L.E.D. (spoiler 200+)

[andrew at cee.hw.ac.uk]

Stephen Deng writes:
> Maybe the emptiness aspect has something to do with properties of Cantor 
> sets, which are fractals.  The Cantor set is the set of points on the 
> interval [0,1] where you first cut out the middle third.  In the next 
> iteration, cut out the middle third of the remaining two pieces and so on 
> for an infinite number of iterations.  The resulting set has dimension 2/3, 
> but it has a probabilistic measure of 0 which means that if you throw a 
> dart at the interval [0,1] (and you are a good enough dart player to his a 
> straight line), there is a 0% probability (in measure theoretic terms) that 
> you will hit one of the points in the set.  Therefore, in essence there is 
> something there which is mostly emptiness.

I believe this is usually called Cantor's Ternary set. And you are
right that measure theory probably provides a better basis for a
redefinition of `empty' than anything based on the space we inhabit,
particularly a naive assimilation with 3-d Euclidean space. Measure
theory provides a conservative extension of the notion of `extent' or
`length' (mathematicians call it `measure').

But note that the set you are describing is very peculiar. It is
*discontinuous* everywhere on [0, 1] i.e. take any point which lies in
the set e.g. the point X, 1/3rd the way along the interval (i.e. the
one with coordinate 0.3333...). If you consider any subinterval [X -
eps, X + eps], where eps is a number between 0 and 1, you will find
that the interval contains points which are in the Cantor Ternary set
and points which are not. Equivalently take any subinterval [a, b]
where 0 < a, b < 1 and the interval will contain both points inside
and points outside the Ternary set no matter how small the difference
between a and b. Clearly the set `extends' through the interval [0, 1]
in a very different way to a conventional interval such as [0.2, 0.4]).

Also, when you say that the set has dimension 2/3 but has
probabilistic measure of 0 you are masking something in the way you
phrase things. What you are actually referring to here is that by
employing one measure you can arrive at an `extent' or `length' for
the set of 2/3 whereas using another measure it has `extent' 0. Which
one is its real length? Only legitimate answer is `neither' or `it
depends'.

Further, the pecularities are all down to the peculiar construction of
the set, not anything wierd in the original notion of measure. For the
usual measures employed to prove this result the interesting thing is
that both measures agree on conventional intervals, they only disagree
on sets which have discontinuities like the Ternary set. So, no
mystery, nothing really surprising, nothing in the result that was not
carefully placed there by adopting the appropriate assumptions and
definitions. That's all this talk of dart playing reduces to, really.
A parlour trick where you substitute the lady for a joker half-way
through.


Andrew Dinn
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And though Earthliness forget you,
To the stilled Earth say:  I flow.
To the rushing water speak:  I am.