Re: Calculus XOR Probability

In article <MPG.1e824df87fe2fdcb98aaf7@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

David C. Ullrich said:
On Wed, 15 Mar 2006 11:01:17 -0500, Tony Orlow <aeo6@xxxxxxxxxxx>
wrote:

David C. Ullrich said:
On Tue, 14 Mar 2006 15:52:34 +0100, Han de Bruijn
<Han.deBruijn@xxxxxxxxxxxxxx> wrote:

David C. Ullrich wrote:

On Tue, 14 Mar 2006 10:40:12 +0100, Han de Bruijn
<Han.deBruijn@xxxxxxxxxxxxxx> wrote:

No, you said that you're taking the sum of infinitely many
zeroes to be non-zero. So either your axioms are different or
you're making an error.

Let's withdraw _that_ statement for the moment being. Informal
question: in your opinion, Mr. Ullrich, do infinitesimal
quantities have a chance in standard mathematics? And if so,
what are the pro's and con's?

The question needs to be more precisely formulated. (For
example, I don't know what it means to say that something "has a
chance in standard mathematics".)

Also I don't see why anyone should care about my _opinion_ on a
mathematical question.

Han de Bruijn


************************

David C. Ullrich


But, David, what IS the chance of any given element being selected
from an infinite set? In standard probability, it's 0%,

Yes.

Well, not necessarily yes, because you're leaving out some
important details. But there will be times when the answer is yes.

indistinguishable from no chance at all.

Depends on what you mean by "no chance at all".

And yet, each element does have some chance, because one of them
is going to be selected.

Depends on what you mean by "has some chance". Yes, for any given
element there is some chance that it will be selected. For any
given element the probability of that happening is 0.

Probability = 0 simply does not mean impossible.

So, how does one represent that kind of chance
mathematically? Isn't that an infinitesimal probability?

No.

************************

David C. Ullrich


0% probability CAN mean entirely impossible, and in fact, does mean
that in any finite set of possible outcomes. So, how do you
distinguish between 0% probability where is is some chance and 0%
probability when there is none? Why do you say the first is not the
same as having some infinitesimal chance, nonzero but smaller than
any finite probability? They are obvious two different kinds of zero.

On a closed real interval, one can define an additive measure function
such that the measure of a subinterval of the interval is the probabilty
that a value from that subinterval will be chosen. Then there will be
non-empty sets with probability zero, but that does not mean that it is
impossible for a member is such a set to be chosen.
.

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