User blog:B1mb0w/Extended Normal Form

Extended Normal Form
This blog extends Cantor's Normal Form to provide precise definitions for ordinals with arbitrary complexity. It follows from my earlier blog on Fundamental Sequences and uses the Aristo Sequence definition from that blog. In turn this blog will be referred to in my other blogs for the J Function.

Basics (Cantor's Normal Form)
Cantor's Normal Form defines the construction of various ordinals. There seems to be some ambiguities when dealing with ordinals of arbitrary complexity.

Constructing Ordinals with Arbitrary Complexity
The following construction procedure can be used to construct any ordinal of arbitrary complexity (up to the size of the Small Veblen Ordinal (SVO) which is defined as follows:

\(SVO = \varphi(1,0_{[\omega]})\)

The construction procedure enforces Extended Normal Form and will be consistent to the Aristo Sequence which is defined on this blog.

\((\gamma + 1)[n] = \gamma\)

\((\gamma + \lambda)[n] = \gamma + \lambda[n]\) when \(\gamma >= \lambda\)

\(\lambda.(\delta + 1)[n] = \lambda.\delta + \lambda[n]\)

\(\gamma.\lambda[n] = \gamma.(\lambda[n])\) when \(\gamma >= \lambda\)

\(\lambda^{\delta + 1}[n] = \lambda^{\delta}.(\lambda[n])\)

and

\(\gamma^{\lambda}[n] = \gamma^{\lambda[n]}\)

I have written another blog to further extend Normal Form to provide detailed definitions for ordinals of arbitrary complexity.