Introduction[]
Absolute Foundation (ט) is a Foundation that exists beyond the Foundation of the Absolute Infinite. It's symbol is Teth, the 9th Letter of the Hebrew Alphabet. It is a "true degree of Absoluteness" unlike the Mathean Absolutes. This is because it's size is sufficiently large to trigger a Foundation Paradox.
Definition[]
Absolute Foundation is the "size" of the constructions that can be formed through hyper-transfinite Induction on Echelon Objects.
For Any Ensemble of Echelon's which may be hyper-transfinitely enumerated we posit the following Extension Axioms:
[1] There exists an Object not reachable from below, which is a fixed point of the Previous Echelon Ensemble
[2] There exists another Cardinal so large that upon repeating any previous operator it forms the next Cardinal
[3] This forms the Basis of a new Operator that can be used to form more Echelon Ensembles
[4] Any concept previously introduced, such an inaccessibility, Mahlo-, as well as Concepts in any higher Echelon may also be applied
Foundation Paradox[]
Since Any Ensemble of Echelons forms the basis of more echelons, There can not be an totality of all Echelons. This is our "Absolute Foundation". If we attempt to speak of the whole of the Echelons, this can be used to create more Echelons, meaning it wasn't the grand total of all Eschelons. Absolute Foundation may only be thought of in potential, not actualization.