Area paradox Math Less Traveled
Area paradox triangle problem: here is another link with hints and discussion
Both of these have connections to Fibonacci numbers: that is not an accident.
Topics:
• Proving that the power set of a set S is always strictly larger in size (cardinality) than the original set S, even when S is an infinite set. This is usually called Cantor's Theorem.
We do not exactly do the proof, but rather we indicate how the proof is done. The problem is that the set we start with may be uncountable, so we cannot assume that we can list its elements. If you are interested to see how an actual proof appears, here is a fairly accessible one (The proof is at the end of the article: but be warned that they use some notation and terms that may be new to you!)
The Wikipedia article on Cantor's Theorem contains a description of how the proof would work if the original set is countable (listable), specifically if the original set is the set of natural numbers. In order to make the proof more general, you should notice that it does not depend on being able to list all of the elements of the one-to-one correspondence: that is only done for convenience (in order to be very concrete). The center of the proof is the rule we use for deciding whether or not each element of the original set should be included in the new set we construct, and that rule can be used whether or not we can list all of the elements of the original set. We just have to give up the idea that we can think of using the rule one-by-one on the elements, and think of using it on all of them "at the same time" so to speak.
I have written up rough notes on the proof as I described it in class which are available here. They are unedited, so please let me know if you find anything that looks wrong or any typographical or other errors!
Next we returned to looking at the sizes of power sets. We already found the following result: For a finite set S, if |S| = n, then the size of the power set of S will be 2n. This is why it is called the power set!
The same thing holds if S is an infinite set, except that then the numbers we get are not real numbers. But there is a way to interpret what 2n means even if n is an infinite numbers, and the point is, we have proved that 2n (the cardinality of the power set) is larger than n (the cardinality of the original set) even when n is an infinite number.
Recall that the cardinality of the set of natural numbers (or of any countable set) is denoted by א0 (aleph-naught or aleph-null) : The power set of the set of natural numbers is the set of real numbers (actually, we should say that the power set of the natural numbers can be put in one-to-one correspondence with the real numbers), so the cardinality of the set of real numbers is 2א0.
The Continuum Hypothesis is the statement that there is no cardinality that is between א0 and 2א0, so 2א0 would be the "next in line" in the infinite cardinalities. It is now known that this statement cannot be proved either true or false in the mathematical system we use. (A somewhat accessible article explaining this, by Juliette Kennedy, is available from the Institute for Advanced Study.Warning: it's long, because it gives some history of the problem and also explains some of the concepts and ideas involved.)
However, with or without the Continuum Hypothesis, we can show that there are in fact an infinite number of "sizes of infinity"! How to do this: we can take the power set of a set S, and form the power set of the power set:
So if |S|=3, we know that |P(S)| = 23 = 8
Form the power set of the power set: its size is |P(P(S))| = 28 = 256
Form the power set of the power set of the power set: its size is |P(P(P(S)))| = 2256 = a really big number! (It has 78 digits)
And we can continue this process indefinitely. This gives us a "tower of powers" where each successive number is 2 to the power of the last one.
The same thing can be done starting with an infinite set, the set of natural numbers. Then we get an infinite "tower of powers" with each one being an infinite cardinality which is strictly larger than the one that came before it. So there are (at least!) countably many sizes of infinity.
Next topic: Stating and proving the Pythagorean Theorem
Note: what we will do through the remainder of the course will be more geometrical, to take a break from the analytical stuff we have done so far. It will still be "Thinking Mathematically", but in a little different way.
For the statement of the Pythagorean Theorem, see the post from Math is Fun, and also the Regent's Prep post
Cut-the-Knot has a list of 118 different proofs of this theorem. Why prove a theorem more than one way? Because every proof sheds a little different light on the theorem. This theorem is so important (maybe the most important theorem in mathematics!) that we keep coming back and looking at it again from different points of view.
The proof we will examine is this one, called "algebraic" by Math is Fun, although it is also geometric.
Next time: We will look more carefully at the picture in that proof, and make sure that all the angles add up as they should! (Being careful after being warned by those area paradoxes.) We will also look at the two proofs given in the original Math is Fun article about the Pythagorean Theorem.(They are near the bottom of the page.)
Homework:
• Go to this Math is Fun article about power sets, and find the Questions at the bottom of the page. Do Questions 1-8 (you can skip #9 and #10). While you do them, record in your journal what work you did to answer the question, what answer you got, was it right or wrong, and what you learned.
• Make sure that you understand how the proof of Cantor's Theorem works, and also how we know that there are an infinite number of sizes of infinity - in other words, an infinite number of infinite cardinalities.
• Read the two links above to Math is Fun on the Pythagorean Theorem and on the proof of the Pythagorean Theorem, looking specifically at the proofs so you are prepared to examine them closely next time.
