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andrewducker ([personal profile] andrewducker) wrote2012-07-17 12:00 pm
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[identity profile] steer.livejournal.com 2012-07-19 10:07 am (UTC)(link)
The cantor set is countable surely?

[identity profile] khoth.livejournal.com 2012-07-19 07:39 pm (UTC)(link)
No, it's uncountable. For any real number between 0 and 1 (of which there are uncountably many), you can express it in binary eg 0.0010101101..., then replace each 1 with a 2 to get a ternary number eg 0.0020202202... which is in the Cantor set.

(This glosses over some technical details about numbers with two binary representations but that doesn't make much difference)

Incidentally, Greg Egan once wrote a short story where it was critical to the plot that the Cantor set was uncountable. I'm pretty sure he did it to prove he could.
Edited 2012-07-19 19:41 (UTC)

[identity profile] steer.livejournal.com 2012-07-19 11:20 pm (UTC)(link)
You are absolutely correct. Indeed, now you mention it, I can recall the very same discussion the first time I heard it about 15 years ago with two number theorists in a ground floor office in York University while they minded the dog belonging to a retired professor of pure mathematics. I think it's the only impressive proof I remember (or I remember now you remind me) involving ternary.

When it comes down to it the properties of the Cantor set are quite remarkable... measure zero, nowhere dense, a complete metric space yet uncountable. No wonder it gave mathematicians of the age fits!

I mentioned elsewhere (not sure if in reply to you) Rudy Rucker's book "White Light" which also hinges on the countable, uncountable and other possible forms of infinity.
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