User blog comment:Scorcher007/S - Large Countable Ordinal Notation. Chapter I, Up to KPm./@comment-31580368-20190912145823/@comment-31580368-20190914012348

Ok, we can freely refuse to use the KP (this is not important), and use the metatheories defined in ZFC. Then I can redefine Θ again.

Θ(x,y) is computable function which assigns to each y∈N the least natural number greater than or equal to the halting times of Turing machines with input 0 whose terminations admit formal proofs of length ≤ y under metatherory M defined in the base theory ZFC, where M such theory that for all statement φ: if x is admissible ordinal: Lx⊧φ; if x is collection of admissible ordinals ("n∈O"-term): ∃φ|Ln⊧φ for every admissible n < O, where x is collection of admissible ordinals < O. if x is limit of admissible ordinal ("n<O"-term): ∃φ|Ln⊧φ for every admissible n < O and exist x=sup(n) such every φ have limit ≤ Lx.