User blog comment:Googleaarex/My number is bigger! (Googology Wiki version)/@comment-1605058-20140920062013

For general ordinal number a, we define a-th order set theory as follows: if a=1, this is simply FOST. If a=b+1, then aOST is bOST with additional variables, which range over bOST formulas. If a is limit, then aOST is exactly the union of languages of bOST for b<a, with sematics naturally extending these of bOST.

To encode many orders of variables, we use xxx...x''...', where ' means it's another variable of the type, and, if there are n x's, we find order of variables as follows: let n encode pair (p,q) with Cantor's pairing function. Consider Sigma_p machines, defined in "Hypermachines" by P. Welch and S. Friedman. Order such machines in a canonical way (e.g. firstly by number of states, and then lexicographically according to transition table). Order of the variable is a number which is eventually written by q-th machine according to that ordering.

Define ordinal M as upper bound of ordinals achievable by above procedure. Because we have only finitely many variables to deal with, the following function is well-defined:

F(n) = the largest number definable in MOST using n characters.

My number is F(F(...F(googolplex)...)) with googolplex nestings.