Re: Infinity......


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. 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.

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.


- 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.
I am not missing your fundemental meaning of size. what I am saying
actually is that this is not fundemental, in reality you are missing
the depth and radicality of my approach.

See William , I am beginning to realize that you and the traditional
mathematicians have a form of what I call "Size Theory" in your minds ,
Perhpas I can translate it as follows.

1) ExAy (x= size y)
2) AxAyAz (y=size x , z= size x -> y=z)
3) AxAyAzAu: z in x, u in y -> size{z}=size{u}
4) AxAyAaAb: a.x={},b.y={}, size x=size y, size a = size b -> size aUx
= size bUy.
5) AxAyAzAuAv: z in x, u in x, v in y -> size {z,u} > size {v}.

I have developed this system more and posted it in a separat topic to
Sci.math.
see:http://groups.google.com/group/sci.math/browse_frm/thread/df5d8ea9d42d034a/23e4fd007d22b535#23e4fd007d22b535

Of course the term "size" in the above system means ABSOLUTE size.
Though I don't hardly disagree with 1), but I say it might not be true.
i.e. not every set has an absolute size. and 1) might be better writtin
as 1)ExSy ( x= size y ) which is read: There exist x for SOME y were x
is the size of y.

The best and most precise way to know weather two sets has the same
size or not is made by using "the serjective state of an injection
between them", so for sets A andB if the serjective state of an
injective function f from A to B is positive, then size A = size B
according to f. this can be symbolized as ( size A = size B)_f or we
can also symbolize it in another manner size A =_f size B.
while if the injective function f from A to B is not serjective, then
this mean
Size A <_f size B. or ( size A < size B )_f.

However how do we know the absolute size comparison between A and B.

If we define size A = size B iff for all f:A->B, f is injective and
serjective.
and define size A < size B iff all f:A->B, f is injective and not
serjective.

Then this mean that infinite sets do not have an absolute set size.,
but of course they have an absolute Range of sizes, since when we
compare any set x by P(x) then the size of x is smaller than that of
P(x).

However for the later definiton of absolute size, we should modefy 1)
as I mentioned.

However if we would keep 1) as most logicians and mathematicians
desire.

then we should change the way we derive the absolute size of a set from
its functional sizes.

a way to do that is the following:

size A R size B <-> OR ( size A R size B )_f , were R might be = or
might be < or might be >.

so for the case of N and E for example were N={1,2,3,....} and
E={2,4,6,...} then
size N R size B <-> size N < size E OR size N = size E OR size N> size
E.

So we should develop a criterion that is fare to choose among those
function which one is the one which matches the absolute size
comparison, I tried " simplicity" (see upthread), however these fare
notions might not be general enough to include all cases of infinite
set comparisons. so some sets would have unidentified absolute sizes at
the end.

All of what I said above is coherent , clear , and correct. to go an
use primitive, non precise notions like valid name change william
always wants us to force our analysis of any definition of size upon
it, would be nothing but feeding the already big mayhem about infinite
size.

If one wants to use Cantorian equality of set size, which is what I
call "ABILITY" type of definition, more specifically two sets have
equal size if we are "able" to have a bijection between them, or
williams version of it "if we are able to name each element of one set
by elements from the other set and vice versa". these capability or
ability, these defintions use the words If there exist, or if we can,
of if there can be, etc...

I don't care for these "can" oriented definitions , I care for what is,
note for what can be.

Anyhow , this thread demonstrated a conflict that is more deep than
william thought.

I am becoming fedup of this discussion. I think it became
confrontational and unvalid.
who wants to choos the "can" definitions of william and cantor, then
let him follow them, I am sure that they will lead to a paradox at then
end, or to great limitations in understanding infinite size.
Who want's to choose my much radical concept of accepting the idea of
infinite sets either having no absolute size, or at best if they have
then many of them can only be determined by a FAIR function the enable
us to choose between different injective fucntions between these
infinite sets, according to the nature of these infinite sets, and not
to our "can" preference to bijection. then he might follow it.

The main message behind all of this, is that I think that there is
another way of looking at this infinite size concept, other than the
standard cantorian biased way. and I think it should be investigated
throught its various ramifications.

This Discussion is ended by my side.Period.

Zuhair

.

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