Re: Probability of picking a positive rational number at random

On 2008-03-16, Ross A. Finlayson <raf@xxxxxxxxxxxxxxx> wrote:
(That could have been quoted inline with the comment above,
in responses I don't reply inline, which while easily indicating
relevant context of response, interrupts.)

Ah, you prefer an uninterrupted monologue rather than a conversational
style. Got it.


It is ridiculous for you to talk about "a uniform distribution over
all uniform distributions over all well-orderings of distributions
over natural numbers."

You're the one who said:
Then, using the distribution that happens to be marked by that
ordinal in a "random" well-ordering of the distributions of the
naturals

I had just noticed that in trying to get from a perfectly well defined
uniform distribution over reals to one over naturals, you were
requiring arbitrary well-orderings followed by random distributions
over well-orderings. I was just saving you a step.


In well-ordering the distributions over natural numbers, in a
bijection to the unit interval of reals, by selecting a real at
random, each distribution has the same probability as any other to
be selected

Only in the sense that every distribution has probability zero.

The biggest problem with your proposal is that for infinitely many
well-orderings (quite possibly all of them), the event of deriving any
given value n corresponds to a non-measurable set in the reals. Hence
its probability is meaningless.


Apart from being a run-on sentence you didn't read it right.

That is probably because you didn't write it right. The quantity of
grammatical errors in your 9-line sentence rendered it nearly
unintelligible. Just what did you mean by the clause "because
well-orderings of the reals are so random"?


It is entirely possible for a well-ordering of the reals and a
well-ordering of the distributions over natural numbers to result in a
99.9% probability of selecting a distribution for which P(n=1) = 0.999.

Apparently to avoid that, you specified that the well-ordering should
be "random". How do you pick a random well-ordering if you can't even
pick a random natural number?


It seems understood: given a well-ordering of the reals, there
exists for the reals mapping to finite ordinals a method to sample
them, trans- finitely

There does not so exist, in any theory I've seen. Perhaps you can
come up with a theory that can model such processes and isn't full of
contradictions, but I doubt it.


- Tim
.

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