"probability functions" to expansions of R

From: Tim Mellor (timm_at_amsta.leeds.ac.uk)
Date: 12/06/04 

Date: 6 Dec 2004 07:25:05 -0800

Apologies for starting a new thread, but I'm having technical
problems....

>Hi,
>
>About the notion of infinitesimal probability I'm glad that you
consider
>it formally acceptible.
>

Well, one has to be a little careful about terminology. I guess, that
the term "probability function" has been defined many times, and these
definitions will probably (at least implicitly) require that the
range is R.

What I was discussing was some extension of this idea, where one maps
to some RCF extension of R, but the function has the same arethmetic
properties as a "normal" probability function.

>Dave Seaman writes in the first result of that query of a/the usenet
>database, comprised of the works of those who wrote in to usenet and
did
>not have the no-archive flag set in their post headers, about
>infinitesimal probability vis-a-vis probability zero. He claims that
a
>random selection from the unit interval of reals has probability
zero,
>instead of a postive, infinitesimal probability. If x is a real on
the
>unit interval, then P(x), the probability of selecting x, from a
uniform
>random distribution, uniform in that each other real number has the
same
>probability of selection, Dave, representing that school of thought,
has
>P(x) being zero. Me, I say its probability is more along the lines
of
>1/2^n, where n is the unit scalar infinity.

"Unit scalar infinity" - meaning aleph_0 or something?

Well, Mr Seaman has a point. Let c = |R| (I guess this is 2^n ?).
There is a problem here with assigning 1/c to the "probability of
picking a via a random selection proceedure for picking elements from
(0,1)".

The problem is that the set of cardinals up to and including c
together with its multiplicative and additive structure doesn't embed
in any field,

e.g. 1.c=c=2.c so in any field extending this structure via
cancelation we have 2=1, and 1=0.

>Then, I also have the number of reals in the unit interval being 2^n,
and
>then, the sum of the probabilities over the entire interval is one.
This
>is a good feature in a probability distribution, for the sum of all
>possible results is equal to unity.

I am not sure how one would in gereral evaluate sums of the form
+_{i \in I} p(s_i)
if I is infinite
addition does not unambiguously extend to infinite sums. I guess you
are thinking of some sort of limit of finite sums, but this will only
work with countable sums, relative to some particular well order
mapping to a complete space.

>If P(x) is zero, and not an infinitesimal, then the corresponding
total
>sum would also be zero, and then selecting a random real from the
unit
>interval is also zero. Pick a number between zero and one. The
problem
>with selecting a number is that it's not random by any means I can
>consider. That's not necessarily so.

>It's simple to begin to sample a real number form the unit interval:
flip
>a fair coin. While not necessarily "random", there are so many
events and
>continua involved in the physical process that it is not necessary to
be
>concerned of external factors that affect the coin flip. My way is
to
>count the number of times the coin flips by the frequency of the
sound it
>makes, whether it be odd or even. Anyways, flip a fair coin, you
have
>started sampling a real number. Assign heads and tails to 1 and 0,
if the
>coin flip is heads then the first (binary) digit is 1, else 0.

>.0

>Flip again. Now, you are sampling two numbers, the one you started
>sampling with the first coin toss, which has two random digits, and
the
>second, with one random digit.

>.00
>.0

>Repeat.

>.100
>.10
>.1

>Etcetera

>.0100
>.010
>.01

>That's an iterative process and unfortunately never for any finite
amount
>of coin flips generates the exact specification of a real number
between
>zero and one.

>What it does show is that for any finite interval bounded by rational
>numbers the denominator of which is even,
>that there is a specific
>combinatorically determined probability that the eventually sampled
number
>would be in that range within the unit interval of real numbers.

True.

>As well,
>as each disjoint interval of the same length has the same exclusive
>probability of containing a sampled value, it would appear that over
all
>of the unit interval that each point was as likely as each other to
be
>selected.

Ok.

>So, for any process that could select infinitely many random bits,

I'm willing to work assuming we have such a thing.

>a
>question is if it generates a random real,

It generates a random sequence f:N -> {0,1}, and we know that this is
bijective with the reals.

> and how many of them. From the
>semi-infinite bit sequence, where semi-infinite means it has a
beginning
>and then trails off infinitely, that can be used as the binary
>representation of a real number, and then toss the first digit of
that and
>then there is a completely different real number, and so on and so
forth.
>In that sense if the infinite sequence ended with an infinitely
repeating
>sequence, which seems remarkably unlikely, then the semi-infinite bit
>sequence has sampled less real numbers.

True. The "semi infinite bit" will get you some subset of Q.

>That is because as the sample is
>retrograded or what-have-you, with the first bit removed and the
result
>string considered a new sample, if it represents a rational then at
some
>point the terminator repeats, at which point after discarding each
bit of
>one iteration of the repeating terminator, then no new samples are
formed.

You've lost me.

>That might have to do with selecting an natural integer m at random,
yet
>another wild improbability, yet for some preferable as it would be of
>probability 1/oo instead of 1 / 2^oo,

Here oo is taking the place of n above yes?

>and then an integer n less than or
>equal to that. Flip the coin m times, and say the n trailing digits
>repeat. That generates a variety of rationals, depending on the
>distribution over the integers.

>Consider if the goal was to get a distribution over the reals with
some
>kinds of numbers more likely to be represented in the population than
>others, statistically.

>If a sequence is semi-infinite, where its probability of being a
rational
>sequence is about as small as its probability of being an infinite
>sequence,

A "semi infinite sequence" will always correspond to a rational. The
set of semi infinite sequences is a subset of the set of infinite
sequences, so the second probability is also 1. Perhaps I
misunderstand you?

>then if it doesn't represent a rational sequence then it
>represents probably an aboslutely normal sequence as does each
subsequence
>derived by discarding finitely many bits from the beginning.

>One of the problems in considering the real numbers on the unit
interval
>is that the regular form of expression of the number is the
expansion, or
>reduced fraction of integers, eg 1/2. That's a problem in the
presence
>of infinitesimals because there is no expansion to represent an
>infintesimal or infinitesimal difference from the real number, and in
many
>"nonstandard" perceptions or models of the real numbers, there are
>infinitesimals among the real numbers.

Let d be some infinitesimal, and S be the real closure of R cup {d}.
Then the subset of S defined by {x: exists y,z in R (y<x<z)} is a
valuation ring V.

We can think of the asscosiated valuation as pointing out
"infinitesimal differences" if I understand you correctly.

Indeed, the set of infinitesimals in S is the maximal ideal of V.

>Briefly, I describe the
>infinitesimal differences form

From? I find this sentence rather confusing.

>definite real numbers represented by a
>convergent expression as representing definite real numbers, and the
>infinitesimal differences from them indefinite real numbers. The
reals
>are gapless and complete and there are only and everywhere reals on
the
>unit interval.

What is your definition of "unit interval"? Normally the definition
would be {x in R: 0<x<1} or perhaps the closure of this.

>There is considerable argument about the acceptance of infinitesimal
>probabilities. Then again, there always has been, in some form,
since the
>ancients and the debate about potential and completed infinities. In
some
>sense that implies both. It's easy to find seeming contradictions
about
>the infinite, even among mathematical sophisticates, the true goal is
to
>extirpate them all.

Some set theoretic results are inconvenient at times, such as 2.c = c.

>Well, I guess I'll go read some more of what I and others wrote
before
>about infinitesimal probabilities. If I think of something good,
I'll
>catalog it into my mathematical ontology.

>Ross

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