Beyond a Naive Absolute Infinite

Class ordinals are first defined as an extension of ordinals, which are called set ordinals. In order to create a satisfactory definition of a maximally large class, this definition needs to quantify over all set-theoretic theories. Consequently, the Axiom of Maximal Strength (AMS) asserts that there exists a class that does not provably exist in any theory that has a theorem-preserving interpretation into a consistent theory that has a parameterless first-order definition with the length of a set ordinal. This property is called non-SO-interpretability. The meta-theory T_M that interprets AMS is presupposed to be sound, non-SO-interpretable, and to extend MK. It is then proven that T_M is consistent with AMS and that the class Ord of all the ordinals of T_M is a consistent version of Cantor's absolute infinite.