Rayo's number

Rayo's number is the largest named number, coined in a large number battle pitting Agustín Rayo against Adam Elga. Rayo's number is, in Rayo's own words, "the smallest number bigger than any finite number named by an expression in the language of first order set theory with a googol symbols or less."

By replacing "googol" with any positive integer, we get a function \(\text{Rayo}(n)\) with a growth rate of at least \(f_{\omega^\text{CK}_\omega}(n)\) in the fast-growing hierarchy, where \(\omega^\text{CK}_\omega\) is the limit of the sequence of \(\omega^\text{CK}_1\), \(\omega^\text{CK}_2\), \(\omega^\text{CK}_3\),..., where \(\omega^\text{CK}_1\) is the. Thus Rayo's number is an uncomputable number &mdash; it is impossible for a computer to calculate it without using infinite time or memory. In fact, even an oracle Turing machine, or any order-\(n\) Turing machine, cannot compute it.

Rayo and Elga's number duel was based on an article by Scott Aaronson.

The second largest named number is meameamealokkapoowa oompa by Jonathan Bowers, which, in contrast to Rayo's number, is computable.