abundance of irrationals

mueckenh_at_rz.fh-augsburg.de
Date: 01/17/05 

Date: 17 Jan 2005 08:08:21 -0800

tinyurl.com/uh3t wrote:
> > From: mueckenh@rz.fh-augsburg.de
> > Do you know of any natural number n which gives an infinite
> > cardinality to the sequence 1,2,3,...,n ?
>
> Please define what you mean by "gives an infinite cardinality to".

Any natural number n, the sequence 1,2,3,...,n of which has an infinite
cardinal number. Remember: The set IN consists of finite natural
numbers only.

> Also, please give two examples of "X gives an infinite cardinality to
> Y", one where it's true, and one where it's meaningful but false.

I can't give a true example, because there is no true (actual)
infinity. Neither it is meaningful anywhere.

>
> > To speak of an infinite set of finite (natural) numbers (as Cantor
> > did) is a contradictio in adjecto.
>
> Do you consider it possible for there to be a collection of objects
> such that each single object is bounded ("limited" in common English
> usage) but the entire collection of objects is unbounded
("unlimited",
> goes on forever)?

If the infinity existed, it might be possible. Consider an infinite set
of similar objects. Or consider the set of all rational numbers. The
set of natural numbers could never be actually infinite, because a
finite ordinal number n is always related to a finite cardinal number
n, describing the cardinality of the set 1,2,3, ..., n. As long one of
the n is finite, the other will be so too. And the ordinal numbers n
are always finite, because the magnitudes of the natural numbers are.

It is not so hard to comprehend: ordinal = cardinal (if at least one of
them is finite)

Regards, WM

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