Conway tetratet

A visual representation of Conway's tetratet, using up-arrow notation.

For the number {4,4,4,4}, see supertet.

Conway's Tetratet, or Conway's four-four-four-four, is a number mentioned by John H. Conway in the Book of Numbers.[1] It is the largest explicitly defined finite number in the entire book. It is also the 4th Conway number.[2]

The number is defined as:

\[4\rightarrow 4\rightarrow 4\rightarrow 4\]

in chained arrow notation. It is the smallest nth Conway number to be larger than the nth Ackermann number.

Using Bird's Proof, it can be shown that \(\{4,5,3,2\} \geq 4 \rightarrow 4 \rightarrow 4 \rightarrow 4\) in BEAF.

Conway's Tetratet is the output of the CG function, by putting the input with 4, CG(4).

\(4\rightarrow 4\rightarrow 4\rightarrow 4 = 4\rightarrow 4\rightarrow (4\rightarrow 4\rightarrow (4\rightarrow 4\rightarrow (4\rightarrow 4)\rightarrow 3)\rightarrow 3)\rightarrow 3 = 4\rightarrow 4\rightarrow (4\rightarrow 4\rightarrow (4\rightarrow 4\rightarrow 256\rightarrow 3)\rightarrow 3)\rightarrow 3\)

Approximations Edit

Notation Approximation
BEAF \(\{4,5,3,2\}\)
Fast-growing hierarchy \(f_{\omega+2}(f_{\omega+2}(f_{\omega+2}(255))\)
Hardy hierarchy \(H_{\omega^{\omega+2}3}(255)\)
Slow-growing hierarchy \(g_{\varphi(1,1,0)}(255)\)
Hyperfactorial array notation \(4![1,2] = 4![4]\)
Notation array \((4\{3,4\}4)\)

See also Edit

Sources Edit

  1. Conway, John Horton. Book of Numbers
  2. Saibian, SbiisLarge Numbers List