N is defined as the positive integers, {0, 1, 2, 3, ...}, i.e. the
*countable* integers.   (I am used to starting with 1 in number
theory.)  N does not include infinity, neither the "countable" infinity
aleph_0 nor any other "higher" infinity.  Infinite length "integers"
fall into this category of infinities.  As you have shown, the infinite
length "integers" cannot be put in a one-to-one correspondence with N.
This is the definition of uncountable.  However, just because the set
of infinite length "integers" is uncountable, or even equivalent in
cardinality to the set of real numbers, doesn't mean they are real
numbers.  There are other sets that have the same cardinality as the
set of real numbers, 2^aleph_0, for instance the set of all subsets of
N.  Granted, there are (undecidable) mysteries involved, as Jesse has
alluded to, when we start trying to sort out all of the possible
infinite beasts, and this is partly why the Continuum Hypothesis is
such a mystery.  But with the given definitions of countable and
uncountable, infinite length "integers" are uncountable, and so not in
N.  Conversely, just because you can *start* counting the reals
(starting with the rationals), and you can *start* counting the
infinite "integers", and it would take "forever" (just like counting
the integers would take "forever") doesn't mean they are countable.  We
need some kind of definition like the one-to-one correspondence
definition of Cantor to distinguish countable/uncountable.


Quentin Anciaux wrote:
> Hi, thank you for your answer.
> But then I have another question, N is usually said to contains positive
> integer number from 0 to +infinity... but then it seems it should contains
> infinite length integer number... but then you enter the problem I've shown,
> so N shouldn't contains infinite length positive integer number. But if they
> aren't natural number then as the set seems uncountable they are in fact
> real number, but real number have a decimal point no ? so how N is
> restricted to only finite length number (the set is also infinite) without
> infinite length number ?
> Thanks,
> Quentin
> On 7/13/06, Tom Caylor <[EMAIL PROTECTED]> wrote:
> >
> >
> > I think my easy answer is to say that infinite numbers are not in N.  I
> > like to think of it with a decimal point in front, to form a number
> > between 0 and 1.  Yes you have the rational numbers which eventually
> > have a repeating pattern (or stop).  But you also have in among them
> > the irrational numbers which are uncountable. (Hey this reminds me of
> > the fi among the Fi.)
> >
> > To ask what is the next number after an infinite number, like
> > 11111...11111... is similar asking what is the next real number after
> > 0.11111...11111...
> >
> > Tom
> >
> >

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