Re: TOE Via Cantor's Transfinite Arithmetic

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 03/23/05 

Date: 23 Mar 2005 01:18:51 -0800

Timothy Little wrote:
> Ross A. Finlayson wrote:
> > for any uniform, random probability distribution over the integers,
> > the probability that a sample is an even integer is one half.
>
> That's true in the same sense that any square prime is less than
> ten.
>
>
> - Tim

Hi Tim,

No, it's true in a different way.

Remember when we were talking about the real numbers, one thing that I
believe was addressed passingly in that conversation was that some
people have an easier time accepting a uniform random distribution over
the reals than they do of one over the integers. Recently I read
something that might have to do with that is called the Bayesian
estimate, which is over a uniform probability distribution of the unit
interval of the real numbers. We talked about sampling the reals by
binary decisions.

That might have been very indirectly addressed, our conversation was
more directly about the well-ordering of the reals, and various
considerations of what it would mean for any sequence to be a
well-ordering of the reals, where one of the consequences of the axiom
of choice, which is a theorem of the axiom-free theory, is that any set
is well-orderable, and talk about iota. We discussed that with regards
to Cantor's first proof, also here called "nested intervals" or
sometimes by myself "Cantor/Megill style", and the transfer principle,
which is about standard and nonstandard results each being true, and
induction in the sense of infinite induction being induction and each
the same thing.

So anyways, I don't say "here's an example of a uniform probability
distribution over the unit interval", where uniform means each value is
equally likely to be a sample, instead I say that for _any_ uniform
probability distribution, regardless of which one it is or that they
would each be the same, for _any_ of them, the probability that a
sample is less than 1/2 is 1/2. For the interval of length two, the
probability that a sample is less than one is 1/2. For a random,
uniform distribution over the integers, the probability that a sample
is an element of a subset is the asymptotic density, where for even or
odd integers that's one half. Given a subset of the set of integers at
random, the probability that a sample from the integers is in that
subset is one half.

Anyways, thanks for reminding me to reread "On Well-Ordering(s) and
Sets Dense in the Reals, Infinity", as we posted on the unmoderated
discussion forum sci.logic and to some extent sci.math, discussion
groups with various amounts of noise about the sciences of logic, and
mathematics, that we read sometimes, and happen to be reading now. The
conversation was left off with consideration of Freidman's 2004 quote
that well-ordering the rationals would require infinite induction.
That was only a couple months ago.

I got some good laughs out of that.

Ross

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