AD. Bigger and Smaller Infinities
Are there only half as many even numbers as counting numbers?
We discuss last week’s No Vacancy puzzle a bit more: Astoundingly it makes sense to say that there really are as many counting numbers as even numbers! This great insight, due to Georg Cantor about 130 years ago, that we can compare the sizes of infinite sets, leads to all kinds of amazing and important ideas in higher mathematics.
And we pose a quick puzzle about running animals.
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jshiau said,
May 2, 2007 at 8:57 am
If you are new to The Math Factor like I am, you’ll hear the answer to the running animals puzzle soon enough (next episode). On the other hand, the Serengeti is a pretty big area… Let’s imagine for a moment that the two animals are running in a giant circle. To them, they are running in “almost” a straight line. If they stay on this circle, sooner or later they’ll meet just as the two hands on a clock do. The slower animal will be able to catch the faster one–not necessarily “catch up”–no matter how slow it is. Better yet, the slower animal can simply sit still and wait for the faster animal to come up from behind.
Now imagine that the whole of a planet were the Serengeti, then the planet could be the shape of a sphere, a toroid, a dodecahedron, a… Then the two animals could literally run in “straight lines” (defined in the topology of the planet surface) and catch each other, but that’s the story of another sci-fi novel.
Matt said,
August 7, 2009 at 10:44 am
I just stumbled on your site while searching for an example of “bigger and smaller infinities”. I remember coming across the idea at some point and wanted to explain this to my brother. I love your site. Thank you for your podcast! :)