Re: Calculus XOR Probability

David C. Ullrich wrote:

On Tue, 21 Mar 2006 10:38:13 +0100, Han de Bruijn
<Han.deBruijn@xxxxxxxxxxxxxx> wrote:

Provided that probability theory can be and has been extended to the
hyperreal numbers.

That last "provided that" strikes me as the one reasonable
thing you've said in all this. [ ... snip ... ]

Seems that you have skipped over all the mathematics I've done with
those hyperreal numbers. Cannot help, but find this somewhat unfair.

But your point is that *I* should give a _precise_ statement of the
transfer principle. I won't. For the reason that I will be building
on top of this. It has no use trying to re-invent the wheel. Anyway,
you could click on "The *-Transform and the Transfer Principle" in:

http://mathforum.org/dr.math/faq/analysis_hyperreals.html

[ ... decent exposure about "Natural Densities" snipped ... ]

Let's suppose for the sake of argument that n is a "random positive
integer", all positive integers equally likely. Let's say p is
the probability that n = 1. Then p is also the probability that
n = 2 and the probability that n = 3, etc. Now it follows from
the axioms of probability theory that

(i) 1 = p + p + p + ...

By definition, what that means is that

(ii) 1 = lim (np),

the limit as n -> infinity of n times p. Where "limit as n ->
infinity" refers to ordinary finite integers. There simply
is no p that satisfies equation (ii). If p = 0 then
lim (np) = 0. If p is a positive real number then
lim (np) = infinity. And if p is a positive infinitesmal,
_if_ there _is_ such a thing as lim (np) then lim (np)
is going to be another infinitesmal.

With standard analysis, you are right. But with non-standard analysis,
you are wrong. Simply because you don't have that rigorous definition
of what an "infinitesimal" is in standard analysis.

Why do I have to repeat it? Let *R be set of all hyperreal numbers.

Let p = (1,1/2,1/3,1/4, ... ,1/n, ...) in *R
Let N = (1,2,3,4, ... ,n, ...) in *R

Then it follows that N.p = 1 in *R .

And you responded correctly to this with: _Right_. See:

http://groups.google.nl/group/sci.math/msg/86ad7b596afac6ea?hl=en&;

Han de Bruijn

.