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 the supremum is taken over all finite subsets 
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.
10 June 2009 at 9:15 pm
Professor Sloughter,
Can you shed some light on why you have such a wonderful nonstandard calculus text but an analysis text that uses epsilon delta proofs, open and closed sets, etc.? Is there no way to extend nonstandard notions beyond basic calculus?
10 June 2009 at 9:49 pm
All the results of standard analysis may be derived in nonstandard analysis as well. In fact, there are a number of nice nonstandard analysis texts, Robert Goldblatt’s Lectures on the Hyperreals being one of my favorites.
Your question is a good one, and one I have been giving some thought to recently. Namely, how much should one’s philosophy of mathematics influence one’s teaching of mathematics? In writing my analysis book, I was writing a book for use in the undergraduate beginning analysis course I taught last fall. Given that this was the first (and, for many, the last) course that these students would see in analysis, I felt it should be a standard epsilon-delta treatment of the subject. If nonstandard methods make more inroads into the mainstream curriculum, I could see myself changing my thoughts on this sometime in the future.