Largest valid googologism

The title of largest valid googologism has belonged to various numbers.

For a long period, Rayo's number, which is defined as the smallest number larger than any number expressible in the language of first-order set theory in a googol (\(10^{100}\)) symbols or less, was uncontestedly considered the largest valid googologism; however, it was later upstaged by Fish number 7, but due to some controversy regarding whether it is significantly larger than Rayo's number, it has never been officially recognized as the largest valid googologism, even after it was concluded it's not a naive extension. Fish number 7 adds an oracle formula to the original microlanguage used to define Rayo's number.

BIG FOOT was created later by Wojowu, and completely dethroned Rayo's number. This number diagonalizes over a generalization of nth-order set theory known as first-order oodle theory, and is enormously larger than Rayo's number. It was fully recognized by the googology community as the new record-holder for largest valid googologism. However, it was later found that BIG FOOT's value was smaller than previously thought. Specifically, first-order oodle theory was considered to be equivalent to first-order set theory with a single truth predicate added. At last, first-order oodle theory turned out to be inconsistent, and hence it is not even equivalent to the set theory. In particular, BIG FOOT is ill-defined.

Little Bigeddon was considered the holder of the title of largest valid googologism before it has been known that it's ill-defined. It was coined by Googology Wiki user Emlightened, and diagonalizes over an extension of set theory which contains an additional kind of variable known as a rank variable. After that, Emlightened herself defined the even larger Sasquatch, which is too difficult for anyone else in the community to understand and therefore is not officially considered to be the title holder yet. Strictly speaking, their definitions include undefined notions, and hence they are ill-defined.

Jonathan Bowers created two numbers, Oblivion and Utter Oblivion, which diagonalize over the fundamental concept of a mathematical system as a whole, and claimed that they were larger than both BIG FOOT and Rayo's number, as well as Fish number 7. However, he only gave a broad outline of his definitions, rather than describing the numbers using proper mathematical rigor. Utter Oblivion would, if taken at face value, be enormously larger than Little Bigeddon and rank as the largest valid googologism, but it is generally not accepted as such. At least, they are not formalised, and hence ill-defined.

The current record holder for the title of largest valid googologism is the Large Number Garden Number, which was defined by Googology Wiki user P進大好きbot. It hasn't been proven that it is ill-defined, and thereby it should be considered largest.

This discussion generally omits numbers created by simply building on other nominees, such as FOOTBIGG(BIGG). See salad number.