The Supra-Ruo Cardinal is a cardinal number that not only surpasses the Ruo Paixu Gendial (Googleaarex) but is also characterized by the following properties:
Inaccessibility by Higher Reflection Principles: While Ruo Paixu Gendial is not reachable by reflection from a 0-sort reducible/ZFC universe, the Supra-Ruo Cardinal cannot be reached by reflection principles even in any higher-order sort reducible universe. This includes n-sort reducible universes, where n is any finite or countably infinite number, meaning it transcends all attempts at reflection from any universe with finite sorting.
Irreducibility Beyond Successor Functions of Any Sort: Not only is the Ruo Paixu Gendial not reachable through successor functions in a 1-sort irreducible universe, but the Supra-Ruo Cardinal is not reachable by successor functions from any n-sort irreducible universe. In other words, no sequence of successor functions can ever reach this cardinal, regardless of how complex or layered the sorting system becomes.
Beyond Any Large Cardinal Hypothesis: While large cardinals like inaccessible, measurable, or even supercompact cardinals lie within certain large cardinal hierarchies, the Supra-Ruo Cardinal is fundamentally beyond these hierarchies. No large cardinal axiom, as typically understood in ZFC and related set theories, can be used to define or reach this number.
Transfinite Independence: The Supra-Ruo Cardinal is independent of any ordinals or cardinals generated from the typical operations within set theory, including powerset, union, or limit ordinals. Its existence is fundamentally tied to principles beyond the current comprehension of standard set-theoretic operations, requiring a meta-theory of cardinals beyond ZFC or any higher-order theories reducible to ZFC.
Beyond Reflection and Successor Universes: Whereas the Ruo Paixu Gendial acts as a limit cardinal for reflection and successor processes in 0-sort and 1-sort universes, the Supra-Ruo Cardinal exists outside of any such definable universes.
Ultimate Unreachable Cardinal: It’s a number so "large" or abstract that no reflection, no chain of successor functions, no known sorting systems, and no large cardinal hypothesis can ever encapsulate it.