List of googologisms/Uncomputable numbers

  List of googologisms
 * Class 0 and 1
 * Class 2
 * Class 3
 * Class 4
 * Class 5
 * Class 6
 * Class 7
 * Class 8
 * Class 9
 * Class 10
 * Class 11 and higher
 * Tetration level
 * Pentation level
 * Hexation level
 * Heptation level
 * Octation level
 * Up-arrow notation level
 * Linear omega level
 * Quadratic omega level
 * Polynomial omega level
 * Exponentiated linear omega level
 * Exponentiated polynomial omega level
 * Double exponentiated polynomial omega level
 * Triple exponentiated polynomial omega level
 * Iterated Cantor normal form level
 * Epsilon level
 * Binary phi level
 * Bachmann's collapsing level
 * Higher computable level
 * Uncomputable numbers

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

Note: The special cases of Oblivion, Utter Oblivion, the iota function, and Hollom's number are not listed due to questionable well-definedness.

Busy beaver numbers
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.


 * Rado's sigma function, Sigma(1919)
 * Xi function, Xi(106)
 * Fish number 4, F463(3)

Rayo numbers
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.


 * Rayo's number, Rayo(10100)
 * Fish number 7, F763(10100)
 * BIG FOOT, FOOT10(10100)
 * Little Bigeddon
 * Sasquatch

Little Biggedon is considered the largest valid googologism as of October 2017. Sasquatch is bigger but the community currently cannot understand it.

Oblivion
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
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
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.