Re: Calculus XOR Probability

Robert Low said:

Han.deBru...@xxxxxxxxxxxxxx wrote:
David C. Ullrich wrote:
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 .

OK. But you should realise that if you do this, the
probability of choosing an even integer less
than N is (infinitely close to) 1/2, but the
probability of choosing a finite integer doesn't
exist, and neither does the probability of
choosing a finite even integer (because the
set of finite integers and the set of even finite
integers are not internal sets).

This sort of thing is why Ullrich was stressing
the importance of the transfer principle.



I thinks that's absolutely true. You cannot really address a set defined for
over all finite values, once you have really accepted infinite and
infinitesimal values, because it becomes clear that there is no clear boundary
where the finites end, and no exact number of them in terms of the overall
infinity allowed with infinite values. It becomes apparent, in that situation,
that the set of finites is finite in range by definition, and as such, cannot
be measured in terms of infinities, but is unbounded in range as well, and
cannot be indentified as a finite number. I think internal set theory has a
point in not considering "the range of all finites" to be a constructive enough
definition to constitute a set.

But that doesn't mean specific infinities and infinitesimals can't be addressed
without causing this problem.
--
Smiles,

Tony
.

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