Day 26, MAT115

• Happy Eclipse Day+1! Was that fun or what?!

• Your quiz Thursday will be over distinguishing knots and links: tricolorability and the Reidemeister moves.

  Before we get back to infinity, let's take a look at your homework, the Knotty Tale.

• You have a new reading assignment over infinity.

• We are having another art contest! Here is the (juried) gallery, and we'll have our vote on Thursday. (All entries)

• Reminder: we're less than two weeks from our project and logo demonstrations (the last week of class).

• We went over the quiz.
• We got started on a really big topic: Infinity....

• We talked about its connection to eternity: what's another ten thousand years when you have an eternity?

• Learned that it's not a number, and tried to figure out a better way to trash talk on the playground.

• Infinity is also not a Size, either, but that's because there is more than one size of infinity! This is the most important thing I want you to remember. (This fact will allow you to win on the playground!;)

• Examined two of Zeno's paradoxes:
  • Achilles and the Tortoise (the Tortoise beats Achilles.)
    

  • The Dichotomy paradox (Motion is impossible.)
    

• Defined Cardinality It is the way that we define the size of sets (finite and infinite):

  Two sets have the same cardinality if there is a one-to-one correspondence between them.

  Intuitively: We will say that two sets have the same size if they have the same cardinality.

• Started in on Hilbert's Hotel, and found ways to accomodate
  • One additional guest couple
  • One infinite school bus
  • An infinite number of infinite school buses
  But we couldn't accomodate a real-numbered bus, as we showed with Cantor's diagonization argument.

Question of the day: Which infinity was Buzz Lightyear headed to (and beyond)?

• One more interesting connection to our work here today, this time from Lewis Carroll:

  "Alice laughed. 'There's no use trying,' she said. 'One can't believe impossible things.'

  I daresay you haven't had much practice,' said the Queen. 'When I was your age, I always did it for half-an-hour a day. Why, sometimes I've believed as many as six impossible things before breakfast."

• Infinity! A really big idea.... maybe impossibly big. It might even "blow your mind", as it may have for Georg Cantor, the mathematician who introduced us to many of the ideas we talk about today.

• The natural numbers are infinite in number (the size of the set is larger than any natural number). It is called a countable infinity (or denumerable), because, well, we can start counting it! (We just never get to the end -- it will take us an eternity).

• Let's review our results from last time (handout):

  In particular I want to share two other ways to think about dealing with the case of an infinite number of school buses.

  • Assigning bus passengers to a unique powers of primes, by bus
  • Assigning bus passengers to unique powers of 2 and 3: \(2^b3^s\)
  • Assigning from a graph.
  What this shows, in fact, is that the rational numbers (ratios of natural numbers, and the negatives of those) are the same size (have the same cardinality) as the natural numbers.

• Here's the upshot of what Cantor discovered:
  1. Some mathematical facts:

     Mathematical fact zero:

     A set S is just a collection of objects; and a subset of S is just what you'd expect: some members of S. It may be all the members, in which we call it an "improper" subset, or it may be none of them (in which case we call it the empty set).

     We often denote a set by using braces, e.g. \(S=\{1,2,3\}\) is the set of the first three natural numbers.

     We say that \(a\) is an element of \(S\) if \(a\) is contained in \(S\), and we write \(a \in S\). So \(1 \in S\), \(2 \in S\), and \(3 \in S\). We deny that an object is in \(S\) this way: \(4 \notin S\).

     Mathematical fact one:

     A subset is never bigger in size than the set itself.

     And if the sets are finite, the proper subset is always smaller, but if the set is infinite, we may actually be able to throw away elements of a set and not change the size of the set!

     Mathematical fact two:

     Let's call the set of all of a set S's subsets its power set, \(P(S)\). Pascal's triangle (that rascal!) shows you how many of each type of subset you have (for finite sets).

     

     

     Mathematical fact three:

     The power set of a set of $n$ elements contains $2^n$ elements.

     (We know that since each row of Pascal's triangle adds to a power of 2.)

  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!

     Here's a silly video to illustrate how the power set grows with sets of increasing size. (Thanks to Dr. Towanna Roller (Asbury University) and her daughter Kristyn Roller (UK) for this one!)

  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).

     And the power set of that set is bigger yet, and so on forever, forever, Hallelujah, Hallelujah!

  4. Upshot: infinity comes in various sizes! Infinitely many, in fact. Crazy sounding, I know -- but it's true.

     That symbol that you've been familiar with for all your lives, $\infty$: you thought it stood for a single thing; but it stands for a whole collection of monstrously big things, all too big to really think about properly. (Well, Cantor did!:)

  5. That's too crazy, it seems; but, nonetheless, mathematicians know that it's true, because we can prove it! The key to determining if two sets have the same size is the one-to-one correspondence -- and that is key for infinite sets, as well.

  6. So, in the end, here's what you say on the playground, when you want to love your friend more:

     "I love you more than the power set of your set of infinite love."

     

     Amen!