- Assignment 3 is due today.
- To hand in Wednesday, 2/18: Section 2.2, problems 16-24.
- New Assignments:
- Reading Assignment: for Wednesday: read sections 3.4: Travels Toward the Stratosphere of Infinities
- Problems for Monday, 2/23: Section 3.2, pp. 156-161, Mindscapes #6-8, 14, 15, 20
- Here's one we should all be able to do now: problem 6 from Test 1 of the fall semester, 2008
- The Fibonacci Museum
- Infinity - so mysterious...
Q: Are mathematicians trying to take all the fun out of it by studying it?! A: Of course not! We relish the mind-blowing mystery. And there's plenty of fun to be had....
- William Blake (from Auguries of Innocence):
-
To see a World in a Grain of Sand
And a Heaven in a Wild Flower,
Hold Infinity in the palm of your hand
And Eternity in an hour.
Here's another one, by Robert Graves: Warning to Children
- As we approach infinity, we start with the very idea of number:
What does "2" mean? - What is infinity? Is it a number? Is it just one thing?
- How do you know that two sets of objects have the same
number of objects in them when you're not allowed to
count?
- What is a one-to-one correspondence, and how is it useful?
- Let's look at some problems:
- II.6
- II.11
- IV.22
- The natural numbers are infinite in number (the size of the set is
larger than any natural number).
- Are there more evens or more natural numbers?
- Are there more primes or more natural numbers?
- Are there more Fibonaccis or more natural numbers?
-
- What is Cardinality?
- Two sets have the same cardinality if there is a one-to-one correspondence between them.
-
- 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!
- Let's look at these curious problems (pp. 157):
- 13 - squaring off
- 16 - Hotel Cardinality
- 17 - Hotel Cardinality (cont.)
- 18 - Hotel Cardinality and the Infinite Life Insurance Co.
- Hotel Cardinality and the Limitless Bus Barn Co.
- The ping-pong ball conundrum
- The ping-pong ball conundrum is related to other paradoxical ideas, such as Zeno's paradoxes