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BIG FOOT is a counterpart of Rayo's number based on an extended version of the language of first-order set theory. 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.[1]

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), and generalizing 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 $$a$$ times (recursion). BIG FOOT is thus equal to:

FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(10100))))))))))

## 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 Rayo's function, where oodle-theory is an extension of set-theory.

Because all the structures involved are elements of the universe of discourse, FOOT has turned out to be equivalent in strength to FOST with a single truth predicate adjoined.[2]

## Sources

1. Wojowu and Nathan Ho. First-order oodle theorysnappizz.com. Retrieved 2014-11-11. [dead link]
2. FOOT is not as strong as I thought, LittlePeng9's user blog