Beyond a Naive Absolute Infinite

This paper proposes an axiomatization of the absolute infinite and argues against width and height potentialism in set theory. It builds on an unrestricted language and a non-recursively enumerable class theory, called MK^meta, that extends the formal MK: Morse-Kelley with global choice (GC). Class ordinals and class cardinals avoid the Burali-Forti paradox and GC is assumed to warrant comparability of class cardinals. Meta-formality subsequently gets a maximal fixed-point definition under consistency filtering of recursively enumerable formality. By showing that the concept of maximal meta-consistent height (MMH) of an axiom is theory-independent, it follows that no Ord can exceed Ord^meta, the proper class ordinal of MK^meta, such that the absolute infinite Ω^meta = Ord^meta. Unlike formal and infinitary formal-based theories, which are fundamentally incomplete, MK^meta achieves completeness by having absolutely infinitely many formal-based axioms. Moreover, potentialism is countered by MK^meta, which accepts those formal axioms that maximize its models, all of which are elementarily equivalent to the representative V^meta. At last, only the meta-formal level can capture the entire mathematical reality in a single theory and thus give definite answers.