User blog comment:Edwin Shade/How High Can You Go ?/@comment-5529393-20170920232027/@comment-5529393-20170921012636

Sure! Just like we define Rayo's Number, we can define Rayo's Ordinal as "the largest ordinal definable in first-order set theory using a googol symbols or less." Now, this requires that we believe that every statement of first-order set theory without free variables has a truth value - but the same was true of Rayo's Number. And we can define a RayoOrd function the just like the normal Rayo function, except it has ordinal values rather than natural number values.

The more you learn about set theory though, probably the less likely you are to believe that every statement of first-order set theory has a truth value. But, I am saying this without being an expert in set theory myself.