Re: infinity
- From: "Randy Poe" <poespam-trap@xxxxxxxxx>
- Date: 1 Nov 2005 10:50:10 -0800
zuhair wrote:
> Randy Poe wrote:
> > > I only asked if you cannot move the sticks what is the comparison
> > > between the number
> > > of sticks in w and in w' , ie can they be bijected, and I simply
> > > received no answer.
> >
> > You didn't? Let me be more clear about my answer then:
>
> what do you mean you didn't????
You said you didn't receive an answer. Here is your answer.
> > If I couldn't move the sticks then I would say, "this arrangement
> > of sticks is exactly the same as one which I can create by inserting
> > a space between every two sticks, neither adding nor removing any
> > sticks. Therefore I conclude that since this arrangement is
> > exactly the same as one which I know to be the same cardinality
> > as w, then this arrangement has the same cardinality as w."
>
> what do you mean arrangement, it seems you are forgetting that we are
> talking about cardinality and not ordinality. the whole statements of
> yours
By "arrangement" here I don't mean order. I mean the spacing
of w' as opposed to the spacing of w.
I would inspect that arrangement, with no need to physically move
anything, and say "w' can be achieved from w without adding or removing
any sticks from w. Therefore the cardinality of w' is the same
as w."
> What got arrangements to do with cardinality here.
I hope my use of the word "arrangement" is now clear.
In w the sticks are arranged with no space. In w' the same
sticks are arranged with a spacing of one stick.
> > You see a map which is not bijective. That does not tell you
> > whether there EXISTS another map that is bijective.
>
> For God's sake , what are you talking about
I'm talking about you looking at one map which is not
bijective and saying "therefore no bijection exists". That
is incorrect.
> I formed w' from
> w by a non bijective function so how can w' be bijected to w,
That's precisely what I mean. You say there's a map that isn't
a bijection, and ask me "how can a bijection exist"? It's easy.
there are infinitely many maps from w to w', and some of them
are bijections, and some aren't. Finding a map which is not
a bijection does not characterize the other infinitely many
maps. Why should it?
> > When I want to compare, I don't ask "is this particular map
> > bijective". The question is "does there EXIST a bijection".
Can you understand that?
> > Do you understand what "there exists" means? Your question is like
> > showing me a white dog and saying, "how can you look at this
> > dog and tell me there exists a black dog?" I don't conclude the
> > existence of the black dog from looking at the white dog. I
> > conclude the existence of the black dog because I've seen a
> > black dog, and the existence of the white dog does not cause
> > the black dog to stop existing.
>
> Yes their can exist a white dog, but it is not the dog we are talking
> about, this is the point.
No, this is not the point. The definition of equal cardinality
does not say "this is the bijection we are talking about". It
asks out of the entire universe of possibilities, whether there
exists one single bijection. It is precisely the same as asking
whether there exists a black dog. There is no "dog we are
talking about" and there is no "bijection we are talking about".
If I ask you "do there exist black dogs" you do not answer my
question by saying "this dog is white".
If I ask you "do there exist seven foot women" you do not
answer my question by saying "this woman is six foot".
If I ask you "do there exist bijective maps" you do not answer
my question by saying "this map is not".
> Since I formed w' by a condition of surjection to w wihtout injection.
> Do you understand the word FORMED , then it is impossible to imagine it
> to be bijected to w since this vilotes its definition.
It is impossible, apparently, for YOU to imagine such a
thing. To you, the existence of one map which is not a
bijection somehow rules out the possibility of any other
map being a bijection.
It does not. It is as simple as that.
- Randy
.
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