We can define \(\text{Rayord}(n)\) as the smallest countable ordinal that is larger than any countable ordinal named by an expression in the language of first-order set theory with \(n\) symbols or less. (Here, we can use the same Gödel-style encoding that \(\text{Rayo}(n)\) employs.) Since the supremum of a countable set of countable ordinals is itself countable, the \( \text{Rayordinal} := \sup \{\text{Rayord}(n) | n \in \mathbb{N}\} \) is still countable, though extremely large.
Recently, I've become interested in googology. The Busy Beaver function, \(BB(n)\), has been of particular interest. It is defined as the maximum number of 1s that can be written by any \(n\)-state Turing machine. A natural extension of the Busy Beaver function is to replace Turing machines with a more powerful construct, such as augmenting a Turing machine with a halting oracle. I have developed such an extension that essentially gives a Turing machine an oracle for Second Order Logic. I outline the extension, a method of iterating the oracle, and how these oracle Turing machines can be used to extend the Fast Growing Hierarchy.