Re:Are real numbers really real?
by filmor on Tuesday June 12, 2018 @04:50PM
(#17297844)
Nope, exactly. The measure of the interval [0, 1] excluding any countable subset (for example all rational numbers) is still exactly 1, that's usually what mathematicians mean by "almost every", here "almost every real number in [0, 1] is irrational" :)
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Re:Are real numbers really real?
by function_seven on Tuesday June 12, 2018 @06:07PM
(#17298520)
I was joking, but to a point.
The problem is that "100%" =/= "All". But most of the time, it is considered to be equivalent. The "almost all" phrase is used instead of "100%" precisely because dividing by infinity doesn't yield a useful result in some contexts.
In more... "conventional" cases, if I tell you that 100% of the items are blue, you can conclude that none of them are red. With the intermingling of countable and uncountable sets, this statement fails. You have a situation where 100% of the items in the superset are blue, and there are also some red ones.
So it's "almost 100%" in the same way it's "almost all". I'm using the word "almost" to mean "infinitesimally close to"