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 <[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 -~----------~----~----~----~------~----~------~--~---