Re: Infinity......
- From: Six Letters
- Date: Thu, 07 Dec 2006 12:22:22 +0000
On 6 Dec 2006 07:38:53 -0800, "William Hughes" <wpihughes@xxxxxxxxxxx>
wrote:
Six wrote:
On 5 Dec 2006 23:06:49 -0800, "William Hughes" <wpihughes@xxxxxxxxxxx>
wrote:
zuhair wrote:
Yes, the have the same size as determined by f when f is bijective.
but when f is not bijective then they don't have the same size_f.
If they do not have the same size then A changes size when
you rename its elements.
Can you be more clear about that.
We have a valid name change for A if for every element a
of A
-we have a new name for a
-if a_1 and a_2 are elements of A
then the new name for a_1 is not the same
as the new name for a_2
Example
A = {1,2,3}
A valid name change is:
change the name of 1 to 4, the name of 2 to 5 and the name of 3 to 6
to get the set
A'={4,5,6}
One invalid name change is
change the name of 1 to 4, and the name of 2 to 5 to get the set
A'= ?
This name change is invalid because we did not specify a new
name for every one of the elements of A (specifically we did
not specifiy a name for the element named 3)
Another invalid name change is:
change the name of 1 to 4, the name of 2 to five and the
name of 3 to 4 to get the set
A' = {4,5,4}
This name change is invalid because we used the same new name
(4) for more two different elements (the element named 1 and the
element
named 3).
Recall, we know nothing about size except for the
fact that if we get A' by applying a valid name change
to A, then A and A' have the same size.
lets see this example.
A= {1,2,3,........}
B={ 2,4,6,........}
Now let f:B->A, f(x)=x
Now according to this f, I cannot rename all of A elements by replacing
them without repetition by elements of B.
It is true that there is a way of using the elements of
B that does not give a valid name change.
However, the fact that there is a way
to use the elements of B so you do not have
a valid name change for A, does not change the fact that
there is a way to use the elements of B that gives a
valid name change for A.
All we need is *one* way of using the elements of B
to give a valid name change for A. Since this
does not change the size of A, we can conclude
that A and B have the same size.
and this is the meaning of
(size E < size N)_f.
so what do you really mean by your statement : if they do not have the
same size then A changes size when you rename its elements. but here we
cannot rename all the elements of A by elements from B when B is
smaller in size than A.
No. There is more than one way to use
the elments of B to rename the elments of A.
The fact that there is one way in which we cannot use
the elements of B to rename all the elements of A does not change
the fact that there is a way to use the elements of B to
rename the elements of A. So B is not smaller in size than A.
see what I wrote. you are not differentiating between reltive and
absolute size.
that's your problem
If the two sets A and B do not have
the same absolute size, then the absolute size of A
changes when you rename its elements.
This is weard, I can't understand that. how I understand it is as
fellows;
if A and B do not have the same absolute size, then we cannot change
all the elements in A by elements in B if A has an absolute size bigger
than the absolute size of B.
we can change all the elements in B by elements from a proper subset of
A but this will not change the the absolute size of B. it will remain
the same.
It seem that you think that for any two sets, if we can change the
elements of one of them by elements from the other without affecting
size, then they are equal in size.
No. We start with the statement (here I quote you)
we cannot change the size of any set
by merelly renaming each of it's elements.
So rather than
if we can change the elements of one of them by elements from
the other without affecting size
we have
if we can change the elements of one of them by elements from
the other *then this does not change the* size
So
if we can change the elements of one of them by elements from
the other then this does not change the size and therefore
they are equal in size.
But if we change elements of one of them
them by elements from the other and this results in size change of the
set who's elements are replaced, then these two sets are inequal in
size.
We cannot do this. If we have a valid name change for A, then
the size of A does not change. Your example above does
not work. The function f:B->A, f(x)=x does not define a valid name
change on the
set A (some elements of A are not included).
- William Hughes
I may not have followed this whole argument about replacing,
swapping or re-naming elements of infinite sets correctly, but it worries
me that your notion of valid name changes is a prescription for banishing
the problem by fiat. Yes, there is a way of renaming the elements of B,
following the bijective relationship between the sets A and B, so that all
elements of A and B are included. But there is another way of renaming or
pairing off the elements (Zuhair's) which does not include all elements of
A. That it does not 'succeed' in using all elements of A is precisely the
point.
No. The fact that there is a way that does not succeed does
not change the fact that there is a way that does succeed. All we
need is one way that does suceed.
Isn't there a danger that you have swapped or renamed or replaced
one problem (a paradox about size) for another (essentially identical )
one, a paradox about renaming elements in infinite sets?
No. All I have shown is that if we apply an obvious constraint
to the term "size", (renaming the elements of a set does not
change the size of the set), then for *any* size we choose
if there is a bijection between sets A and B, then A and B have
the same size.
The statement
there may be two ways of using the elements of
the set B, one way that suceeds in renaming the elements of A
and one way that fails to rename the elments of A
is true but why would you call it paradoxical?.
The root "paradox" of inifinite sets remains.
There can be a bijection between an infinite set A
and a proper subset of A.
This only becomes a paradox when some natural-looking implications
about size are included. There is an important difference.
However, since this is the defining characteristic of an infinite
set, it is not going away.
- William Hughes
What bothered me was wondering what more there is in the notion of
renaming than in this pairing off notion of bijection. On the face of it,
there is something more. Would this work as a counter-argument?
We have:
N = 1, 2, 3, 4,......
which we suppose to be bigger than the even numbers:
E = 2, 4, 6,........
We use the elements of N to rename the elements of E, producing:
1, 2, 3, 4,........
But this isn't N. It is code for 2, 4, 6, .......
Or we may replace the elements of N with the elements of E, and
vica versa. But replacement doesn't preserve identity of the set. N
becomes E, E becomes N, and it is still the case that N > E.
On a different tack, I personally don't find the idea of sets
changing size in response to renaming that implausible. It has the added
attraction of being highly amusing. Here we are pinning this platinum
standard |N| against these larger and larger (or smaller) sets, and all the
time it is stretching and shrinking. Elastic infinity. Or is there a
standard N kept at constant pressure and temperature at the French Academy
of Science?
Six Letters
.
- Follow-Ups:
- Re: Infinity......
- From: William Hughes
- Re: Infinity......
- References:
- Re: Infinity......
- From: zuhair
- Re: Infinity......
- From: William Hughes
- Re: Infinity......
- From: zuhair
- Re: Infinity......
- From: William Hughes
- Re: Infinity......
- From: zuhair
- Re: Infinity......
- From: William Hughes
- Re: Infinity......
- From: Six Letters
- Re: Infinity......
- From: William Hughes
- Re: Infinity......
- Prev by Date: Re: Cantor Confusion
- Next by Date: Re: Cantor Confusion
- Previous by thread: Re: Infinity......
- Next by thread: Re: Infinity......
- Index(es):
Relevant Pages
|
