Re: abundance of irrationals!)
- From: mueckenh@xxxxxxxxxxxxxxxxx
- Date: 1 May 2005 07:49:34 -0700
imaginator...@xxxxxxxxxxxxx wrote:
> aeo6 Tony Orlow wrote:
> > The fact is that one cannot have an infinite set of finite numbers,
> if the difference
> > between any two is at least 1, whiich is true of natural numbers.
>
> That's a _fact_, is it? I mean, not just something you worked out for
> yourself?
>
> Anyway, given this 'fact', suppose I've tried to make an infinite set
> of finite numbers at least 1 apart - to keep it absolutely safe, I'll
> make the set of even integers {0, 2, -2, 4, -4, ... }. Now I think
> there are an infinite number of even integers,
Do you really? Where starts infinity?
because I don't think
> that if I start counting them, that I will ever stop.
That s potential infinity. But that is not mant by set heory, because
it does not supply a cardinality, in particular it does not lead to
aleph_0.
Does your 'fact'
> mean that actually some of these even integers are actually infinite?
> Suppose it must, no?
As long as the number of your numbers counted is finite, the magnitudes
will be finite. If one of them changes, the other will change too
simultaneously.
>
> OK, how many of them are infinite?
Precisely as many as do not fit into a finite set.
{1,2,3,...,n} do not include any infinite magnitude because that set
has not an actually infinite cardinality.
> Can you see why your 'fact' is nonsense?
Best ask yourself.
Regards, WM
.
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