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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
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