Re: infinity
- From: "zuhair" <zaljohar@xxxxxxxxx>
- Date: 30 Oct 2005 01:47:02 -0800
Randy Poe wrote:
> zuhair wrote:
> > Randy Poe wrote:
> > > zuhair wrote:
> > > > Randy Poe wrote:
> > > > > zuhair wrote:
> > > > > > Randy Poe wrote:
> > > > > > > zuhair wrote:
> > > > > > > > Randy wrote:
> > > > > > > >
> > > > > > > > Oh yes? Name one.
> > > > > > > > ---------------------------------
> > > > > > > >
> > > > > > > > Before I relpy to that point let me be sure of your interpretation of
> > > > > > > > bijection
> > >
> > > You said you were goint to answer this question. You never did.
> >
> > I am going to do that , but wait a little. The picture will be clear to
> > you.
> >
> > of coarse You know very well I don't regard the bijection you are doing
> >
> > a real bijection , it is a pseudo-bijection.
>
> I presume by now you know the definition of a bijection from
> set A to set B.
>
> 1. For every element of B, there is some element of A mapped
> to that element of B (surjection).
>
> 2. No two elements of A are mapped to the same element of B
> (injection).
>
Randy Poe wrote:
> zuhair wrote:
> > Randy Poe wrote:
> > > zuhair wrote:
> > > > Randy Poe wrote:
> > > > > zuhair wrote:
> > > > > > Randy Poe wrote:
> > > > > > > zuhair wrote:
> > > > > > > > Randy wrote:
> > > > > > > >
> > > > > > > > Oh yes? Name one.
> > > > > > > > ---------------------------------
> > > > > > > >
> > > > > > > > Before I relpy to that point let me be sure of your interpretation of
> > > > > > > > bijection
> > >
> > > You said you were goint to answer this question. You never did.
> >
> > I am going to do that , but wait a little. The picture will be clear to
> > you.
> >
> > of coarse You know very well I don't regard the bijection you are doing
> >
> > a real bijection , it is a pseudo-bijection.
>
> I presume by now you know the definition of a bijection from
> set A to set B.
>
> 1. For every element of B, there is some element of A mapped
> to that element of B (surjection).
>
> 2. No two elements of A are mapped to the same element of B
> (injection).
>
I think I should elaborate on this notion bijection which is repeatedly
used
in this group. Bijection is used as similarity criterion, when two
things are
bijected then they are similar in what they are bijected.
I want only to speak alittle bit about what cardinality mean.
Cardinality is an index of comparison between the multiplicities of
discriminated
collection of items .
Now for example { Lion , elephant , dog} this is a discriminated
collection of items
usually called set which is a group of things fulfilling a specific
propositional
function and having no specific relation between them other than
discrimination.
when we say the cardinality of { Lion , elephant,dog} = x then this
means also
that it is the same for any arrangment of that set and also it is the
same as
other sets like {mouse, planet, zeebra}
In order to determin similarity of cardinality we use the concept of
one-one correspondance.
Before we go to one-one correspondance we should know what is one-one
relation
A relation is said to be"one-one" when,if x has the relation in
question to y, no other term x' has the same relation to y,and x does
not have the same relation to any term y' other than y.
if we have X={a,b,c}, Y={e,f,g} and their is one-one relation between
all their members then this is one-one correspondance ( bijection)
Now the above is cardinal bijection and it implies similarity in
cardinality.
However their is another kind of bijection which implies similarity of
multiplicity of items collected according to a certain ordinal rule.
For example:A= 4,5,6 bijected to B=1,2,3 , this kind of bijection is
not
cardinal bijection.It is ordinal bijection it denote similarity between
the multiplicity of items in the first ordered collection ( series A)
and
the second ordered collection ( series B).
In general: when you biject to a series you are implying Ordinal
similarity.
For example if a,b,c is bijected to 1,2,3 this means that a,b,c are of
the
same sequence as 1,2,3 , It DOESN'T imply that a,b,c are of the same
cardinality
as 1,2,3!!!!!! Specifically speaking this bijection means that the
ordinal number
of a,b,c is the same as that of 1,2,3.
In order to assert that the cardinality of a,b,c is the same as the
cardinality
of something else then we should compare it whith a set of dicriminated
collection
that has no order.
for example a,b,c if bijected to Lion,elephant,monkey ===> a,b,c has
the same cardinality as lion,elephant, monkey. This is a right
statement
But a,b,c if bijected to 1,2,3 ===> a,b,c has the same cardinality as
1,2,3
this is a wrong statement , the right statement is a,b,c has the same
ordinal number
as 1,2,3.
Most functions mentioned in that discussion group are in reality
ordinal bijective
functions and not Cardinal bijective functions.
