- 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 that many of them are ill-defined (i.e. their original descriptions do not define a number), and such comparisons do not make sense.

## Warning[]

A reader should be **extremely careful** that **there were many abuses of Sam's Number in personal websites, videos, and user blogs** in the past years, which was previously listed in the wiki and later got removed in March 2022, due to the fact that Sam's Number is only just a "described" number without any definitions and is **actually not considered to be a defined number at all**, and merely remains as an in-joke among googologists. In addition, Sam's Number is not only completely ill-defined but also **not considered to actually be an uncomputable number in any sort and is completely unclassed**. The original definition was also intended to be larger than Oblivion and Utter Oblivion (but later it no longer does as DeepLineMadom pointed out in his blog post). Moreover, when assuming that Sam's Number to be well-defined, it could theoretically have infinitely many possible definitions.

## Turing theory 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 are believed to have growth rates greater than or comparable to \(\omega_1^\text{CK}\) of the fast-growing hierarchy with respect to a certain reasonable choice of a system of fundamental sequences, but such a system of fundamental sequences is not known.

- \(S(643)\), the smallest known number undecidable in ZFC generated by maximum shifts function in the sense that the equality \(\underbrace{1+ \cdots +1}_m = S(643)\) is unprovable in ZFC set theory for any meta-theoretic natural number \(m\) as long as it is consistent.
- \(\Sigma(745)\), Smallest known busy beaver number that cannot be proved in ZFC
- \(\Sigma(1919)\) (busy beaver function), Upper bound for smallest unprovable Busy beaver in ZFC
- Naive extensions: Grand theft duty, Great theft duty, Gigantic theft duty, Gorged theft duty, Gulp theft duty, *Gasp* theft duty, Ginormous theft duty, Gargantuan theft duty, and Googondo theft duty (all naive extensions of each other)

- Fish number 4, \(F_4^{63}(3)\)
- \(\Xi(10^6)\)
- \(\Sigma_\infty(10^9)\)

## Set theory numbers[]

These numbers diagonalize over first order set theory. They are currently the largest named numbers in professional mathematics.

### Rayo numbers[]

- Rayo's number, \(\text{Rayo} (10^{100})\)

- Fish number 7, \(F_{7}^{63}(10^{100})\)

#### Ill-defined descendants[]

These numbers are descendants of Rayo's number, and are considered as significant works compared to the rest of the other ill-defined uncomputable numbers. As Rayo's function diagonalizes over first-order set theory, the derived FOOT function was intended to diagonalize over nth-order set theory.

- BIG FOOT, \(\text{FOOT}^{10}(10^{100})\)
- Its naive extensions, INSANE FOOT, HYPER FOOT, ULTRA FOOT, And OMNI FOOT

- Little Bigeddon
- Sasquatch / Big Bigeddon

BIG FOOT was regarded as a well-defined number, but is actually an ill-defined number due to several issues. Roughly speaking, there is no "reasonable" choice of axioms which makes BIG FOOT a well-defined number. The issues are explained in that article.

The original definitions of Little Bigeddon and Sasquatch (also known as Big Bigeddon) include several obvious errors, and nobody in the community currently understands the way to fix them because the original descriptions lack sufficient information which help us to consider the creator's intention. At least, Little Bigeddon was considered as the largest valid googologism of October 2017. Sasquatch was even guessed to be an even bigger number, but it is also not even known if Sasquatch can be proven to actually really exist; it relies not only on the way to fix the definition, but also on some conjectural statements.

### Large Number Garden Number[]

This number is currently the largest valid googologism, defined using an extension of first-order set theory, far beyond any reasonable higher-order extensions of regular first-order set theory. It is also not yet explained whether this is a well-defined number or an ill-defined number.

- Large Number Garden Number, \(f^{10}(10 \uparrow^{10} 10)\)

### Salad numbers[]

## Larger ill-defined numbers[]

Since the comparison of the ill-defined numbers are unknown, the order of entries does not easily imply the order of magnitude, but to the intended values when assuming these numbers to be well-defined.

### Hollom's number[]

Hollom's number is just introduced as a thought experiment, and hence is not originally intended to be well-defined.

- Hollom's number, \(\approx \mathrm{I}_{6.895 \times 10^{104}}(200)\), using Iota function

### Oblivion-based numbers[]

This number is a number coined by Jonathan Bowers, but are just "described" in an informal explanation which does not characterise any specific number. If it were well-defined, it would be greater than all the previous numbers. Its extension Utter Oblivion is intended to be an even larger number.

#### Original numbers[]

- Oblivion, the largest number defined using no more than a kungulus symbols in some K(gongulus) system
- Utter Oblivion, the largest finite number that can be uniquely defined using no more than an oblivion symbols

#### Naive extensions[]

There are multiple Oblivion-based naive extensions (including salad numbers. See also how Hayden's "Ultimate Oblivion" appears to be a naive extension).

- The True Borderline, Oblivion
^{oblivion} - Oblivious Plex – by Garrett Wilkinson

#### DeepLineMadom's extensions[]

DeepLineMadom coined three Oblivion-based numbers in the Flenary of Ukrainian Town.

#### Hayden's extensions[]

HaydenTheGoogologist2009 coined multiple Oblivion-based numbers beyond Utter Oblivion, including:

- True of Real
- Ultimate Oblivion, the largest finite number that can be uniquely defined using no more than an utter oblivion symbols

#### Nihilustheabsolutist’s extensions[]

- Hyper Oblivion, the largest finite number that can be uniquely defined using no more than an Ultimate Oblivion symbols
- Ultra Oblivion, the largest finite number that can be uniquely defined using no more than an Hyper Oblivion symbols

## Infinity[]

*Main article: Infinity*

Infinity is not a Real number, it is the name for a concept or idea. It is not considered as 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. For example Fast-growing hierarchy.