Integration of Game Theoretic and Tree Theoretic Approaches to Conway Numbers

In this article, we develop our formalised concept of Conway numbers as outlined in [9]. We focus mainly pre-order properties, birthday arithmetic contained in the Chapter 1, Properties of Order and Equality of John Conway’s seminal book. We also propose a method for the selection of class representatives respecting the relation defined by the pre-ordering in order to facilitate combining the results obtained for the original and tree-theoretic definitions of Conway numbers.