Re: The meaning of 0/0

Bob wrote:

"You confuse a number, a particular thing, with a set of things. Sets
are
distinct from their members. The set of books in your library is not a
book."

Answer:

In reality this is the strongest argument posed in that discussion
thread, and it is at the heart of the subject. Bravo Bob!

In reality their is no consensus on the definition of number. However I
am attracted to Bertrand Russell's definition which is also outlined
clearly in my web site.

According to that definition "number" is not in reality what you think
" a particular thing".
a particular cardinal number is the class of all similar classes of
particular multiplicities.

So numbers like 1 , 2 , 3 ,.... are in reality classes or sets, they
are not particular thing
they are particular sets of sets of things , and things here are
multiplicities.

Now other sets of numbers like integers, rational, irrational,.....
are all defined as number
because they represent sets of simiar relations happening between the
cardinal numbers.
See( Introduction to Mathematical Philosophy -Bertrand Rusell), so in
fact they are Relation numbers .

Now your basic argument is strong what you want to say is that if 0/0
is the class of all
consistent numbers , then it should be different from it's members ( as
I was always saying that throughout the discussion thread) ,
accordingly 0/0 is " Not(consistent number)"

Now the phrase_ not( consistent number) is different from ( not
consistent) number.
in reality the later is a subgroup of the former.

Now we have two main groups :

1] not( consistent number) that are ( not consistent) number
2] not( consistent number) that is different from ( not consistent)
number

a Parallel example is like when we say not (a comdy book) , this can
be a tragedy book
or it can be something else other than book.

For simiplicity lets call the kind of refutation involved in 1] as the
first degree refutation
and in 2] the second degree refutation

Now when we say that a class is something different from it's members,
what kind of
refutation we should follow first or second degree.

When I read your sentence

"Sets are distinct from their members. The set of books in your library
is not a book."

It implies that you are calling for a second degree refutation, ie that
if 0/0 is the set of
all consistent numbers then 0/0 is something else other than an number
, so you want
to say to me that according to my definitions 0/0 is not inconsistent
number , rather it is not a number either, it is something outside the
sphere of mathematics.
But yet even if you are right still 0/0 is definable but withing the
field of Logics not mathematics and it can be regarded as the Exit of
mathematics to something which is either simpler than it or parallel to
it in simplicity ,that is Logics .

According to that one may even be tempted to say that 0/0 can have
Juleus Caeser as
a member ( although this is not right becase 0 is the set of sets of
particular nil multiplicity, and 0* Julus Caeser is not equivalent to 0
, it is rather a member of 0 ) , and
extend the definition of 0/0 to include all precise terminology in
human knowledge.

Now what determins the degree of refutation involved in the sentence "a
class is something distinct from its members". Is it an internal issue(
determined by the nature of the members) or an external one (
determined by definitions set irrespective of the nature of the
members)

Lets set few examples to clarify the above :

when I say a set of books , it is clear here that their is no external
definiton here
and the set of books is entirely defined by its members, Now the kind
of refutation here
is full refutation ie the 2nd degree refutation , ie the set of books
is different from the member book, simply speaking it is not a book.

when I go to a library and I am searching their for " not a comedy book
" , here in that situation their is a kind of external context
from which we conculude that "not a comedy book" means a book but of
other kind like trajeday or politics book etc..

However when I in general say "Not a comdy book" , then its difficult
to know what degree of refutation is operative , but the trivial
conculsion is it means a book that is not comdy in kind, because
otherwise the word comdy in "Not a comdy book" will be un-necessary
word.

Now lets take a more closer example to what I meant by external
definitions. For example if we define a collection of papers to be a
File , and a collection of Files to be a Folder , now if we set a
threshhold that any collection of papers larger than a file is to be
called a folder , then accordingly a folder of folders would also be a
folder.

In a similar manner if I adopt the view that Numbers are sets of
classes of similar particular multiplicities and any larger set also is
to be called a number , to clarify the analogy a set of particular
multiplicities like in saying the set of two cows , such a set
is like a File level in the example above, now number 2 is the set of
all similar sets of particular multiplicities ( similarity is defined
by one-one correspondance) then
number 2 is the set of two cows, two boys, two trains,...... , this is
like Folder level .
Accordingly the set of what is similar between 1,2,3,.... , z were
z:z-z=0 , this set
is also a Folder level ( see the threshhold definition of folder above
)
Now if we define the word Number as any Folder level set of particular
multiplicites , then
by that definition 0/0 will be Folder level and accordingly a Number.

However my main argument behind believing in 0/0 being a number is the
equation below which I refered to in many places in the discussion
thread.

0/0 = 1/0 =2/0 = ...............z/0=..... , z:z-z=0

Since accroding to set theory x/0 , x<>0 is well defined to be a number
that represents the reciprocal of the relation number_ zero of all
rationals( See Introduction to mathematical philosophy).

Then since 0/0=x/0 , then this equality is a type of an external
definition of 0/0 as a number. In other word 0/0 is inside the
mathematical field of numbers.

Second it is known that a/b is a number were a is any consistent number
and b is a consistent number other than zero.

But what is in zero to say that division by it would not yield a
number. I think that
zero is like the other numbers and division by it also yields a number
but of coarse of an other kind ( inconsistent number).

In summary saying 0/0 is the class of all consistent numbers means that
0/0 is " not a consistent number "

But since x/0,x<>0 is well defined as a number.
and since 0/0 = x/0 ,x<>0
then 0/0 should be a number.

Accordingly 0/0 is a number but it cannot be a consistent number since
this is refuted by the class definiton of 0/0.

Then 0/0 is an inconsistent number.

And from previous discussion thread posts I layed arguments supporting
the view that
their is only ONE inconsistent number.

Then 0/0=x/0 is The one inconsistent number.
For simplicity purposes its better to convert x/0 to 0/0 always.

You wrote:

Your theory of inconsistent numbers
is literally about nothing. Like the Seinfeld Show.

Two corrections here

First : it is my theory of The inconsistent number ( not numbers).
Second: Seinfel Show makes money! mine makes mathematicians angry!

Best
Zuhair

.

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