User blog comment:Googleaarex/My number is bigger! (Googology Wiki version)/@comment-2033667-20140919220815

first-order set theory is defined as first-order logic over the Von Neumann universe and a membership relation.

(n + 1)th-order set theory is defined as taking nth-order set theory, and allowing designated variables to be collections of the valid variables in nth-order set theory (and, consequently, allowing quantification over these).

higher-order set theory is defined as the union of nth-order set theory for all n > 1.

the smallest positive integer bigger than any finite positive integer named by an expression in the language of higher-order set theory with a googol symbols or less.