Re: Transfinite Ordinal Multiplication


Stephen Montgomery-Smith wrote:
zuhair wrote:
Hi All

I am reading Introduction to Mathematical Philosophy by Bertrand
Russell.

It says that Trasnfinite Ordinal multiplication is non commutative.

2*Omega is different from Omega * 2

so 2+2+2+............. = Omega

While Omega + Omega = 2.Omega>Omega

I see this a little bit fabricated! or let's say fixed.

Why?

Because I can sum number 2 Omega of times and result in 2.Omega, and on
the other hand
I also can sum Omega twice to result in Omega.

How?

To ease visualization of these ordinal summations let us use the unary
numeral system.

Zero = =Empty raw of stars
One = *
Two = **
Three = ***
.
.
.
n = ****...nth*
.
.
.
.
.
Omega = *****......

Now Omega + Omega = ****.... + ****.... = ****.... ****.... > ***.....
This is clear

While ** + ** + ** + ** +............ = ****...... = Omega this is also
clear.

But it seems as if there is something fishy out their!

I can visually sum ** Omega of times and obtain ****....*****.... and
not ***.... see below

* * this is the first double of
stars
I will add the next double also wide apart each to the right of the
star of the first double
as below:

** **

Also the third double can be added in a similar manner to get

*** ***

If I continue for that infinitely the result would be ****.....
*****.... = 2 Omega.

From the other hand, I can sum two Omegas to result in one Omega as
below

For simplicity let us denote one of the Omegas as (*)(*)(*)......

Now ****..... + (*)(*)(*).......... = *(*)*(*)*(*)....... = Omega

This last summation can be called Inter-digital summation.

Also If we examine the example of adding the two stars Omega of times
to result in
2.Omega we see it is also a form of Inter-digital summation.

So It seems that there are two kinds of multiplication operator.

One is Inter-digital multiplication, and the other is extradigital
multiplcation.

Of coarse both are non-commutative and each one is the converse of the
other.

Let (*) be intradigital multiplication and *( ) be extradigital
multiplication then in Summary:

Omega (*) 2 = Omega
2(*) Omega = 2.Omega> Omega

Omega *( ) 2 = 2.Omega > Omega
2 *( ) Omega = Omega

Any Comments?

Another way to define all this:

a*()b = b(*)a.

Yes

.

Relevant Pages

  • Transfinite Ordinal Multiplication
    ... It says that Trasnfinite Ordinal multiplication is non commutative. ... 2*Omega is different from Omega * 2 ... Zero = =Empty raw of stars ... 2.Omega we see it is also a form of Inter-digital summation. ...
    (sci.math)
  • Re: Transfinite Ordinal Multiplication
    ... It says that Trasnfinite Ordinal multiplication is non commutative. ... 2*Omega is different from Omega * 2 ... Zero = =Empty raw of stars ... 2.Omega we see it is also a form of Inter-digital summation. ...
    (sci.math)
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