Re: Probability of picking a positive rational number at random

On Mar 15, 12:51 am, Tim Little <t...@xxxxxxxxxxxxxxxxxxxxxxxxxx>
wrote:
On 2008-03-15, Ross A. Finlayson <r...@xxxxxxxxxxxxxxx> wrote:

Maybe that's a misperception. Maybe it just is so that there's an
infinitesimal iota, greatest among infinitesimals

I'm sure such a thing could be defined, but I bet you wouldn't be able
to do arithmetic with it. Is iota+iota infinitesimal?

The notion is basically about using uniformity and symmetry as first
principles in describing densities for probability density
functions, instead of standard measure theoretic ones.

Good luck with that. The rest of us like being able to prove things
like P(A or B) = P(A) + P(A but not B).

Now, that's rather unstructured but those are actually technical
words with specific meanings.

There's a word for sentences that use technical words with specific
meanings in ways that contradict those meanings: technobabble.

In general, given a well- ordering of R, there is absolutely no
idea, no algorithm to conclude, what ordinal o maps to each element
of r.

You already went way beyond the bounds of algorithms in the first
step. But just for laughs, let's see how we get from our real number
to a natural number.

So, by sampling a real (and figuring the order type of c was the same
as the order type of the set of distributions, where having it not be
is funny)

Fortunately the cardinality of distributions over N is c, so it is at
least *possible* to have the same order type. Though the two
well-orderings are completely unnecessary: any bijection will do.
Requiring well-ordering just obfuscates what's going on.

Then, using the distribution that happens to be marked by that
ordinal in a "random" well-ordering of the distributions of the
naturals

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?

- Tim

Can't you just use classical probabilities to show that? Classical
probabilities are widely applicable, and there are ways using them to
define conditional probabilities. Functions of random variables don't
have to use measure theoretic foundations for analysis.

That iota is great among infinitesimals represents that summing
infinitely many of them yields one. Any bigger infinitesimals than
iota don't represent anything else but multiples of iota. Are the
multiples of three a smaller subset than the naturals? It's easy to
define induction over them, and they're only a third of the naturals.

Those words have those meanings. I enjoy their meaning.

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.

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. In ZFC, most samples would be discarded, almost all. Then,
that ordinal value is a natural integer.

Given a well-ordering of the reals, sample reals until one of them
matches to a finite ordinal. In ZFC, there's no argument to be made
except that there are reals that biject to finite ordinals, and of all
those, each is uniformly likely to be sampled by infinite fair
Bernoulli trials.

Universe is own powerset.

Ross
.

Relevant Pages

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