In one version of Zeno’s arrow paradox, the paradox is that the distance traveled by the arrow is supposed to be the sum of the lengths of all the points it passes through. But if a point has no length, then the arrow has not moved at all: the sum of an infinite number of zeros is still zero.

In the paper “A Consistent Conception of the Extended Linear Continuum as an Aggregate of Unextended Elements,” (Philosophy of Science, Vol. 19, No. 4. (Oct., 1952), pp. 288-306), Adolf Grünbaum contends that Zeno’s problem lies in his ignorance of the issues of cardinality. Specifically, Zeno did not know that there are an uncountable number of points in any line segment. And, Grübaum contends, although mathematicians have devised ways to add up a countable number of number, there is no way to add up an uncountable number of numbers. Hence Zeno’s mistake was in assuming that the all the zeros could indeed be added up.

However, Grünbaum was mistaken. Even within standard mathematics, there is a way to describe the sum of a set of numbers of any cardinality: if , where is of any cardinality, then one may define

,

where the supremum is taken over all finite subsets of . With this definition, Zeno is correct: the sum of even an uncountable number of zeros is zero.

One response is to say that Zeno is still wrong because Lebesque measure is only countably additive, not uncountably additive. But, as far as I can tell, the only reason measures are required to be at most only countably additive is to bar Zeno’s paradox. That is, this argument against Zeno is nothing but a petitio principii.

It seems to me that the way around this version of Zeno’s arrow paradox is to understand that we are not adding up an infinite number of zeros, we are adding up an infinite number of infinitesimals. And that sum need not be zero.