Abstract
<jats:p>Class ordinals are first defined as an extension of ordinals, which are called set ordinals. The axiom of maximal ordinality (AMO) then asserts that there exists a class ordinal Ω that cannot be interpreted by any recursively enumerable extension of ZFC. This property is called non-ZFCR-interpretability, while the property of being consistent with the existence of Ω is called Ω-consistency. The meta-theory TM that interprets AMO is presupposed to interpret every Ω-consistent ZFCR-theory while preserving theorems, to be Ω-consistent and non-ZFCR-interpretable, and to extend ZFC. In this way, TM interprets 'extension of ZFC' to mean 'Ω-consistent extension of ZFC'. Three theorems prove that this meta-theoretically defined Ω is consistent, unique, and maximal, such that it is a consistent version of Cantor's absolute infinite.</jats:p>