Re: Question

In article <1146146691.912732.76490@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
zuhair <zaljohar@xxxxxxxxx> wrote:

[.snip.]

I confess I have scarce mathematical knowledge, and I don't insist on
my beleives, I am only questioning to know the truth.

From what I know from Bertrand Russell's books Number x is the class
of all similar classes containing x members ( the definition seems
cyclic but it is not ). similar classes means the existence of one to
one relation between their members ( bijection).

Try reading Halmos's "Naive Set Theory" instead.

A binary relation <= on a set S is called a partial order if and only
if it satisfies the following properties:

(i) Reflexivity: a<= a for all a in S.
(ii) Antisymmetry: if a<=b and b<=a, then a=b, for all a,b in S.
(iii) Transitivity: for all a,b,c in S, if a<=b and b<=c, then a<=c.

A partial order is a total order if for all a,b in S, either a<=b or
b<=a.

A total order on S is a well order on S if and only if for every nonempty
subset X of S, there exists x in X such that x<=y for all y in X
(i.e., every nonempty subset has a least element).

An ordinal is a well-ordered set S such that for all a in S, the
subset {x in S : x<a } is equal to the element a.

Some examples of ordinals: the empty set, ordered by set inclusion, is an
ordinal. This is usually called 0.

The set whose only element is the empty set is an ordinal; {0} is
usually called "1".

The set {emptyset, {emptyset}} is an ordinal, ordered by set
inclusion. This is {0,1}, and is usually called "2".

The set {0,1,2} is an ordinal (ordered by inclusion), and usually
called "3".

Continuing in this manner you obtain all the natural numbers: if you
have already defined all the natural number 0 through n, then "n+1" is
the ordinal {0, 1, 2, 3,..., n}, ordered by set inclusion.

In general, if a is an ordinal, ordered by set inclusion, then the set
a U {a}, which contains all elments of a, plus a itself, is also an
ordinal, denoted a+1. Such an ordinal is called a "successor ordinal".

omega is the union of the ordinals 0, 1, 2, 3, 4, ..., n, ...

And so on.

A cardinal is an ordinal which cannot be bijected with any ordinal
strictly smaller than itself.

So number two is the class of all doubles and number three is the class
of all triples. Review " Introduction to mathematical philosophy".

Please read some MATH, not some philosophy. These definitions don't
really work inside set theory, as Russell himself demonstrated. You
need to go to class theory, and even there they are a bit dodgy.

An Ordinal number is the number terms in a series, it is not the
series. It is the class of all similar series.( similar has the same
meaning above).

A cardinal number is the number of members in a set, ie the class of
all similar sets, and it is not affected by order.

Both of these notions are at odds with the standard
definitions. Ordinals are specific sets, as are cardinals; they are
not "quantities" or "number of elements" or any such animal. No wonder
you sound like a loony.


What is a real number ? I don't have an exact definition , but from
what is writtin in this forum it seems to be a number which possess
inductive properties, the kind of a number Bertrand Russells refers to
as inductive number or finite number, perhaps?.

No. None of these things.

An ordinal is finite if and only if it cannot be bijected with a
proper subset of itself. The "natural numbers" are the finite
ordinals.

One can define an addition and a multiplication of natural numbers
inductively. For any natural number n, we define

n + 0 = n
n + (a+1) = (n+a) + 1

for every natural number a.

Then one can define multiplication also inductively:

n * 0 = 0
n * (a+1) = (n*a) + n

for every natural number n.

Once you have the natural numbers, you can define the integers. One
way is to consider the collection of all pairs (n,m) of natural numbers;
define an equivalence relation so that (a,b) ~ (n,m) if and only if
a+m = b+n. The integers are the equivalence classes under this
relation.

One can show that every equivalence class contains either a pair of
the form (n,0) or a pair of the form (0,n), with n a natural
number. If n=0, we call this equivalence class "0". Otherwise, we
denote the class of (0,n) also by "n", and the class of (n,0) by
"-n".

(Intuitively, the pair (a,b) represents the solution to the equation
"a+x=b"; define addition and multiplication accordingly)

It is then an easy exercise to show that the map that sends the
natural number n to the class of (n,0) respects addition and
multiplication and order, so that we can view the natural numbers as
contained in the integers, and this abuse of notation is justified.

Once we have the integers, you can define the rationals. Consider the
set of all pairs (a,b) of INTEGERS, with b nonzero. Define the
equivalence relation (a,b) ~ (x,y) if and only if ay=bx. (Intuitively,
the pair (a,b) represents the solution to the equation ax=b).

