Technically, I should say that countable means that the set can be put
into a one-to-one correspondence with *a subset of* N, to include
finite sets.
Tom
Tom Caylor wrote:
> 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.
>
> Tom
>
> 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 <Daddycaylor.domain.name.hidden> 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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Received on Thu Jul 13 2006 - 13:54:26 PDT