BIG FOOT

BIG FOOT is a large number that is a derivative of the famously extremely large Rayo's number. As a result, it is among the largest named numbers. It was defined in October 2014 by an author under the pen name "Wojowu" or "LittlePeng9", and was given its name by Sbiis Saibian.

Its definition is almost identical to Rayo's number, another well-known large number which diagonalizes over first-order formulas in the von Neumann universe (which is the universe of discourse for first-order set theory). BIG FOOT, extends first-order set theory by making use of a unique domain of discourse called the oodleverse, using a language called first-order oodle theory (FOOT), which is an extension of nth-order set theory (of arbitrarily large n).

Letting \(\text{FOOT}(n)\) denote the largest natural number uniquely definable in the language of FOOT in at most \(n\) symbols. We define BIG FOOT as \(\text{FOOT}^{10}(10^{100})\), where \(\text{FOOT}^{a}(n)\) is \(\text{FOOT(n)}\) iterated \(\text{a}\) times.

Oblivion and Utter Oblivion could be considered larger than BIG FOOT, but it is questionable if they are sufficiently well-defined, and sufficiently compliant to the basic rules of googology, to take that title.

Definition of FOOT
The language of first-order oodle theory is defined as the language of set theory augmented with the symbols \([\) and \(]\). The universe of discourse consists of oodles, which are subject to the Tarskian definition of truth for a set theory. We call \(\in\)-transitive oodles oodinals, and consider \(\in\) as the ordering relation amongst them (so that we can speak of "larger" and "smaller" oodinals).

The FOOT function is the oodle-theory analogue to the Rayo's function.