How big is infinity? - Dennis Wildfogel
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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.
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Additional Resources for you to Explore
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 this video.
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. Learn more about Georg Cantor's Infinity Paradox in this video here.
Do a web search on the book One, Two, Three,... 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. Also look for Gödel, Escher, Bach by Douglas Hofsteader. This is a deep, intricate, and thought provoking book.
For videos on other math topics, check out Dennis Wildfogel’s website here.
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. Learn more about Georg Cantor's Infinity Paradox in this video here.
Do a web search on the book One, Two, Three,... 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. Also look for Gödel, Escher, Bach by Douglas Hofsteader. This is a deep, intricate, and thought provoking book.
For videos on other math topics, check out Dennis Wildfogel’s website here.

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