Class 5: Introducing Infinity

In class today,

  1. We started by talking about infinity and the different images and ideas that come up when we talk about the infinite. Then we watched a Vi Hart video “Infinity Elephants” that provided an introduction to some important ideas, while looking back at fractals. Particularly, Vi mentioned that there were different sizes of infinity which is what our next few classes will explore. Also, if you like apollonian gaskets like the ones in the video and you want to know more about the math, here is a link to start with.
  2. Next, I talked about some of the weird things that happen when we start thinking and talking about infinity. You want re-watch the Hilbert Hotel problem we worked o in class here, and you can watch a another cute animated video about it here. You can read a New York Times Opinionator piece about it as well.
  3. After tackling the Hilbert Hotel, I defined what it means for two sets (a set being a collection of things) to be the same size.  If two sets can be paired evenly, with pairs featuring one object from the first set one from the second, with no leftovers, then the two collections are in “one-to-one correspondence.” We say that two sets are the same size if we can create a one-to-one correspondence between the sets.Note that if we try to put two collections into one-to-one correspondence, we can decide if the collections are the same size without even knowing how many objects are in each collection.Children learn to count using one-to-one correspondence and it is an important early numeracy activity.
  4. A collection of objects (a set) is finite if it can be put in one-to-one correspondence with a set {1,2,3,4,…, n} for some n – that number n tells us how big the set is.The easy way to say this is that a set is finite if you can count it! A set is infinite if it is not finite.
  5. Infinite sets have this really weird feature that they can be put in one-to-one correspondence with a proper subset of themselves. A proper subset is just a part of a set that contains something but not the whole set. So given the set {1,2,3,4,5,6}, one proper subset is {1,2,3,4}. Note that we can never put those in one-to-one correspondence, but the infinite set {1,2,3,4,….} can be put into one-to-one correspondence with the set of even numbers {2,4,6,8,10, …}.
  6. We learned just a little about Georg Cantor, a mathematician that explored and discovered many of our modern mathematical ideas about the infinite. Cantor discovered that there are diffferent sizes of infinity. I found a better video for this that doesn’t cut people’s heads off. We will watch more of Cantor’s story as we explore his ideas about infinity.
  7. Finally, we talked briefly about different kinds of numbers, in particular irrational numbers, which so disturbed Pythagoras. They’ll come up more as we go in search of a larger infinity.

For next week, there’s a more concrete exploration of infinity for you to do before class. We will start class with talking about your explorations.


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