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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) Edit

  • 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
  • Infinity is known to be greater than both, Fish number 7 and Fish number "Fish number "...Fish number 7"..."" ("Fish number" repeated "Fish number 7" times with "Fish number 7" at the end.).
  • log1(2)
  • -(log10(0))
  • 1/0
  • 0-1

North’s Number Edit

North’s Number is... Very Complex. Take Infinity and mutiply it and...

  • Aleph-0

Edit

ALEPH NULL

ALEPH NULL IS AN ORDER OF INFINITY...aleph one,aleph two,etc. would exceed it but are not all defined.

Isaac Asimov ended his book Realm of Numbers with "there is no end,there is only aleph infinity".

Paratrix, Superfinity and Subsemantix Edit

Two extensions of Sam’s Number. The smaller of the two terms is the Subsemantix. It means the largest number which cannot be expressed as “(x - 1) + 1”, which is not infinity. Thus, this term’s actual value must be above infinity, as x - 1 = x.

Next is a Superfinity, the definition of which is an infinitely long string of any notation, the start of which is also infinity. Any notation, it’ll still be a Superfinity.

A paratrix is a more specific definition of Sam’s Number, which is the smallest number which cannot be expressed using any notation in the universe, placing it a long way above a Transfinity, as it is defined using arrow notation. In fact, using any notation with the beginning number and ending number both being infinity would still not be possible of being a Paratrix. However, if you use a certain notation an infinite amount of times, you would get a much stronger variant of infinity that still is not a Paratrix. This means a Paratrix is somehow bigger than an exponentiated Paratrix, making a Paratrix smaller than a Paratrix. This starts off a series of the Paratrixical Numbers, the smaller of which is the reciprocal of Little Bigeddon, added to a Superfinity.

Largest Paratrixical Number Edit

The largest Paratrixical Number is also the Little Bigeddon’th Paratrixical Number, which is the largest number of which any notation whose product is Transfinity can be repeated again and again, with an exponentially growing product, a Little Bigeddon number of times.

Absolute Infinity Edit

The largest number possible.

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