Re: infinity
- From: "Randy Poe" <poespam-trap@xxxxxxxxx>
- Date: 27 Oct 2005 19:18:21 -0700
zuhair wrote:
> Ok good I think here we have mutual agreement:
I doubt it.
> Now when we take set P and calculate its cardinality , my argument is
> that the cardinality of that set , is the same as the cardinality of
> the field of any series having the same members of that set.
I told you before that I am not familiar with the term
"field of a series". Repeating it will not make me more
familiar with it.
> Now let's take the series S = (0,0), (1,0) ,
> (2,0),................,(0,1) , ( 1,1) , (2,1),.....
I know what you mean, but in fact I can't think of a way
to define this as a sequence in that order, since a sequence
*IS* a bijection between N and a set of numbers.
> lets P be the field of the serial relation of S.
>
> Now P is a set .
>
> Now I can calculate the cardinality of P by bijecting with N.
If a bijection exists, then card(P)=card(N).
> first I will try to biject P with N while P is having the same order as
> S
I guess we aren't in agreement, since that's not a meaningful
statement. A bijection of N to P merely means that every element
of P is assigned some label from N. Any order that accomplishes
this will do.
> I dont mean that I am saying that P should have an order,
You don't? We'll see.
> but it is only a trial to biject P at that order
So you are going to ask, "if a bijection exists, does it begin
this way? f(n) = (n,0), n=1,2,..."
And you will get the answer, "No, if a bijection exists, this
isn't it."
> lets say it's chance related that I had P with it's members ordered
> like S)
>
> the result is that P cannot be bijected to N.
No, the result is that this is not a bijection of P to N, not
that "P can't be bijected to N".
> but it can be bijected to N,N
There's no such thing as N, N.
However, what you have rediscovered is that the infinite set
P can be bijected with a proper subset of itself, something
you claimed to be nonsense.
> Since:the cardinality of that set , is the same as the cardinality of
> the field of any series having the same members of that set.
I still don't know what you mean by that.
> then the cardinality of set P is Aleph-0,Aleph-0
>
> and not Aleph-0 as you thought or as it is known.
The cardinality is defined to be Aleph_0 if a bijection to
N exists. You try one mapping out of the infinite set of
possibilities, find it is not a bijection, and conclude that
there is no bijection possible.
Yet you already know there *is* a bijection possible. Therefore
your statement "the cardinality is not Aleph_0" is wrong, and
you know it to be wrong before you type it.
> Randy you think that the function you presented bijects set P to N but
> in fact it doesn't
>
> it bijects another set although it looks similar to setP but it
> contains other members in it.
Oh yes? Name one.
- Randy
.
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