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Re: Skepticism, mysticism, Jewish mathematics
Posted:
Aug 6, 2006 1:31 PM


> (1) This business with the nested intervals is generalized in a theorem > which is called "Cantor's Theorem" in "Modern Theories of Integration" > by H. Kestelman [1960]. IIRC, the statement of the theorem (in > Chapter 1) is (pretty close to) this: > > For a [infinite] sequence of nonempty, closed, bounded pointsets {S} > such that each S_n includes S_(n + 1), the intersection of all the > sets in the sequence is nonempty. [The inclusions need not be proper.] >
Well, 0 is the limit of both the point on the left and the points on the right. Perhaps the result that the intersection is nonempty is eqivalent to the fact that the limit exists.
I was puzzled by this result for a while because it seems devoid of any meaning. I.e. if an interval has zero lenghth who cares if there is anything in it or not? But the existence of a limit has a rather significant meaning.



