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The Rayordinal

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.

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