**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(10^{100}))))))))))

## 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

- ↑ Wojowu and Nathan Ho. First-order oodle theory.
*snappizz.com*. Retrieved 2014-11-11. - ↑ FOOT is not as strong as I thought, LittlePeng9's user blog

## See also

- Rayo's number
- Oblivion
- Utter Oblivion
- Uncomputable function
- Little Bigeddon
- Sasquatch
- Largest valid googologism

**Bignum Bakeoff contestants**: pete-3.c · pete-9.c · pete-8.c · harper.c · ioannis.c · chan-2.c · chan-3.c · pete-4.c · chan.c · pete-5.c · pete-6.c · pete-7.c · marxen.c · loader.c

**Channel systems:**lossy channel system · priority channel system

**Uncomputable functions:**Busy beaver function · Maximum shifts function · Doodle function · Betti number · Xi function · ITTM busy beaver · Rayo(n) ·

**FOOT(n)**