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.

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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 <[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 > > > > --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~----------~----~----~----~------~----~------~--~---