Re: Earth 8??

Joe Sewell wrote:
In article <1141535224.377757.241600@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"The Watch Dog" <tirhuan@xxxxxxx> wrote:


Joe Sewell wrote:

Actually, the size of the set of positive integers is exactly the same
as the size of the set of integers (they can be put into a 1-to-1
correspondence), and that is referred to as "countably infinite." The
size of the set of odd integers, prime numbers, and rational numbers
(fractions) are each also countably infinite.

Very well. That's not I heard the phrase "countably infinite" defined when I first heard it, but I defer to consensus.

What consensus? What's your definition?

(I agree with another reply, though, in disagreeing that the set of integers is not the same size as the set of positive integers. The set of positive integers is a *subset* of the set of integers, therefore it must be smaller; specifically, the cardinality of the set of integers = 2 * card(set of positive integers) + 1 ... remember, 0 is neither positive nor negative.)

I agree with you here. I was earlier agreeing that a 1:1 corresponance _can_ be made between these sets, AND that both are 'countably infinite'.
I hope I wasn't taken as agreeing that the two sets were both the same size.

Interestingly, Dog's original premise can be applied to most sets.
(I'd say all, but someone might find an exception.)

For example:
The set of all imaginable animals.
We'll start with:
1 a 2" high duck
2 a 3" high duck
3 a 4" high duck
etc.

Clearly going this way we never get to 6.5" high duck, let alone 'pink elephant'. But I could easily assign a 1:1 correspence, say I stop counting ducks for a while and go:

4 Pink elephant
5 Unicorn
6 Dragon

and then resume with:

7 a 6.5" high duck.

Every element of my imaginable animals set CAN be assigned a positive integer.

So what Dog was saying, I'd assume, is not merely that there are enough integers to assign to _any_ set, which of course, there are, but that some sets will have an ordered corresponence. Further, to be relevant, those sets will have a predictive relationship. If I know an element of set I, I can then derive the corresponding element of set II. Dog's correspondence between all integers and all positive integers does have such a correspondence. But I'm not conceding that they are the same size.
.

Relevant Pages

  • Re: Earth 8??
    ... the size of the set of positive integers is exactly the ... numbers are each also countably infinite. ... a 2" high duck ... set II. Dog's correspondence between all integers and all positive ...
    (rec.arts.comics.dc.universe)
  • Triples correspond to sequences
    ... For every triple of positive integers there is associated a sequence { ... Is this a one-to-one correspondence? ... With the addtional condition gcd, how about now? ...
    (sci.math)
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