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?