Re: Probability of picking a positive rational number at random

On 2008-03-15, Ross A. Finlayson <raf@xxxxxxxxxxxxxxx> wrote:
So now you're going from a probability distribution over [0,1] defined
by Lebesgue measure to a completely undefined probability distribution
over the set of all well-orderings of distributions of natural
numbers? How do you know that one with the properties you want even
exists?

Can't you just use classical probabilities to show that?

Classical probability is more limited than measure-theoretic
probability, so it can't help you. Can you even explain in terms of
mathematical properties what you mean by your desired uniform
distribution?

Simply asserting "for all x,y: P({x}) = P({y})" isn't enough, since
there are infinitely many distributions satisfying that relation.
Feel free to try to define a uniform distribution over all uniform
distributions over all well-orderings of distributions over natural
numbers. That won't get you any closer, and I'll feel free to call it
ridiculous.


That iota is great among infinitesimals represents that summing
infinitely many of them yields one.

Originally you said "greatest".


In a theory like ZFC with well-orderable reals, and the ability to
"sample" an element of the population of real numbers by infinitely
many Bernoulli trials implying the ability to select an ordinal at
uniform random from their initial ordinal, because well-orderings of
the reals are so random, there is the ability to sample from the
naturals a value such that for any other value, the probability of
its selection is exactly the same, and the sum of their
probabilities, their whole, in that they are mutually exclusive
events one of which occurs, censored to countable, is one.

Apart from being a gross run-on sentence, this is rubbish.
Well-orderings aren't random.


Simply, sample a real by infinitely many Bernoulli trials and
discard it if a particular well-ordering has that real not mapping
to a finite ordinal.

So your "algorithm" is: Generate a countably infinite number of binary
digits. Do this (on average) *uncountably* many times, until you get
a countable ordinal. I don't know what you're on, but it's not
mathmatics.


- Tim
.

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