However Cardianlity can be infered from Ordinality:
for example the set {a,b,c} if I want to say that it is of the same
cardinality
as 1,2,3 then I should fulfill the following:
a,b,c bijected to 1,2,3
a,c,b bijected to 1,2,3
b,a,c bijected to 1,2,3
b,c,a bijected to 1,2,3
c,a,b bijected to 1,2,3
c,b,a bijected to 1,2,3
Or
a,b,c bijected to 1,2,3
a,b,c bijected to 1,3,2
a,b,c bijected to 2,1,3
a,b,c bijected to 2,3,1
a,b,c bijected to 3,1,2
a,b,c bijected to 3,2,1
Of coarse all of these bijections are Ordinal bijections.
Now iff all these ordinal bijections are fulfilled then we
say that their is Cardinal bijection between a,b,c and ,1,2,3
At finite level Cardinality goes hand by hand with Ordinality
ie the cardinal number of a finite set or series is the same
as it's Ordinal number.
In similar situations to the finite congrouence between cardinality
and ordinality one needs only to establish an ordinal bijection
in order to assure similarity of cardinality.
However at infinite level their is Ordinal- Cardinal
discripancy(Cantor)
In general when the cardinal number of A is not the same as its ordinal
number, like below
A= { a,b,c,.... }
Ord A' =w
Ord A" =L
Ord A"'=Q
..
..
..
(Ord for ordinal number of, A',A",.. are a specific arrangments of A)
In that case max(w,L,Q,...) = Card A
If their is no maximum upper bound then
lim(w,L,Q) = Card Q
If their is no limite then the card A is undefined.
Their is another propblem with estimation of cardinality of infinite
sets, that is it cannot be really estimated , it can only be
assumed to be estimated. Because for an infinite set {a,b,c,d,....}
to have cardinality that is the same as N, one should take all
rearrangments of N and biject each with the above set
( or the converse) , Now all arrangements of N is N! and this
is bigger than N itself in ordinality and possibley even in
cardinality.
But such a series of ordinal bijections between { a,b,c,d,....}
and N is never completed and so one cannot assure that for example
which is the max(ord N) , or even if every ord N is similar .
Assuming that bijection of any series to the series N=1,2,3,4,...
means that the cardinality of that series is the same as N cardinality
ie w =Aleph-0
Now one can see the rule of max (ord N) .
For example set S={ (0,0),(1,0),(2,0),.........,(0,1),(1,1),(2,1),....}
in which all (x,0)pairs are befor all (x,1) pairs.
Max Ord S = 2w
min Ord S= w
Ord S= w when S is rearranged such that the i-th (x,1) pair comes
after the i-th (x,0) pair .( Cantor's way of Ordinality).
I call that merging of (x,0) pairs with (x,1) pairs
So Card S = 2w=2Aleph-0
while if we have a set like M {S,(0,2),(1,2),(2,2),......}
we have three orderinal numbers for M
Ord M = 3w
Ord M=2w
ord M= w
2w when we merge (x,0) pairs with (x,1) pairs, or merging (x,1)pairs
with (x,2)pairs , or merging (x,0)pairs with (x,2)pairs.
w when we merge all types of pairs such that the i-th (x,2) pair
comes after the i-th(x,1) pair which comes after the i-th(x,0)
pair.
Why I think that Cardinality of M should be 3Aleph-0 and not
simply Aleph-0 as Cantor stresses.
The reason behind that is fairly logical : I cannot have an ordinal
number that is bigger than the cardinal number of the field of
the series the ordinal number is counting it's terms.
Because when we say that series S in its original order has a number
of terms that is bigger than the number of terms in series S
in another order , then from were the series put in the first order
brought it's terms, from the smaller series????? the series is the same
series and so should have the same cardinal number, so the biggest
ordinal
number which contain extra numbers is the real cardinal number of the
series.
My explanation to that is the series having the smaller order
is a form of merged series, were the ordinal bijection here is
not of a full discriminative power to decide the real cardinality
of the series.
In general day after day I am convinced that their is no need
for the concept of Cardinality in infinite mathematics, it only
adds confusion.
Ordinal transfinite mathe. is better, cardinal infinite math.
is never really confirmed it is only an assumption.
Ordinal mathematics even that of Cantor is better.
Of coarse I call for a fully inductive ordinal mathematics
with zero's as a number which has a placeholding role on the comparison
( this is different from the ordinary zero which is not placeholding
on comparisons)
In that group I was repeatedly asked about if their is a number
in the set N/1 where N= 1,2,3,4,......
that is not present in the set 2,3,4,5,...........
The answer is yes , if you only allow me to call the first set N-1
and the second set N'
Now : N-1 = (0),2,3,4,...........
while N' = 2 ,3,4,5,..............
Now (0) { the placeholding zero} is not present in N+1. of coarse
(0)<>0
However if you do not accept my invention of a placeholding zero
and you accuse me of not discriminating between numeral and number
as it would be expected from you to say, since you think I am very
ignorant of mathematics. Just in case of the very expected refusal
of that then I will say that a mathematics dealing with the infinite
would be The Ordinal Transfinite Cantorian mathematics, were
the priniciple of reflexion here have partial influence unlike
the math. of cardinality were it is dominant.
Transfinite Cardinality is a myth.
Zuhair
.
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