Day 27, MAT115

Last Time

Next Time

Announcements:
- Last time we moved from fractals to infinity. Our first formal encounter with infinity was rather surprising observation that it is possible to have a curve of infinite length in a space of finite extent.
  But an even more remarkable and astonishing result awaits, and we'll get started on it today.
- You have a final reading from Strogatz's book: The Hilbert Hotel (p. 249). The book is subtitled "from One to Infinity", and today we're getting to infinity!
- Homework(s) graded:
  - A knotty problem
  - Distinguishing Knots (key thanks to Emilys K. and B.) and and a video.
- Materials for today:
  1. A video covering today's material: The Hilbert Hotel
  2. My Hilbert Hotel scribbles
Let's review what we learned about last time:
1. making fractals out of almost anything (A knot fractal and an infinity fractal).
2. Fractals leading to infinity (a focus on the Koch fractal one last time)
  - The Koch fractal grows in length each time: the generator is 4/3 times as long as the initiator.
  - This means that the fractal grows by a factor of 4/3 with each step.
  - If the initial stick has length 1, then the first iteration of the fractal has length 4/3. The second iteration has length $(4/3)^2$, the third $(4/3)^3$, and so on.
  - This is an example of exponential growth. If we compute the length of the Koch fractal after 10 iterations, it will have length \[ L = \left(\frac{4}{3}\right)^{10} \approx 18 \]
  - If we do it forever, the Koch fractal has length $\infty$! It's infinitely long, yet lives in a finite space.
    This is only our very first example of something really mysterious about infinity....
1. Today: we want to focus on another strange fact about infinity: that when you get to sets that are infinitely sized, you find that their subsets can be exactly the same size as the set itself. This is very strange, and doesn't work with finite sets. So it's a very mysterious property.
  We'll illustrate with something called "The Hilbert Hotel"
  1. First of all, the Power Set of a set is the set of all subsets; and Pascal's triangle shows you how many of each type of subset you have (for finite sets).
  2. Even the set containing no elements has one subset -- the set itself. So the power set has one element -- has size bigger than the set itself.
    This property holds true for all finite sets -- and it turns out to be true for infinite sets, too!
  3. This means that, although the natural numbers $1,2,3,\ldots$ is infinite as a set, there's a bigger set (the power set of the natural numbers). Upshot: infinity comes in various sizes!
  4. That's too crazy, it seems; but, nonetheless, mathematicians know that it's true. The key to determining if two sets have the same size is the one-to-one correspondence -- and that will be key for infinite sets, as well.
  5. So we now begin considering "The Hilbert Hotel". The Hilbert Hotel has natural numbered hotel rooms -- there's an infinite number of rooms, labelled 1, 2, 3, 4, .... for every natural number.
    1. The place is packed, and one person shows up;
    2. The place is packed, and an infinite school bus shows up;
    3. The place is empty, and an infinite number of infinite school buses show up!
2. Links
  - Nothing to report.

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