Using the fundamentals of set theory, explore the mind-bending concept of the “infinity of infinities” -- and how it led mathematicians to conclude that math itself contains unanswerable questions.
We mentioned that the square root of two is irrational. Prove that this is so. Here are some hints. Start by supposing that the square root of 2 is a fraction. Put that fraction in lowest terms, say, p/q, where p and q have no factors in common. Square both sides and conclude that p must be even. Then conclude that q must be even, violating the assumption that p and q have no factors in common. You can see the proof in detail in the video http://youtu.be/HiWXvSbzdqI. (The video will tell you when to pause to try to deduce the next step yourself before it’s shown to you.)
We mentioned that the set of all subsets of an infinite set constitutes a larger infinity than the original set. Prove this. Here are some hints. Start by supposing that there is in fact a match between the given set and the set of all its subsets. Call any element that is matched to a subset that contains that element included; call all other elements omitted. Now consider the subset A of all omitted elements. Show that no element of the original set could possibly be matched to A by considering whether any such element is included or omitted.
You can see the proof in detail in the video http://youtu.be/ms1nboFLMnc. (The video will tell you when to pause to try to deduce the next step yourself before it’s shown to you.)
Do a web search on the book 1,2,3,... Infinity by George Gamow. You will find some great testimonials by people saying that the book had a big impact on their lives. It’s an easy book to read, it’s inexpensive, and it’s terrific.
Look for the book Gödel, Escher, Bach by Douglas Hofsteader. This is a deep, intricate and thought provoking book.
For videos on other math topics, see Dennis Wildfogel’s website: http://www.dennisw.com.