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Han de Bruijn wrote:
> stephen@xxxxxxxxxx wrote:
>
> > Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> wrote:
> >
> >>Why a "non standard" integer? Why is "large enough for our purpose" not
> >>good enough?
> >
> > You are the one who claims that it is not good enough. If you
> > want to talk about the probability of choosing an even integer
> > out of the set {1, 2, 3, ... n} where n is some very large integer,
> > then you will get no argument from anybody. However that is
> > apparently not good enough for you, and so your definitions keep
> > bouncing back and forth, apparently just for the sake of argument.
>
> Sure. Taking arguments out of context always works in 'sci.math'.
>
> The probability that an integer out of the set {1, 2, 3, ... n } is
> relative prime to 2, 3 and 5 is:
>
> = (n - floor(n/2) - floor(n/3) - floor(n/5)
> + floor(n/(2.3)) + floor(n/(2.5)) + floor(n/(3.5))
> - floor(n/(2.3.5)) - 1) / n .
>
> Right?
>
> For n -> oo we find:
>
> = 1 - 1/2 - 1/3 - 1/5 + 1/6 + 1/10 + 1/15 - 1/30 = 4/15 .
>
> Therefore the probability that a natural is relative prime to 2, 3 and 5
> is equal to 4/15 . But this is _different_ from the finitary result. And
> it is _different_ from the result in mainstream mathematics that there
> do not exist such probabilities. (It agrees with "asymptotic densities",
> I hope). Following the style of reasoning a la Max Planck, we could say:
>
> The chance of picking an arbitrary natural from all naturals N is 1/N .
> This is an infinitesimal. Integration of these infinitesimals is delayed
> until we are asked for a finitary result.
>
> Now we'll wait.
>
> Pompedompedompedompedom.
>
> Still waiting.
>
> But hey!
> What is the probability that a natural is relative prime to 2, 3 and 5?
>
> Integration starts.
>
> And the outcome is 4/15.
>
> Han de Bruijn
Hey, that's the same thing as the asymptotic density of numbers not
multiples of those, in the natural integers.
That's easily determined by multiplying them together, and then
calculating how many numbers less than that are not having as factors
one of those, and dividing it by the product, eg there are eight
numbers less than 30 not divisible by 2, 3, or 5 (1, 7, 11, 13, 17, 19,
23, 29) so 8/30 = 4/15 of all the natural integers are not multiples of
2, 3, or 5. What's true for the first block of 30 positive integers is
true for each successive block, via induction it's true for all
integers.
Then, if there ever existed a random natural integer generator at
uniform random, the best estimate of the probability of its output not
being a multiple of those small primes is 4/15.
If you integrate f(n) = 1, defined on the natural integers, over the
positive reals, it evaluates to two. That's why discrete reals on a
2-D line are called two-sided, or about the impulse function in the
semi-infinite. On a plane they're four- or five-sided.
(Some of) the arguments against a uniform distribution over the
naturals, and thus the existence of a "nilpotent" infinitesimal, seem
to result from measure theory, and Vitali's result predicated on there
not being a nilpotent infinitesimal, that is a fallacy of circularity.
Where there is such a beast that result does not hold.
That gets into NCD sets in the reals, in terms of well-ordering and
natural orderings of the reals, and frequentist didn't used to be a bad
word for probabilists.
There are a variety of non-standard measure theories. Most and if I'm
not mistaken all applied results of measure theory don't need anything
but a minimal set of statements about continua.
One of the first items in "Counterexamples in Real Analysis" is that
there exists s, or iota, in the reals.
Speaking of counterexamples, delta's infinitesimal. With unrestricted
transfer the hyperreals are no better than the reals, and no different,
there are only reals in the reals, the reals are already complete, and
the naturals are implicitly compact. Completion of induction requires
a maximal element in the naturals, that there is is a synthetic
deduction.
Regards,
Ross F.
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