are \(2^N\) pieces. But it turns out that for any natural or infinite number, \(N\), \(2^N \gt N\), and so the number of (supposed) parts obtained by the infinity of divisions described is an even larger infinity. This result poses no immediate difficulty since, as we mentioned above, infinities come in different sizes. The number of times everything is divided in two is said to be ‘countably infinite’: there is a countable infinity of things in a collection if they can be labeled by the numbers 1, 2, 3, … without remainder on either side. But the number of pieces the infinite division produces is ‘uncountably infinite’, which means that there is no way to label them 1, 2, 3, … without missing some of them—in fact infinitely many of them. However, Cauchy’s definition of an infinite sum only applies to countably infinite series of numbers, and so does not apply to the pieces we are considering. However, we could consider just countably many of them, whose lengths according to Zeno—since he claims they are all equal and non-zero—will sum to an infinite length; the length of all of the pieces could not be less than this.