Re: Probability of picking a positive rational number at random

On Mar 15, 6:04 pm, Tim Little <t...@xxxxxxxxxxxxxxxxxxxxxxxxxx>
wrote:
On 2008-03-15, Ross A. Finlayson <r...@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

About the suggestion of the utility of classical probabilities and
analysis without regard to measure theoretic foundations, that was
about proving those laws of probability that you deferred to measure
theory. (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.) Besides that measure
theory isn't necessary for the reason you mentioned, there are as well
nonstandard measure theories in consideration, besides nonstandard non-
measure-theoretic foundations for analysis, where continuous,
discrete, and mixed distributions can be analyzed without standard
"measure theory", albeit using theories of measure. (There are other
types of analysis than standard and Nonstandard. Some modern
probabilities that aren't classical kind of defy measure theory as
well, in for example parastatistics.)

About the iota comment, yeah I did say "greatest" and later
equivocated for it to be that. That's rare. It was just to draw
parallel in notion that where there's an infinity, that's in a way
inaccessible from finity and counting, yet suitable as a prototype of
infinite objects, that it's symmetrical in concept, and thus elegant
and even natural, to consider that there's an infinitesimals, that's
in a way inaccessible from finity and dividing, particularly where it
would be effervescent differential, from infinitesimal analysis
(integral calculus, analysis). You accept "smallest infinity",
although infinities are defined often by having a smaller infinity,
where's "greatest infinitesimal"?

It is ridiculous for you to talk about "a uniform distribution over
all uniform distributions over all well-orderings of distributions
over natural numbers." 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, and it's not obvious that there's some
way to correlate adjacency in reals to adjacency in the distributions
over the naturals.

Apart from being a run-on sentence you didn't read it right. 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, by sampling the reals "until" one of them is a sample.
(That's basically another well-ordering of the reals, in discarding
duplicate samples, some least element of which map to a finite ordinal
in the original's.) Because a well-ordering of the reals is so random
(in terms of Kolmogorov complexity and Pi_1^1 or what it is) unless
the reals are actually in a contiguous sequence the real indicating a
finite ordinal has that the ordinal is exactly as likely as each other
ordinal to be the sample, regardless of the development of initial
segments of the path that is the real.

And: that very well satisfies the meaning of "uniformity".

(While it leads to obvious contradictions in some entrenched uses of
those words doesn't affect that given a fair coin and transfinite
induction a random integer is a probabilistic event, as one of the
natural integers, quite uniform: constant, in its probability among
the others.)

Asymptotically, a random integer from UNIF(0,N) doesn't have an
expected value.

Where a natural integer is sampled, and the probability of each is a
constant, that's uniform.

Ross

--
Finlayson Consulting
.

Relevant Pages

  • Re: Probability of picking a positive rational number at random
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  • Re: Probability of picking a positive rational number at random
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