User:Ubersketch/Drafts

Define an ordinal notation O as a set of strings with a well-ordering and fundamental sequence function a[n]: PxN -> O where P is a subset of O. 

a+1 is the minimum ordinal larger than a

A limit ordinal is an ordinal, a, such that there does not exist b such that b+1=a

a is in C(b) iff An ordinal, a, has singularity b iff it is in C(b) and not in any C(c) for any c>b
 * b[n]=a for some n
 * c[n]=a for some c in C(b) and some n

An ordinal notation O is well-behaved iff:
 * 1) For all a in O, a[n]=b[n] iff a>=b
 * 3) For all a in O, there does not exist b such that the limit of b[n]=limit of a[n] and b=/=a  
 * 4) For all a in O, a+1[n] is not defined
 * 5) For all limit ordinals a where a[n] is not defined in O, a is a substring of some b such that b[n] is well-defined
 * 6) There exists a with singularity b not in O, such that the union of O and a is well-behaved, and c is in C(a) iff c has singularity b.