# [FOM] A New Ordinal Notation

Lew Gordeew legor at gmx.de
Sun Aug 28 19:39:49 EDT 2005

```Dmytro Taranovsky <dmytro at MIT.EDU> wrote on Wed, 24 Aug 2005:
>
> In my description of the comparison relation for the ordinal
> notation I wrote about in my previous FOM message, in item (1)
> (a, g) < (b, e) should have been (a, g)  < (d, e).
>
> The corrected form is:
> C(a, b, c) < C(d, e, f) iff C(a, b, c)<=f or c<C(d, e, f) and
> (a, g) < (d, e) where g is the largest ordinal such that
> C(a, g, c) = C(a,b,c).

I implemented in Maple the revised recursive definition of standard form
notations that is obtained as suggested on Tue, 16 Aug 2005, via

> g = b whenever C(a, b, c) is in the standard form.

The resulting system is presumably consistent. However, it is incomplete.
Here are the first five pairs of incomparable standard forms X, Y, i.e. such
that neither X < Y nor Y < X is the case:

X:=C(C(0,0,C(0,0,0)),0,0)     Y:=C(C(0,0,C(0,0,C(0,0,0))),0,0)

X:=C(C(C(0,0,0),0,0),0,0)    Y:=C(C(C(0,0,0),0,C(0,0,0)),0,0)

X:=C(C(C(0,0,0),0,0),0,0)    Y:=C(C(C(0,0,0),0,C(0,0,0)),C(0,0,0),0)

X:=C(C(0,0,C(0,0,0)),C(0,0,0),0)    Y:=C(C(0,0,C(0,0,C(0,0,0))),0,0)

X:=C(C(C(0,0,0),0,0),C(0,0,0),0)    Y:=C(C(C(0,0,0),0,C(0,0,0)),0,0)

Regards,
LG

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