Nonstandard Transfinite Digraphs

TL;DR

This paper develops a framework for constructing nonstandard transfinite digraphs across all natural-number ranks, including special cases like the arrow rank and limit ordinal ranks, extending previous work on nonstandard and transfinite digraphs.

Contribution

It introduces methods to construct nonstandard transfinite digraphs for all natural-number ranks and the arrow rank, advancing the theory of transfinite digraphs.

Findings

Constructed nonstandard transfinite digraphs for all natural-number ranks.

Developed a method to include the arrow rank $oldsymbol{ ightarrow}$.

Extended the construction to higher ordinal ranks.

Abstract

Nonstandard digraphs and transfinite digraphs have been defined and examined in two prior technical reports. The present work examines digraphs that are both nonstandard and transfinite. This requires a combination in certain ways of the techniques used in the prior two works. We first construct herein nonstandard transfinite digraphs for all the natural-number ranks. Then, a special kind of nonstandard transfinite digraph having the "arrow rank" $\vec{\omega}$ needs to be constructed. Once this is done, the first limit-ordinal rank $\omega$ can be attained. This procedure can be continued on to still higher ranks.