Infinity: Beyond the Beyond the Beyond
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Welcome to the wild world of Georg Cantor's Theory of Transfinites where infinity actually exists! What is the difference between potential infinity and actual infinity? What are cardinal numbers? What are ordinal numbers? What cardinal and ordinal numbers represent the natural numbers, the rational numbers, and the real numbers?
Are there infinities bigger in size (or cardinality or "power") than the real numbers? How do you add and multiply cardinal numbers? How do you add and multiply ordinal numbers? What are simply ordered and well ordered sets? Can you explain the famous problem of the race between the tortoise and Achilles, Zeno's paradox, using Cantor's theory of transfinites?
This month's discussion will be based on the fun book Infinity: Beyond the Beyond the Beyond by Lillian R. Lieber. The book includes proofs and sophisticated arguments. Although it might be too easy for some members, I hope that that means we will be able to discuss each argument in detail. Although reading the book is optional (we will explain the content needed to follow the discussion), we hope that many of you will have time to read at least part of it before the discussion (although the first 100 or so pages are an easy read, the book has some very sophisticated and subtle passages).
Unlike most mathematics books (even those for the general reader), Lieber's book closes with a chapter entitled "The Moral". Throughout the book Lieber builds a moralistic theory of mathematics based on SAM (Science, Art, and Mathematics) as a principle of right thinking based on "honesty and patience". Does Lieber's morality of mathematics enhance or detract from the book? Is there a moralistic quality to mathematics? Since Math Counts is bold enough to endeavor to "Cultivate Mathematics Culture" and if members are interested, we might discuss the thesis of the morality of mathematics that Lieber advocates in the book. We will certainly discuss as much of the mathematics of infinity discussed in the book as we can fit into a 2+ hour meetup.
Fernando Q. Gouvêa's good MAA book review of Lillian Lieber's Infinity (http://www.maa.org/publications/maa-reviews/infinity-beyond-the-beyond-the-beyond).
Note: This is the shortest book for the general reader that I could find which pretty thoroughly covers Cantor's Theory of Transfinites.
Note: Last month's meetup event (click the "see all" link to read the whole description) (https://www.meetup.com/MathCounts/events/218584403/) included four highlighted videos (just over one hour's worth) on infinity. In addition, links to an additional two hours of video with Michael Starbird that explore the cardinality of the integers, rationals, and reals in depth were referenced (see the fifth bullet under infinity videos). If you cannot read the book, watching those videos will provide a good background for the discussion. Finally, in the comments for last month's discussion, CJ wrote summaries of each video which might help you find the highlights.