Comments on: Most Disturbing Math Theorem Ever

By: ricktaylor

ricktaylor — Tue, 20 Sep 2005 08:23:30 +0000

This bothered me too, but a related fact that bugged me more was that the Cantor set has an uncountable number of connected components, even though you’ve cut out only a countable number of intervals. More generally, it’s curious that the structure of an arbitrary open set in R^1 is relatively simple (an at-most countable union of disjoint open intervals), but closed sets in R^1 can be so complex; even though a closed set is just the compliment of an open set.

–Rick Taylor

By: Walt

Walt — Fri, 16 Sep 2005 03:16:10 +0000

Banach-Tarski depends on Axiom of Choice, so it doesn’t count.

By: octracker

octracker — Fri, 16 Sep 2005 02:56:55 +0000

what about banach-tarski? chop an orange up into tiny bits – rearrange the bits and voila! two identical oranges.

By: easwaran

easwaran — Sun, 11 Sep 2005 05:08:03 +0000

The related result that disturbs me more is that the reals are the union of a set of measure 0 with a meager (in the sense of Baire) set. Just take the countable intersection of dense open sets of arbitrarily small measure containing all the rationals.

By: sigfpe

sigfpe — Sat, 10 Sep 2005 14:41:01 +0000

I agree. This one has always bothered me too. The proof is so short you can just see why it’s true. And yet it’s so counterintuitive.

By: ComplexZeta

ComplexZeta — Sat, 10 Sep 2005 06:23:21 +0000

I no longer really find this surprising, but when I first saw it in the study of Lebesgue theory, I found it somewhat pathological. I think I’ll spend some time before I go back to school next week trying to construct a covering of the rationals that excludes some fixed irrational (or, dually, finding an irrational not covered by a given covering of the rationals). That seems like an interesting challenge.

By: citylight

citylight — Sat, 10 Sep 2005 05:37:42 +0000

Hmm… to me it only feels equivalently ‘strange’ to the fact that the reals have a countable dense subset.

Once you know that it doesn’t seem too surprising, you can just take, say, the union of open balls of radius epsilon/2^(n+1) around the nth rational, right?