Here's an easy question: What whole number comes immediately after 1? You've known since grade school that the answer is 2. In fact, if I give you any whole number, you can always tell me what the next one is.

Here's a more interesting question: What decimal number comes immediately after 1? There seems to be a problem - if you say "1.1", I'll say "You skipped 1.01". No matter what decimal number you think comes next, I can always find another one that's even closer to 1.

In the 19th century, a mathematician named Georg Cantor thought about this strange property of decimal numbers. He succeeded in proving that we cannot count the decimal numbers one-by-one (like we can with the whole numbers) because they constitute a strictly larger infinity than the one we're used to. Not only that, but he discovered infinitely many different infinities, each one larger than the last.

During the seminar, we will think carefully about what it could possibly mean for one infinity to be larger than another. With this understanding in hand, we will be ready to see Cantor's clever proof that the infinite collection of the decimal numbers really is larger than the infinite collection of whole numbers. We will also look briefly at how to take any infinity and create a larger one, thus producing the infinite sequence of progressively larger infinities first discovered by Cantor.