Re: Probability of picking a positive rational number at random

On 2008-03-19, Ross A. Finlayson <raf@xxxxxxxxxxxxxxx> wrote:
elements being sequences a of of R_[0,1]^N such that sum a_n = 1 ( a
very, very small subset of R_[0,1]^N

What makes you think such sequences are a very small subset? They are
equinumerous with the set of all real sequences.


Selecting one of those distributions such that it is only as likely
as being selected as each other is the same problem as selecting an
ordinal alpha (less than some given ordinal) such that each ordinal
has the same chance of being selected as any other, and no greater,
and no lesser.

In other words, a problem that has no solution.


A notion I put forward is that given a well-ordering of the reals,
which exists in ZFC, if not necessarily ZF, which is basically a
bijection between ordinals and reals, that by selecting one of those
reals at uniform random, that there is thus indicated an ordinal at
uniform random.

If the set of ordinals is countable, then countable additivity of
probability proves that either the distribution is not defined for
singletons, or some ordinals will have greater probability than
others. Which result holds depends upon which well-ordering you start
with.


That is where well-orderings of the reals have no structure

There exist well-orderings of the reals with all sorts of structure.
E.g. there exist well-orderings of [0,1] where the subset of all
elements mapped to a limit ordinal has Lebesgue measure (and thus
probability) 1.


Then, admittedly and surely outside the "standard" framework of
definitions of "uniform", it is still in a perhaps more "universal"
sense, quite "uniform" in a sense that subsumes the standard.

If you want to go outside the "standard" framework of uniform, you
don't need to go for arbitrary well-orderings that are themselves
"uniform", you can just use probability distribution over N:
P(n is even) = P(n is odd) = 1/2.

There you go, a perfectly uniform probability distribution over N.
Who cares that it's only defined for 2 nontrivial subsets? Your
supposed definition generally even defined for one.


There's much more to say about the density of limit ordinals, and
elements of the "cumulative limit hierarchy", within themselves, yet.

What is the order type of the set of limit ordinals in omega^omega?


- Tim
.

Relevant Pages

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