Beyond a Naive Absolute Infinite

The aim of this paper is to show that Cantor's absolute infinite Ω can consistently be proven to exist by introducing a consistent class theory NBG_Ω that is sufficiently strong. Gödel's first incompleteness theorem asserts that an axiomatic theory cannot be simultaneously (1) arithmetical, (2) axiomatizable, (3) consistent, (4) complete, and (5) countable. When axiomatizability is generalized to non-countable theories, a self-sufficient theory can be defined as possessing the first four of these five properties. An Ω-expanded theory is a countable theory that is extended with Ω-many omni-independent axioms, where Ω is axiomatized as a class cardinality so large that it cannot be proven to exist as a set cardinality in any Ω-consistent first-order set theory. ZFC_Ω, an Ω-expansion of ZFC, is subsequently shown to be self-sufficient. NBG_Ω, which extends ZFC_Ω to the class level, can then consistently prove the existence of Ω.