Grounded for Math

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 \(1\)s written by any halting \(n\)-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 develop such an extension below.

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.