Comments on: Cardinality of infinite sets, part 1: four nonstandard proofs of countability

By: Mathematical surprises « Division by Zero

Mathematical surprises « Division by Zero — Wed, 25 Aug 2010 17:57:59 +0000

[...] The rational numbers are countable [...]

By: Web Worker Example: The Rationals « Extremely Satisfactory Totalitarianism

Web Worker Example: The Rationals « Extremely Satisfactory Totalitarianism — Sun, 07 Feb 2010 09:15:42 +0000

[...] and its connection to the rational numbers before, but to recap: I first came across it on Division by Zero (still waiting for the rest of that series) and the original paper by Calkin and Wilf is [...]

By: Hyperbinary numbers and latex plugins « An Extremely Satisfactory Totalitarianism

Sat, 24 Oct 2009 08:19:31 +0000

[...] appearing once and only once, in reduced form. I originally saw it mentioned on Division by Zero here (along with a bunch of other great non-standard proofs), Calkin and Wilf’s original published [...]

By: Travis

Travis — Mon, 14 Sep 2009 19:05:58 +0000

If for no other reason than to further whet your appetite, the surreal numbers can actually be used to model types of combinatorial games (i.e. games with finitely many moves and perfect information). Then “surreal arithmetic” can be used to solve several game theory problems… and vice versa!

By: Dave Richeson

Dave Richeson — Mon, 14 Sep 2009 16:46:00 +0000

That’s a great analogy.

In our particular course we aren’t going to construct the real numbers. Instead, we just assume the axiom of completeness. Abbott’s book does have the construction (using Dedekind cuts), but sticks it at the very end of the book. We’re not going to get there.

I have actually never read about Conway’s surreal numbers. It is on my to-do list. I know that they are an extension of the reals to include infinity and infinitesimals, but I don’t know much more than that.

By: Dan M

Dan M — Mon, 14 Sep 2009 13:21:41 +0000

I’ve often thought that the process of learning about countability, uncountability, and transfinite numbers is a lot like the Zen story of the bowl and tea:

“Before enlightnement, the bowl is a bowl and tea is tea. In the process of achieving enlightenment, one discovers that the bowl is not a bowl, and tea is not tea. After enlightenment, the bowl is again a bowl, and tea is again tea.”

When you first start thinking about numbers like this, confusion sets in – the familiar numbers that you’ve known since childhood are suddenly changed.

Personally, I didn’t find countability to be too much of a problem, but what I did struggle with was how we construct numbers out of sets and cuts and equivalence classes etc. Recently I’ve been reading Conway’s ONAG (http://books.google.com/books?id=tXiVo8qA5PQC), and bowls are not bowls again for me right now. What do you think of using Conway’s approach in teaching undergrads? Is it ever done?

By: Dave Richeson

Dave Richeson — Mon, 14 Sep 2009 13:19:26 +0000

I’ve never heard of that book. Thank you for letting me know. I remember learning about Hilbert’s Hotel when I was in high school or junior high school from Martin Gardner’s “Aha Gotcha!” book. At the time it seemed like he was cheating, but I couldn’t put my finger on the problem.

I must admit that I’m a little disappointed to have been scooped on this idea. I’ve been saying to my friends for the last few years that I wanted to write a children’s book about infinity (although I think it would be highly unlikely that I’d ever follow through on the idea).

By: Sue VanHattum

Sue VanHattum — Mon, 14 Sep 2009 04:05:45 +0000

Check out a children's book called The Cat in Numberland, where the Hilberts are the innkeepers at the Hotel Infinity, and the cat wonders how it happens.