Homework for sections 3.2 and 3.3

In discussing these sections we had two main results:

1 There are many sets with the same “cardinality” as the counting numbers. I.e. There are many sets that can be placed in 1-1 correspondence with the counting numbers.

2 But the “decimal numbers” can not be placed in 1-1 correspondence with the counting numbers.

Your assignment is to write an essay on these two sections covering the following topics:

· Surprising examples illustrating item 1. from above. To get full credit, you need to give some examples not done in class and some of the “harder” examples. There are lots of ideas for this in the exercises at the end of section 3.2.

· Some ways in which infinite sets differ from finite sets.

· Some ways in which mathematicians' thinking differs from other scientists. Though we haven't discussed this very much, the whole discussion of the infinite gives us a good example of this.

· Cantor showed that any attempt to correspond the “decimal numbers” with the counting numbers must fail. Thus Cantor showed that there is a cardinality larger than that of the counting numbers. Use problems 8 and/or 9 from page 169 to show you really understand this proof by giving Cantor's proof in your own words using different numbers than I used in class and different numbers than the text uses in their version (a tiny bit different than mine).

Extra credit problem:

Suppose we had an unordered list of all of the counting numbers, such as:

1. 17654321

2. 734

3. 89876

4. 912345

.

.

.

Then suppose you tried to apply the same argument as in “Cantor's Diagonal Proof” hoping to reach some sort of contradiction. What would go wrong? I.e. Why would the argument fail?