[FOM] Throwing Darts, Time, and the Infinite
Joe Shipman
JoeShipman at aol.com
Wed Apr 18 16:07:14 EDT 2012
This is erroneous.
The smallest cardinality set of positive measure might have a larger cardinality than a nonmeasurable set, in which case the argument below fails.
These matters are discussed in my thesis, "Cardinal Conditions for Strong Fubini Theorems", published in the October 1990 Transactions of the AMS.
-- JS
Sent from my iPhone
On Apr 18, 2012, at 7:37 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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>
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