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Han de Bruijn a écrit :
Virgil wrote:
In article <MPG.1e9dd8e096dfe76398abee@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
Han's original point is that calculus is very precise and works,
whereas the case where you have a uniform probability distribution
over an infinite set of possibilities is not well handled by the
classical notion that all individual probabilities sum to 1, because
the individual probabilities are considered equal to zero. It is the
opinion of both of us that this can be resolved, among other ways, by
assigning infinitesimal nonzero probabilities to each possibility,
leaving intact the notion that the sum of the individual
probabilities sums to 1. I am not sure why this is roundly rejected.
Can you address that?
Because there is no model of the reals, either standard or
nonstandard, in which the sum of countably many equal values can equal 1.
You can repeat this a thousand times, but Tony and I don't get it. In my
not so humble opinion, you must also reject then the integral(0,1) dx ,
because it is derived from the Riemann sum n.1/n , which is exactly the
same as summing up (n) probabilities with 1/n chance for each. dx = 1/n.
Now take the limit for n->oo and you're done. What's the problem? (Well,
I _can_ understand that mainstream mathematics can't drop the whole wide
world of calculus because of this little issue)
Han de Bruijn
Here is what comes closest to HdB "ideas" : take a non standard integer
N ("infinite", ie greater than all standard ones, of course). Define
the measure of a standard set of integers A as p(A)= the sum of 1/N for
all n <N in A* (A* is the extension of A to the non-standard integers)
For instance, p("n is even")= 1/2 or 1/2-1/N according as N is even or
odd; and the shadow of p(A), p°(A), is a real having "almost" the
properties of a probability (could it be the density :-)), while p(A)
*is* (of course) a discrete measure... and so additive. Note that with
this definition, p(n=42)=1/N and the shadow p°(n=42)= 0, according to
HdB "intuitions". Note also that the main point of contention (ie p(A),
or p°(A)are nor probabilities (ie not countably additive)) is somewhat
shunted...
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