- Class 0 and 1
- Class 2
- Class 3
- Class 4
- Class 5
- Tetration 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 the iota function and Hollom's number are not listed due to questionable well-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.
- \(\Sigma(1,919)\)
- \(\Xi(10^6)\)
- Fish number 4, \(F_4^{63}(3) \approx f_{\omega^\text{CK}_{(\omega^{\omega+1}) 63}+ 1}(63)\)
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.
- Rayo's number, \(\text{Rayo} (10^{100})\)
- Fish number 7, \(F_{7}^{63}(10^{100})\)
- BIG FOOT, \(\text{FOOT}^{10}(10^{100})\)
- Little Bigeddon
- Sasquatch
Little Bigeddon is considered the largest valid googologism as of October 2017. Sasquatch is even bigger but the community currently cannot understand it.
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.