[FOM] Throwing Darts, Time, and the Infinite

[John Allsup]

You can equally argue that a real chosen at random will be one of the
reals never identified by humanity in all its history (the countability
of the 'humanly nameable numbers' being a well-known undergraduate
exercise in set theory), which again seems paradoxical.  The confusion
lies in the difference between probability 1 and logical certainty.

Consider the Lebesgue integral over the real line of a function that
takes 1 for every rational and 0 for all other numbers.  It is 0, and at
no point is the function negative, yet the function is not identically
zero.

Consider now a function on [0,1] that is 0 for every rational and 1 for
every irrational, so that the Lebesgue integral over this interval is 1.
 What 'results' about the non-existence of rationals could you concoct
if you try to consider this as a probability distribution?

To get to your example, with probability 1 but not certainty the dart
hits a real higher in the well-ordering.  It may still encounter a
'probability 0 event' of a real before being hit.  In everyday life,
functions as nasty as a well-ordering of the reals won't turn up, so our
intuition can lead us astray with examples such as the one given below.

John

On 17/04/12 18:40, Jeremy Gwiazda wrote:
> Hello,
> 
> Chris Freiling’s Axioms of Symmetry have, I believe, been discussed on
> FOM at least twice. In ‘Axioms of Symmetry: Throwing Darts at the Real
> Number Line’, Freiling considers two darts thrown at [0, 1]. He
> writes, ‘the real number line does not really know which dart was
> thrown first or second’, which leads to one of his axioms of symmetry.
> In a recently published paper, I suggest that a well-ordering of [0,
> 1] does know the order of the darts under certain assumptions. Fix a
> well-ordering of [0, 1]. Let r1 be the real hit by the first dart.
> Then assuming ZFC and CH, there are only countably many reals less
> than r1 in the well-ordering. Thus with probability 1 the second dart
> hits a real greater than r1 in the well-ordering. (Put again slightly
> differently: working in ZFC, Freiling demonstrates that assuming that
> the reals can’t tell the order of the darts proves not CH; I argue
> that assuming CH means that a well-ordering of [0, 1] can tell the
> order of the darts.) I go on to create a puzzle using special
> relativity. In case it is of interest, the paper is here:
> 
> http://www.springerlink.com/content/e2746w1kn6913580/
> 
> 
> An earlier, countable version of a similar puzzle is available here:
> 
> http://www.springerlink.com/content/v1n12t200jv2553u/
> 
> 
> Best,
> 
> Jeremy Gwiazda
> jgwiazda at gc.cuny.edu
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