- Your fractal assignment is due this Friday, as is your funhouse mirror image (after which judging will take place): make your
funhouse mirror image using my web
interface and your own image, and I'll post them
on our "website
gallery".
- Hope you've read Idea #7: Infinity (p. 28).
- Question of the day: "What set could be larger than the natural numbers?"
- What does this image have to do with our topic?
- Alaina came up with something similar:
- What does this line from Amazing Grace's sixth stanza have to do with our class?
When we've been here ten thousand years...
bright shining as the sun.
We've no less days to sing God's praise...
than when we've first begun.
John Newton 1725-1807; ironically Newton was a slave trader (who ultimately saw the error of his ways).
- What we've learned so far:
- Infinity is NaN: Not a Number
- Infinity is NaS: Not a Size, either! The reason is that 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!;)
- The natural numbers are infinite in number (
).
- Cardinality
- Two sets have the same cardinality if there is a one-to-one correspondence between them.
- A sensible rule you can be sure of: 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!
- Infinity is NaN: Not a Number
- Let's look at some curious problems about the Motel Infinity:
- Welcome to the Motel Infinity
- Motel Infinity and the Infinite Schoolbus
- Motel Infinity and the Limitless Stable of SchoolBusses
- Are the rational numbers a bigger infinity than the natural
numbers? (The natural numbers are a subset of the rational numbers.)
- The real numbers are bigger than the natural numbers, and we can
prove it -- by contradiction.
- We'll assume that they're not bigger -- that there is a one-to-one correspondence -- then show that that is impossible.
- We'll need this fact: every real number can be expressed by an infinite decimal.
- We need to be aware, however, that some real numbers are expressed by more than one decimal (for example, 1=.99999.... -- did you know that?)
- We'll show that just the real numbers between 0 and 1 are too numerous to be put in one-to-one correspondence with the natural numbers.
- We show this using a procedure that we might call "dodge ball".
- The power set: showers of infinities!
There are infinitely many sizes of infinity. It turns out that the power set of a set is always of larger cardinality than the set itself. Thus every infinite set is smaller than its power set, which is an infinite set, which is smaller than its power set, etc....
- We'll end on a really strange note: The ping-pong ball conundrum (which is related to other paradoxical ideas, such as Zeno's paradoxes).