Song-writer Bill Calhoun is a faculty member in the Department of Mathematics, Computer Science and Statistics at Pennsylvania's Bloomsburg University (where I also hung out for many years). He belongs, along with colleagues Erik Wynters and Kevin Ferland, to a band called "The Derivatives." And Bill has granted permission for me to include several of his math lyrics (parodies) here. (In this previous post, we consider the connection between song parodies and mathematical isomorphism.) My first Calhoun selection deals with difficult mathematical questions concerning classification of infinite sets and decidability. Following that, later lyrics consider proving theorems and finding derivatives.
Questions You Can’t Ever Decide* by Bill Calhoun
Picture yourself in a world filled with numbers,
But the numbers are really just words in disguise.
Gödel says “How can you prove you’re consistent,
If you can’t tell that this is a lie?”
Georg Cantor (1845-1918), a German mathematician, first dared to think the counter-intuitive notion that not all infinite sets have the same size--and then he proved it: The set of all real numbers (including all of the decimal numbers representable on the number line) cannot be matched in a one-to-one pairing with the set of counting (or natural) numbers -- 1,2,3,4, . . . . Sets whose elements can be matched one-to-one with the counting numbers are termed "countable" -- and Cantor's result showed that the set of all real numbers is uncountable.
Cantor developed an extensive theory of transfinite numbers -- and poet (as well as philosopher and professor) Emily Grosholz reflects on these in a poem:
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