Zeno’s Paradox

Some infinities are bigger than others.

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A paradox to motion:

Zeno described a paradox of motion, which helps describes the one type of many infinities. Zeno’s paradox is described below (Stanford Encyclopedia of Philosophy, 2010):

“Imagine Achilles chasing a tortoise, and suppose that Achilles is running at 1 m/s, that the tortoise is crawling at 0.1 m/s and that the tortoise starts out 0.9 m ahead of Achilles. On the face of it Achilles should catch the tortoise after 1s, at a distance of 1m from where he starts (and so 0.1m from where the Tortoise starts). We could break Achilles’ motion up … as follows: before Achilles can catch the tortoise he must reach the point where the tortoise started. But in the time he takes to do this the tortoise crawls a little further forward. So next Achilles must reach this new point. But in the time it takes Achilles to achieve this the tortoise crawls forward a tiny bit further. And so on to infinity: every time that Achilles reaches the place where the tortoise was, the tortoise has had enough time to get a little bit further, and so Achilles has another run to make, and so Achilles has in infinite number of finite catch-ups to do before he can catch the tortoise, and so, Zeno concludes, he never catches the tortoise.”

This paradox was used to illustrate that not all infinities are the same, and one infinity can indeed be bigger than another.  An interpretation of this paradox was written poetically in a eulogy for the book of The Fault in Our Stars (Green, 2012):

“There is an infinite between 0 and 1. There’s .1 and .12 and .112 and an infinite collection of others. Of course there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities. … There are days, many days of them, when I resent the size of my unbounded set. I want more numbers than I’m likely to get, and God, I want more numbers for Augustus Waters than he got. But, Gus, my love, I cannot tell you how thankful I am for our little infinity. I wouldn’t trade it for the world. You have me a forever within the numbered days, and I’m grateful.” (pg. 259-260)

So to my readers out there, I want to thank you in advance for the little infinity(ies) I will get to share with each of you through this blog, and for that I am grateful.

Resources:

  • Green, J. (2012). The fault in our stars.  New York, New York: Penguin Group (USA) Inc.
  • Stanford Encyclopedia of Philosophy (2010). Zeno’s Paradoxes. Retrieved from http://plato.stanford.edu/entries/paradox-zeno/#AchTor

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