 The question of which parts the division picks out is then the question of which part any given chain picks out; it’s natural to say that a chain picks out the part of the line which is contained in every one of its elements. Consider for instance the chain \(\{[0,1/2], [1/4,1/2], [3/8,1/2], \ldots \}\), in other words the chain that starts with the left half of the line and for which every other element is the right half of the previous one. The half-way point is in every one of the segments in this chain; it’s the right-hand endpoint of each one. But no other point is in all its elements: clearly no point beyond half-way is; and pick any point \(p\) before half-way, if you take right halves of [0,1/2] enough times, the left-hand end of the segment will be to the right of \(p\). Thus the only part of the line that is in all the elements of this chain is the half-way point, and so that is the part of the line picked out by the chain. (In fact, it follows from a postulate of number theory that there is exactly one point that all the members of any such a chain have in common.) The problem is that by parallel reasoning, the half-way point is also picked out by the distinct chain \(\{[1/2,1], [1/2,3/4], [1/2,5/8], \ldots \}\), where each segment after the first is the left half of the preceding one. And so both chains pick out the same piece of the line: the half-way point. And so on for many other pairs of chains. Thus Zeno’s argument, interpreted in terms of a repeated division of all parts into half, doesn’t divide the line into distinct parts. Hence, if we think that objects are composed in the same way as the line, it follows that despite appearances, this version of the argument does not cut objects into parts whose total size we can properly discuss.