Beyond a Naive Absolute Infinite

This paper proposes an idealized, philosophical axiomatization of the absolute infinite in a meta-formal class theory, called MK\( ᵐᵉᵗᵃ \), that can be back-translated to the formal Morse--Kelley with the axiom of global choice (GC). First, class ordinals and class cardinals are introduced, which avoid the Burali-Forti paradox. Second, GC is assumed to make class cardinals well-orderable. Third, the Hamkinsian multiverse \( M_h \) is defined as the meta-formal collection of all the models \( v \) of any relatively consistent, formal theory. Fourth, a meta-formal theory is rigorously defined by ranging over all the sets \( x∈ v∈ M_h \). Fifth, \( Vᵐᵉᵗᵃ \) is the unique model of any meta-formal theory. At last, the absolute infinite \( Ωᵐᵉᵗᵃcard \) is the proper class cardinality of \( Vᵐᵉᵗᵃ \). Moreover, truth relativism can be countered in a GC-consistent branch of \( M_h \), by accepting the axioms that maximize \( Vᵐᵉᵗᵃ \). Consequently, the definition of \( M_h \) can be used as a rebuttal of both height and width potentialism, when combined with the argument that only the meta-formal level can capture the entire mathematical reality.