Quote:
Originally Posted by GeeTwo
... I specifically addressed your concern that the even integers and all integers are of the same cardinality.
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FTFY. When discussing cardinality, distinction between aleph-0 and aleph-1 is key. The integers (aleph-0) have measure 0 in the real line (aleph-1), so any "proportion" there would be 0.
Quote:
Originally Posted by GeeTwo
A far as I am aware, any argument that decides the statement "Half of the integers are even." as meaningless can also be used to decide the concept of "average length of a line segment located within the unit square" as meaningless -- or worse.
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Rather, what composes half of an infinite set is not well-defined at all, since there is no well-defined limit of half of a finite set. The average of an infinite set, on the other hand, is well understood as the limit of the average of a finite set (an infinite series or an integral).
Quote:
Originally Posted by GeeTwo
OK, I'll go with that -- and it's a remarkably fast asymptotic function, as of every pair of consecutive integers, exactly one is even.
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The error bounding condition is still error <= 1/2n, so not quite that fast. You just happen to hit the limit exactly on every other term.