Larry Gonick’s cartoon guide to Calculus February 6, 2012Posted by larry in Logic.
If you go to the link below, and then click on the book image, you can look inside the book. Then go to the discussion of Zeno’s motion paradox. He represents Zeno’s paradox well, but the solution isn’t quite what he suggests it is. Newton and Leibnitz got around Zeno’s motion paradox but Gonick doesn’t quite say what the error in Zeno’s argument was.
The solution to Zeno’s paradox is to realize that there is no such thing as a next instant. Between any two given instants, there is a nondenumerable (uncountable) number of instants. Think of the real number system as a time line. Between any two numbers, or instants of time, however close together you pick them, there are an infinite number of numbers, or instants of time, between them. Hence, no next instant or number.
This is in contrast to the set of integers where there is a next number – this set of numbers is countably infinite. The real line is uncountably infinite, otherwise referred to as the continuum. There are at least two kinds of infinite sets, those that are denumerable (even = odd = rationals in size) and those that are nondenumerable (reals = irrationals in size).
Newton and Leibnitz got around Zeno’s paradox by conceiving of infinitesimals in their independent developments of the differential calculus. The notation dy/dx indicates an infinitesimal change of y and x with respect to one another. A differentiable function is deemed to be continuous, which means that we are dealing with the real line, that is, the real number system. Hence, on a line of a graph, there is no next point, just as there is no next instant of time.
To assume that there was a next instant was Zeno’s error. So, in Zeno’s case, an arrow travels through an uncountably infinite numbers of instants on its way from a to b, no one instant of which can be claimed to be next to another.
I hope that is clear. Whew!
The mathematical theory of infinitesimals wasn’t formally resolved until the latter half of the 20th century by Abraham Robinson.