[FOM] Throwing Darts, Time, and the Infinite

[Kenny Easwaran]

I would just add - on the particular point at question here (that is,
Freiling's claim that "the real number line does not really know which
dart was thrown first or second" vs. Gwiazda's claim that the order
that the darts are thrown in really does matter), it seems that
Freiling is totally right.  The joint distribution of two dart throws
gives a space in which the event of the first dart hitting an earlier
number in the ordering than the second is unmeasurable, and so is the
converse.  It's true that conditionalizing on the location of the
first dart gives probability 1 to this event, but conditionalizing on
the location of the second dart gives probability 0 to this event.

If the probabilities are subjective probabilities, then there is no
privilege to one of the conditionalizations over the other.  If the
probabilities are objective chances, then whether one of them is
privileged depends on the metaphysics of time.  Given the most
plausible view for relativity (on which there is no distinguished time
axis), neither one will be privileged, and so there's no reason to
give up on the unmeasurability of this event, and thus no reason to
think it will almost certainly go one way rather than the other.

Kenny Easwaran

On Wed, Apr 18, 2012 at 4:37 AM, Thomas Forster
<T.Forster at dpmms.cam.ac.uk> wrote:
>
> This is an old chestnut.  I am taking the liberty of inflicting
> on listmembers the following email from my colleague Imre leader, with his
> permission. (slightly edited).
>
> Dear Thomas,
>
> I was thinking about that thing you told me, that was supposedly
> against CH: that one bijects $\Re$ with the set of countable ordinals
> and then throws one dart and then another at the board.
>
> My reply then was (correctly) that this was just silly, as it confuses
> what conditional probability means (one cannot condition on an event of
> zero probability, as in the phrase `given that I throw $\alpha$'), and
> also it forgets that not all sets are measurable.
>
> That reply was entirely correct, but I have had two further thoughts. The
> first one is: the fact that the event `second dart beats first' is in
> fact one of the absolutely most standard examples of a non-measurable set.
> Indeed (assuming CH) one takes the subset of $[0,1] \times [0,1]$ given by
> those points $\tuple{x,y}$ for which $x<y$ in the ordering induced from
> $\omega_1$. Then each row is countable but each column is cocountable!
>
> My second thought is more important. It is that, even ignoring the
> fact that the `paradox' is rubbish because of conditional probability
> and nonmeasurable sets, even then, it is {\bf not} against CH. What I
> mean is, I will hereby run the {\bf exact} same paradox without any CH
> assumption. Ready? Here goes \ldots
>
> Let $\kappa$ be the least cardinality of a set of positive measure. Let
> such a set be $A$, and let us well-order $A$ in such a way that all
> initial segments are smaller than kappa. [he means $\kappa$-like]. Note
> that all initial segments have measure zero, by definition of $\kappa$.
>
> OK, our experiment is: throw a dart at the set $A$. Twice. [Note: as $A$
> has positive measure, we can do this by throwing a dart at $[0,1]$ and only
> counting it when it lands in $A$.] Then all the paradox still applies.
>
> Conclusion: there is no way, not even intuitively or anything, that this
> `paradox' has anything to do with CH.
>
> Imre
>
>
> On Tue, 17 Apr 2012, 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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>>
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