| abstract | - Rayo's number is one of the largest named numbers, coined in a large number battle pitting Agustín Rayo against Adam Elga. Rayo's number is, in Rayo's own words, "the smallest positive integer bigger than any finite positive integer 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 very quickly growing function \( ext{Rayo}(n)\) (sometimes denoted \( ext{FOST}(n)\)). Rayo's number is \( ext{Rayo}(10^{100})\). Rayo's function is uncomputable, which means that it is impossible for a Turing machine (and, by the Church-Turing thesis, any modern computer) to calculate \( ext{Rayo}(n)\). But order-\(n\) Turing machines are able to compute this function. Rayo's function is one of the fastest growing functions ever to arise in professional mathematics; only a few functions, especially its derivation, the FOOT (first-order oodle theory) function surpasses it.
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