Re: infinity ...


David R Tribble wrote:
> David R Tribble said:
> >> So I'm giving you set S, which obviously does not contain any
> >> infinite numbers. So by your rule, the set is finite, right?
> >
>
> Tony Orlow wrote:
> > If it doesn't contain any infinite members, it's not infinite. Those terms
> > differ by more than a constant finite amount, but rather a rapidly growing
> > amount greater than 1. There is no way you have an infinite number of them
> > without achieving infinite values within the set.
>
> Yes, you and Albrecht keep saying that repeatedly. Please demonstrate
> why it must be so, because it's not.


Your argumentation is not fair, but I don't wonder about that.
_You_ has to show, that in the case of the whole set there is no
natural number as big as the whole set.
You argue: there is no infinite natural number since the peano axioms
don't allow an infinite natural number.
That's right. I agree with you.
But that's no proof about sets. That's only an aspect of the definition
which contradicts with the fact, that every set has a number of
elements.

You misinterpret totally when you say, I think there must be an
infinite natural number. I don't think so. I only argue that, if there
are infinite sets, there must be infinite natural numbers (since nat.
numbers are sets).
I don't say: there are infinite sets. You say: there are infinite sets
and there is no infinite number. And I say: If there are infinite sets
there must be infinite numbers.

My argumentation is very easy:
Every nat. number represents a set. If you look at the first 100 nat.
numbers, the 100th nat. number "100" represents the set {1, ... , 100}.
As this holds for every nat. number, if there are infinite nat. numbers
there must be a infiniteth nat. number representing this set.
But the definition of the nat. numbers with complete induction leads to
the consequence, that there could not be an infinite nat. number.

That's the contradiction.

So either the definition of nat. numbers must be changed or there is no
infinite set of natural numbers.
Or infinity must be interpreted in a completely other way. Not as a
size like you do. Infinity is just an unability to count it with
numbers because it runs out of all what we can know.

All this is shown very expressive in my sketches at the start of this
thread.

Why do you misinterpret all the time? Maybe my ability to express my
thoughts in english is too bad.
But why do you misinterpret Tony also? I think he is native english
speaker and you should be able to understand him.

In this state there is no real problem with all this. aleph_0 is just
onother symbol for infinity.
The problems occure in that moment if someone declares, that aleph_0 is
a size, which is greater than any nat. number.
But there is no "greater" or "less than" or something like this. There
is just something other, something out of the things we could measure,
wigh or count.
The possibility of bijection don't say anything about the size of
infinity, since infinity is something sizeless, endless, countless.
That's all.

Regards
AS

.

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