Georg Cantor

A Song About Georg Cantor

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• published: 30 Sep 2012
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Daniel Rose - A Song about Georg Cantor -------------------------------------------------------------- Georg Cantor: I could maybe rant for, Days about how his ways were so non-standard. Wanna learn set theory? Take a gander, At the versitile works of one Georg Cantor. Let's hear it for Georg Cantor, That man I demand you go and raise your hands for! Get Ready for the exciting wonders, Of Cantor's theory of transfinite numbers. ------------------------------------------------------- Late 19th century Mathematicians, When faced with the infinite acted suspicious. Here comes Cantor now with the highest affinities, for specifics intrinsic in Different Sized infinities. Philosophical obstacles, jests and jeers, Resistance from half of his mathematical peers. "Infinity's really silliness," They all clamber, "Man, Look at that fool, there, Georg Cantor!" But Sets of the Same size implies an assumption, There's defined among them bijective functions. Example: Integers and rationals, I promise, Can be thought of as in 1-1 correspondance. That thought is awfully profound in full, The fact that the rationals are actually countable. The reals on the other hand: a vastly larger set. See: Cantor's Diagonal Argument. ------------------------------------------------------ Let's hear it for Georg Cantor, That man I demand you go and raise your hands for! Get Ready for the exciting wonders, Of Cantor's theory of transfinite numbers. ------------------------------------------------------ Starting with I, the unit interval, observe, We're about to take out just the little middle third. Two disjoint intervals are what's left and soon, We'll take out the middle third of those next sets too. One becomes two and, yup, two becomes four then, Four becomes eight as we're removing a portion. Soon we'll have more than a billion quadrillion items, As we continually itterate it ad infinitum. That's a limit, now consider what's left, After choppin' off a lot of our original set. Certainly no interval could at all survive. 'Cuz it would have to be infinitely small in size. Singletons, namely, is all that now remains. Uncountably many, in fact, if that seems strange, Well, Consider this, although it's not a proof: For every one step you take you walk off with two. ------------------------------------------------------ Let's hear it for Georg Cantor, That man I demand you go and raise your hands for! Get Ready for the exciting wonders, Of Cantor's theory of transfinite numbers. ------------------------------------------------------