Originally posted by FabianFnas
There are an infinit amount of primes, that is easily proven, so we can continue as far as we like. Trillions, quadrillions, even zillions of primes and 1/n gets smaller and smaller. If we break the last barrier and think of all primes there is, all infinite of them, we have 1/inf which gives zero as a result, right?
No,
wrong I'm afraid. What exactly is this 'barrier' you're 'breaking through'?
Read my first post above. To make a statement like
Q( { p is odd } ) = 1
needs a probability measure Q on the set of primes p. If this is to be got from a limit of statements
Q( { p is odd } given { p is one of the first n primes } ) = 1 - 1/n
then Q needs to be uniform on the set of primes. But no uniform probability measure exists on an infinite countable set.
To see why this is true, suppose P is a uniform probability measure on the set N of natural numbers 1, 2, 3, 4,...
Let P( { 1 } ) =: c a number between 0 and 1 inclusive. Since P is uniform P( { n } ) = c for all natural numbers n. So for every k in N we have, since measures are finitely additive:
k*c = P( { 1, 2, ... , k } ) < P( N ) = 1
and it follows that c = 0. But now, since measures are countably additive:
1 = P( N ) = sum_{k=1 to infinity} P( { k } ) = sum_{k=1 to infinity} 0 = 0
a contradiction. So no such P exists.
I thought also Fat Lady made a very good point with the googol, although what (s)he said wasn't probabilistic.
Anyway, I assume that by now you're joking with us!