Re: Does a "pure" real valued probability function mak

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 12/05/04 

Date: 5 Dec 2004 06:50:45 -0800

I apologize for baiting Herc with the nutcase comment. Hell, even
Cantor spent some time in a mental hospital, as long as I'm taking
swipes at insane people. Then again, "Uncle Louie was crazy." I
don't put much stock in the voodoo methods. I'm a sink.

I'm not Uncle Al. I like and respect Uncle Al regardless of some of
his biases and prejudices. I'd like to meet him in person and have
beer with him or something. I respect Al, because Al's a hard
scientist, and he'll happily answer your honest question about
chemistry and he's probably right. I use more innocuous pejoratives,
which I used to spell perjoratives, eg, you're incompetent. I
disagree with some of Al's opinions.

I further analyze the notion of the sequence representing the rational
being infinitely many sequences of the same or a similar rational.

Basically a rational number will exhibit a repeating terminating
sequence. Then, the sample sequence of 111 (01)... would also
represent the sample sequence 11 (01)..., and then (10)... and (01)...
infinitely many times. You can get as many rational sequences as
subsequences as there are digits in the repeating terminator,
infinitely many times, plus as many as are in the prefix, once each.

One and zero and the reduced fractions with prime denominator not
equal to the base would appear to be sampled more often, because a
fraction with a composite denominator would have a prefix and only be
sampled finitely many times. They also might appear to be sampled
less often, because the probability of getting the same number from
consecutive rolls of a fair die diminishes, but they don't, because
it's a fair die and independent.

Then I encounter a problem that while the irrational sequence contains
only unique sequences instead of duplicates that the sample sequence,
just one of them, would have infinitely many subsequences where each
would be unique.

Obviously enough the probability to get any particular sequence,
representing either rational or irrational, is the same. So I'm
considering how those probabilities coalesce. I almost had it there:
the probability of the sample sequence being any particular sequence
is the same as that of any other. These plain language statements are
often easy to find. That's not quite it, but the idea here in terms
of probability is that one sample is infinitely many samples in varied
ways.

Here the consideration is of the sample digit, ie a coin flip or
random counting number, and the sample sequence, an infinite sequence
of those digits.

I had a glimpse of the plain language statement above, if it's worth
anything I'll think of it again, I want to discuss the normality of
the numbers. With an irrational sequence, it's probably normal to any
base. While that is so, the digits of pi are not within the digits of
e somewhere. Where the sequence is normal, so is any subsequence,
where subsequence here means discarding finitely many digits from the
prefix.

So, the probability of a sequence sample includes all of the
subsequence samples. By sampling a real number by sampling some
infinite sequence of digits, there was thus sampled infinitely many
real numbers, if it is irrational, or finitely many if rational.
Where it is rational as many digits as there are in the terminator
indicate how many rationals are sampled infinitely many times each by
the one sample.

The probability of getting .000... all the way out to infinity is the
exact same as that of getting pi/4, or sqrt(2)/2, or .111.... When
pi/4, an irrational number between one and zero, is sampled as a
sequence, so is (pi/4 *2)-1, and that times two minus the first bit of
that subsequence, ad infinitum. When zero is sampled as a sequence,
only zero is sampled, infinitely many times. Then, where basically
this is about some nonstandard probability, when (pi/4 * 2)-1 is
sampled, so are all of its subsequences, each different, and it still
has the same probability of being sampled as zero, which when sampled
is sampled infinitely many times. There are infinitely many sequences
that have as a subsequence the digits of e, and there's only one
sequence that has the digits of zero, and all of its subsequences have
the digits of zero, and it is a subsequence of many sequences.

So, each subsequence of zero gets a chance over and over again, and
when it's sampled so is itself infinitely many times over again.

With finite sequences there is no problem, this is all trivial and
well-known in the finite.

While that might be so, in complex systems there might be continuistic
tendencies in the large.

I guess basically I got a good notion for some definitions to surround
a context of a "nonstandard probability." I think the idea is that
the probabilities would still sum to one, unity. Then I can catalog
that into my mathematical ontology.

Search for "nonstandard probability":
http://www.google.com/search?q=%22nonstandard+probability%22

Thus, there might be an avenue into the comprehension of the
nonstandard probability from that. Here's to conciliation.

Warm regards,

Ross "Dolt" Finlayson 

--
"Say hello to my little pen."

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