Darren New wrote:
> Warp wrote:
>>   *After* you have thrown the dart, the probability to hit the number
>> the dart did hit grows to 1 (or 100%).
> 
> But is the probability that it hit any rational number (vs any 
> irrational number) non-zero at that point?  If you do it countably many 
> times, will you *ever* hit *any* rational number?
> 
>>   (I really wonder if this has any relation whatsoever to the so-called
>> axiom of choice.)
> 
> That's pretty much exactly what the axiom of choice is. Given an 
> infinite set of sets, can you create a set by picking one element from 
> each subset?
> 

I don't see how it directly applies completely directly, but the notion 
of choosing elements from sets without need a rule by which to make the 
choice is certainly the main feature of both.

As a more mind bending thing than the probability of 0 that you'll hit a 
rational number, if you assume that the axiom of choice is true then you 
can show that there exists subsets of the real numbers in the unit 
interval such that it it's *impossible* to define *any* probability that 
the dart will hit an element of it (zero or otherwise).

Essentially the real numbers are numerous enough that with the help of 
the axion of choice it's possible to define subsets of the reals that 
are so bizarrely structured that the notion of length, area, volume etc. 
don't even apply to them. (more specifically, they're known as 
non-measurable sets since they don't have a well defined measure).

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