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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 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, or related systems, given the condition that they will halt. They have growth rates of at least 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.

  • Rayo's number, Rayo(10100)
  • Fish number 7, f763(10100)
  • BIG FOOT, \(\text{FOOT}^{10}(10^{100})\)
  • 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 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.

  • Ultrathree, 3↑Sam's Number3

There are a Sam's Number of arrows (↑)

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.

  • Zero's Large Number, H(1)+10