How big is infinity? - Dennis Wildfogel
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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.
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