Re: Earth 8??

Ophidian wrote:
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.

Any. But not every. There's a distinct difference.

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.

In the example given of integers and positive integers, _every_ integer
eventually shows up paired with a positive integer. No matter what integer
you come up with, there's a place where it's paired up, uniquely (for that
particular ordering) with a positive integer. That's what a 1:1
correspondence means: that _every_ member of one set can be paired up with
a member of the other set with none left over.

David L. Burkhead


.

Relevant Pages

  • Re: Earth 8??
    ... and that is referred to as "countably infinite." ... reply, though, in disagreeing that the set of integers is not the same size as the set of positive integers. ... Clearly going this way we never get to 6.5" high duck, ... 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. ...
    (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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