Re: Continuum hypothesis
- From: Han.deBruijn@xxxxxxxxxxxxxx
- Date: 3 Jan 2006 12:16:00 -0800
cbrown@xxxxxxxxxxxxxxxxx wrote:
>
> So you're saying that there exists an infinite sequence (i.e., it makes
> sense to talk about it as a mathematical object)? For example, does the
> sequence (1, 2, 3, 4, ..., n, ...) "exist" "in the limit"? Isn't that
> just the same as the naturals?
That's what I hoped for. But the "standard" naturals seem to have
additional
properties. And these do not automatically show up as the above
sequence
becomes "infinitely large" (whatever that "means" according to "you").
> > If not, what's your problem?
>
> Consistency?
Consistency indeed, but within mathematics AND beyond.
[ ... reasoning with Archmedian property involved skipped for now;
should be
comprehended later on; thanks anyway; own reasoning repeated
here: ]
Take the finite sequence of naturals (1 ... n) . When I count the even
and
odd numbers (cumulative) _within_ that sequence, I find the following:
If n = even then the number of even integers in (1..n) is n/2
and the number of odd integers in (1..n) is n/2
If n = odd then the number of even integers in (1..n) is (n-1)/2
and the number of odd integers in (1..n) is (n+1)/2
Now suppose that the number of elements in (1..n) is very large. Then
we
can define the probability P that a natural in (1..n) will be even/odd:
If n = even then P(even) = (n/2) / n = 1/2
and P(odd) = (n/2) / n = 1/2
If n = odd then P(even) = ((n-1)/2) / n -> 1/2
and P(odd) = ((n+1)/2) / n -> 1/2
In the limiting case, of an infinitely large interval (1..n), quite as
expected, we find:
P(even) = P(odd) = 1/2
I'm still at lost what precisely would be the "flaw" in this reasoning.
A generalization could be the theorem that P(a natural n is divisible
by a) = 1/a . I _know_ that mainstream mathematics finds it wrong
to say this, but I can not comprehend why.
Han de Bruijn
.
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- Re: Continuum hypothesis
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- Re: Continuum hypothesis
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