Week 2: What is Infinity? Plus Zeno’s Paradoxes

The lemniscate, ∞, in several typefaces.
The lemniscate, ∞, in several typefaces. (Photo credit: Wikipedia)

This week, we talked about the mathematical memories that everyone came up with — thanks to all for the powerful discussion! Then we moved on to

  • One-to-one correspondence. This is a fancy name for something more basic than counting — matching up two collections to decide whether or not they are the same size. See this video: http://ed.ted.com/lessons/how-big-is-infinity, which also introduces lots of other ideas about infinity.
  • Sets, subsets, and proper subsets. A “set” is a collection of things. If we take a set, and we make a group of just some of the things, that’s a “subset.” If we make sure that group we make doesn’t contain everything in the original set, then that’s a “proper subset.” So, for instance, in the set that contains all of the letters of the alphabet, one example of a proper subset is all of the consonants. Another would be the set {a, b, c, d}. There are lots of proper subsets of the alphabet!
  • Mathematicians say that a set is “infinite” if we can put it in one-to-one correspondence with a proper subset of itself. Whoa. More talk about this in class. We also did some practice with these ideas in class.
  • More strangeness happens once we start to think about infinity. See this video about the paradoxes of the philosopher Zeno of Elea: https://www.youtube.com/watch?v=MbNNFtuwA0k
  • If there’s enough time, in Wednesday’s class we’ll do a little coloring exercise that relates to Zeno’s dichotomy paradox.
  • Here we have an exploration of Zeno’s dichotomy paradox through film, depth, and movement: https://vimeo.com/13790985

Homework due Monday. A short mathematical autobiography and two goals for this course! The assignment details are posted here.

 

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