Re: Probability of picking a positive rational number at random

On 2008-03-16, Ross A. Finlayson <raf@xxxxxxxxxxxxxxx> wrote:
By "random" well-ordering I meant "any."

Then your assertion is false. There exist well-orderings that
generate a distribution over N which is not uniform. Of the rest,
none assign a probability to all singleton subsets of N.


I'm surprised that you describe the putative uniform distribution
over reals "perfectly well defined."

Uniform over reals in [0,1] obviously, since you talked about
generating binary digits in the fractional representation via a
Bernoulli process. Such a definition is present in many textbooks.
If that is not what you meant, say so.


About the well-ordered reals (not arguing that they're naturally
well- ordered, although it's been so stated) that correspond to
finite ordinals being a non-measurable set

I never claimed that they were. The statement about non-measurable
sets was in relation to your idea of mapping reals to distributions
over n, not directly to finite ordinals. But since you don't reply in
context, you get confused about which relate to what.

In the context of mapping reals directly to natural numbers via
conditional replacement in some well-ordering, the conditional
probability is simply undefined: 0/0.


Well-orderings of the reals are random because some of them map to
nonrecursive ordinals (in ZFC).

You're equivocating on the meaning of "random". The sense you're
using in this sentence has nothing to do with probability.


what it doesn't say is that for almost all well- orderings of reals
and well-orderings of natural distributions, nothing at all like
that would be the case.

What - precisely - do you mean by "almost all"? The sets have the
same cardinality so you can't mean that. It sounds like you're
talking about measure. What measure?


In omega^2 there are omega many limit ordinals, omega^3 there are
omega^2 many limit ordinals, Past epsilon is it clear how many limit
ordinals there are?

You don't even need to go to epsilon to find ordinals that are
isomorphic to their subsets of limit ordinals. omega^omega is one,
and that's far short of epsilon. What's more, every cardinal greater
than aleph_0 shares the same property.


So, the first real "generated" by flipping fair coins that maps to a
recursive ordinal

First you have to prove existence of any first such real before
arguing about its probability distribution. Any finite number of
flips (of infinitely many digits each) will with probability 1
generate no suitable real. In fact, even conducting an infinite
number of such infinite sequences of flips *at the same time* will
have probability 0 of ever generating a suitable real.

So your first mistake is assuming without justification that such a
thing exists. And the rest is predicated on that mistake.


Of course, another method was described early in the development of
this thread, which began about positive rational numbers at random,
and is now about how given a natural integer, and another natural
integer with no other information about either, that the rational
gambler would wager a dollar to win one that they were coprime,
because the probability of those two numbers being coprime is
greater than one half.

Are you a rational gambler? If so, then according to your argument
you will accept a 1:1 bet on coprimality without any information about
the numbers I intend to generate.


The concept of probability, its meaning, has that for two anonymous
distributions of the integers, the probability of two independent
samples being coprime is 6/pi^2

OK, I'll even make sure that the numbers are generated independently.
The first two numbers are ... 2 and 2. You owe me a dollar, or you're
not a rational gambler.


- Tim
.

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