A "rational number" is an equivalence class under this. We denote the
equivalence class of (a,b) by "a/b"; each equivalence class contains
one and only one pair (a,b) such that gcd(a,b)=1, and then "a/b" is
the "expression in least terms". Define addition by (a,b) + (x,y) =
(ay+bx,xy), and multiplication by (a,b)*(x,y)=(ax,by).


It is then also an easy exercise to show that the rationals contain a
copy of the integers, namely the integer n corresponds to the rational
n/1 (or to the class of (n,1)).

Once you have the rationals, the reals can be constructed in any
number of ways.

A "Dedekind Cut" of the rationals is a partition of the rationals into
two sets, (A,B), such that:


(i) A U B is all the rationals.
(ii) A/\B is empty.
(iii) for each a in A and b in B, a<b.
(iv) A and B are each nonempty.

These partitions come in three flavors: either A has a largest element
(in which case B has no smallest element); or B has a smallest
element (in which case A has no largest element); or A has no largest
and B has no smallest element.

For example, the partition A = {x : x <=0}, B = {x : x>0} is of the
first type.

The partition A = {x : x<1}, B = {x: x>=1} is of the second kind.

The partition that has B = { x: x>0 and x^2 >= 2}, and has A =Q-B
is of the third kind.

The real numbers can be defined in terms of equivalence classes of
these Dedekind cuts. Intuitively, in either case 1 or case 2, the cut
represents a rational; in case 3, it represents an irrational.

Read "Continuity and irrational number" by Richard Dedekind; in
"Essays on the Theory of Numbers" by Richard Dedekind, translated by
Wooster Woodruff Beman. Dover Publications, Inc.

Alternatively, one can define a "distance" of rational numbers by the
usual means: the distance between a/b and c/d is |ad-bc|/bd, where
|ad-bc| is the absolute value of ad-bc.

A sequence of rationals is function from the natural numbers to the
rationals.

We say a sequence (a_0, a_1,...) (usually denoted {a_i}) converges to
the rational number Q if and only for every N>0 there exists M>0 such
that if n>M, then |a_n-Q|<1/N.

We say a sequence (a_0, a_1, ...) is a "Cauchy sequence" if and only
if for every N>0 there exists M>0 such that if n,m>M, then
|a_n-a_m|<1/N.

It is easy to verify that if a sequence converges to some rational,
then it is Cauchy, though the converse does not hold.

We can define an equivalence relation among sequences by saying that
the sequence {a_i} and the sequence {b_i} are "equivalent" if and only
if the sequence {a_i-b_i} is a Cauchy sequence.

It is an easy exercise to show that if {a_i} converges to q and {a_i}
is equivalent to {b_i}, then {b_i} also converges to q. And if {a_i}
is Cauchy and {b_i} is equivalent to {a_i}, then {b_i} is also Cauchy.

The "real numbers" can be defined to be the set of all equivalence
classes of Cauchy sequences of rationals. Those that converge
correspond to the rationals; those that do not converge to a rational
correspond to the irrationals.

It is this latter definition that gives rise to the numerical
representation. A decimal expansion

N.d1d2d3....

with N an integer, di an integer between 0 and 9, is short hand for
the sequence

(N, N+d1/10, N+(d1/10)+(d2/100), ..., N + (d1/10) + (d2/100) + ... + (dn/10^n),...)

which can easily be verified is a Cauchy sequence; so the decimal
expansion represents the EQUIVALENCE CLASS of cauchy sequences
corresponding to this sequence. It is again a trivial exercise to
show, for example, that the Cauchy sequence represented by
1.000000... (which is the constant sequence (1,1,1,1,...)) and the
Cauchy sequence represented by 0.9999.... (which is the sequence
(9/10, 99/100, 999/1000, ....)) are equivalent Cauchy
sequence. Therefore, a fortiori, they represent the same "real
number".



I heard that division is not defined for ordinals , but I don't know
why? Division of transfinite cardinals by finites is defined,

No, it is not defined. At least, not in STANDARD cardinal theory. You
can define anything you want, naturally. But for example, the USUAL
meaning of division is that "a/b" represents the UNIQUE element c such
that b*c = a. This does not work for cardinals, except in the very
limited situation in which a and b are both finite, and b divides a
(in the usual sense of natural numbers).

Do read some set theory, man.


why division and subtraction is not defined for ordinals?

Because division is defined in terms of the inverses for
multiplication, and subtraction is defined in terms of the additive
inverses. Neither of them exist for ordinals nor for cardinals.

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx

.

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