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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
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