Re: Infinity......


William Hughes wrote:
zuhair wrote:
William Hughes wrote:
zuhair wrote:
William Hughes wrote:
zuhair wrote:
William Hughes wrote:
zuhair wrote:
William Hughes wrote:
zuhair wrote:
William wrote:
Note that there is a bijection from A to B. Thus there
is a way to rename the elements of B to the elements
of A. Thus B must have the same size as A or we can
change the size of B by changing the names of the elements.

This finitisim. it doesn't apply to the infinite.

I will divide this up.

Note that there is a bijection from A to B.

This applies to infinite sets.

Thus there is a way to rename the elements of B to the elements
of A.

Call the bijection f
For every elment in a in A we rename it using f(a) an element from B.
This applies to infinite sets
[Note that the fact that there may be another way to use the
elements of B does not change the fact that we can use the elements of
B to
rename all of the elments of A]

Thus B must have the same size as A or we can change the size
of B by changing the names of the elements.

Since we can rename all of the elments of A to the elements of B
for infinite sets, this applies to infinite sets.

Nope, no finitism here.

Your ideation using
this name change prinicipal is primitive way of representing things ,
it will only confuse you nothing else.

You are missing the fundamental point that you
*cannot* have a second meaning of size in
which the sizes of A and B are different. If there is
a bijection from A to B then we can use the elements
of A to rename the elments of B and we can use the
elements of B to rename the elements of A. It does
not matter whether or not we can do anything else with
the elements of A and B.


For *any* definition of size: if the sizes of A and B are different
then we must be able
to change the size of a set by renaming the elements.

You didn't give me an example of this. A={1,2,3} have a different size
from B={4,5}
can you tell me how you will change the size of A by renaming its
elements from elements of B.


You changed the sets A and B from two sets that have
a bijection between them to two sets that do not have
a bijection between them Unsurprisingly you got nonsense.

Let me see Replace 1 by 4 and 2 by 5 then A'={4,5,3} were is the change
in size of A.
Now replace 4 in B by 1 in A and 5 in B by 2 in A then B'={1,2} , were
is the change in size of B. In reality what you said above can only be
made with what I call it serjective name change, i.e if we Replace 1 by
4 and 2 by 4 and 3 by 5 then A'={4,4,5}={4,5} , Yes this type of name
change will result in what you are saying above, but you already
rejected this type of name change, you call it invalid.

Frankly speaking according to your definition of valid name change, I
can't see how such valid name change between any two sets would lead to
the change of any of them weather these two sets are equal in size or
not.


Giving details of potential new defintions of size will not change this
fact.



what is this fact, you didn't even demonstrate it in a clear manner.

{Note that there is a bijection between A and B.] The fact is:

For *any* definition of size: if the sizes of A and B are different
then we must be able
to change the size of a set by renaming the elements.



- William Hughes
No, what you said above is nothing but a biased definition of size. I
rejected it already.
I don't believe in your "cannot". see william I know what you mean very
well. I reject it.

Give an example of "size" under which

- there exists a pair of sets A and B with a bijection
between them for which the size of A is not equal
to the size of B

-there is no set that changes size when you
rename the elements.

Ok, A={0,1,2,3,.........} and B={0,2,4,6,...........} , this is a pair
of sets with a bijection between them.

Let me use the simplist injective function to compare between their
absolute size.

f:B->A, f(x)=x is the simplist injective function.

Now according to my definitions size B < size A, since f is injective
and not serjective.

Let me rename the elements of A by elements from B to get A', i.e every
f(x) in B will replace an element x in A.

A'={ 0,1,2,3,4,5,....} , so A'=A.



You want to show that the size of A cannot change when
you rename its elements
You need to show that the size A does nott change under
*any* change of names of the elements. Showing that a specific
name change does not change the size is not enough.

- William Hughes
I already showed that

We have

A={0,1,2,3...}

Form A' by using the name change a-> 2a

A'={0,2,4,6,...}

A' does not have the same "size" as A

It is not true that "size A does not change under
*any* change of names of the elements.

- William Hughes

William go read what I wrote before, obviouselly you are understading
nothing of my dichotomy between Absolute and relative size., you say
that A' doesn't have the same size as A, which size you mean, the
relative or the absolute size. If you are talking about the relative
size then you are wrong, because A has the same size as A', we have (
size A = size A')_f:A->A', f(x)=ex.

Perhpas you mean the absolute size of A' will be different from the
absolute size of A.

Yes. You claim there exists a "size" (you call it "absolute size")
such that the "size" of A is not equal to the size of A'.


This replacement of you above is not the Absolute replacement of
elements of A by elements of B.

Irrelevent. The claim is not that any renaming of the elements of A
will change the size, but that there is at least one renaming
of the elements of A that will change the size. Above I give
one renaming of the elements of A that changes the size.

Note: The negation of

size A does not change under any change of names of the elements.

is not

size A will change under any change of names of the elements

but

there is at least one change of names of the elements, under which
size A changes.

Yes, size A does not change under the "Absolute replacement of elements
of A by elements of B". However, there is a replacement of elements
under which size A does change. So "size A does not change under any
change of names of the elements" is false.

NO, it is not.

- William Hughes

Absolute size of A doesn't change under any absolute change of names of
the elements.
Relative size of A doesn't change under any relative change of names of
the elements.
what you are saying is that absolute size of A change under a relative
change of names.

Yes. This means that there is at least one change of names
(a relative change of names is certainly a change of names) under
which the "Absolute size" changes.

- William Hughes

no, there is not. you are still confused. absolute size doesn't do not
change according to any name change made by any function between A and
B other than the fair function. absolute size is determined by one
function only. what other functions dictate is a local issue private to
these functions it doesn't affect the absolute size. I am telling you,
there is an essential point in my system that you are missing.

Zuhair

.

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