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Smaller numbers

The term "uncomputable number" here refers to the numbers defined in terms of uncomputably fast-growing functions.

Note: The special cases of the iota function and Hollom's number are not listed due to ill-definedness.

Busy beaver numbersEdit

These numbers arise from functions that eventually dominate all computable functions, and are based on the unsolvability of the halting problem. They exploit the maximum scores of a particular Turing machine, or related systems, given the condition that they will halt. They have growth rates of at least \(\omega_1^\text{CK}\) of the fast-growing hierarchy.

Rayo numbersEdit

These numbers diagonalize over nth-order mathematical theories: Rayo's function diagonalizes over first-order set theory, and the derived FOOT function diagonalizes over nth-order set theory. They are currently the largest well-defined named numbers in professional mathematics.

Little Bigeddon is considered the largest valid googologism as of October 2017. Sasquatch is even bigger but the community currently cannot understand it. However, it is also not even known if Sasquatch can be proven to exist; it relies on some conjectural (although likely true) statements.

Oblivion Edit

Jonathan Bowers defined a number called "Oblivion", but the well-definedness is debatable, but if it was well-defined, it would be greater than all the previous numbers. Even larger is "Utter Oblivion".

Sam's Number Edit

A user by the name SammySpore created a page called "Sam's Number", but the "number" described isn't defined, only "described". It is obviously not well-defined, but it remains as an in-joke among googologists.

Infinity Edit

Infinity is not a number. It is not considered a googologism of any sort, and googologists don't like people messing with it in googology. However, transfinite ordinals (a set-theoretic type of "infinity"), are sometimes used to index functions.