- 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 Oblivion, Utter Oblivion, 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 given the condition that they will halt. They have growth rates of at least \(\omega_1^\text{CK}\) of the fast-growing hierarchy.
- \(\Sigma(1,000)\)
- Fish number 4, \(F_4^{63}(3)\)
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.
- \(\Xi(10^6)\)
- 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
- Largest valid googologism, currently Sasquatch