List of functions

This page lists various googological functions arranged by growth rate according to the fast-growing hierarchy.


 * \(\approx\) means that two functions have comparable growth rates.
 * \(>\) means that one function significantly overgrows the other.
 * \(\geq\) means that it is not known exactly whether one function overgrows the other or not.
 * (limit) means that the function has many arguments, and the growth rate is found by diagonalizing over them.

From \(f_0(n)\) to \(f_\omega(n)\)
These functions are primitive recursive.


 * Addition \( = a+b > f_0(n)\)
 * Multiplication \(= a*b > f_1(n)\)
 * Exponentiation \(= a^b \approx f_2(n)\)
 * Factorial (and most of its extensions) \(= n! \approx f_2(n)\)
 * Exponential factorial \(\approx f_3(n)\)
 * Tetration \(= {^{b}a} \approx f_3(n)\)
 * Pentation \(= a \uparrow\uparrow\uparrow b \approx f_4(n)\)
 * Circle notation \(\approx f_4(n)\)
 * Hexation \(= a \uparrow^{4} b \approx f_5(n)\)
 * Heptation \(= a \uparrow^{5} b \approx f_6(n)\)
 * Octation \(= a \uparrow^{6} b \approx f_7(n)\)
 * Enneation \(= a \uparrow^{7} b \approx f_8(n)\)
 * Decation \(= a \uparrow^{8} b \approx f_9(n)\)
 * Undecation \(= a \uparrow^{9} b \approx f_{10}(n)\)
 * Doedecation \(= a \uparrow^{10} b \approx f_{11}(n)\)
 * Tredecation \(= a \uparrow^{11} b \approx f_{12}(n)\)

From \(f_\omega(n)\) to \(f_{\omega^\omega}(n)\)

 * Weak goodstein function \(= g(n) \approx f_\omega(n)\)
 * Ackermann function \(= A(n,n) \approx f_\omega(n)\)
 * Ackermann numbers \(\approx f_\omega(n)\), the limit of the hyper operators in general
 * Steinhaus-Moser notation \(\approx f_\omega(n)\)
 * Arrow notation (both variants) \(= a \uparrow^{c} b \approx f_\omega(n)\)
 * Hyper-E notation \(= E\# \approx f_\omega(n)\) (limit)
 * Graham's function \(= g_n \approx f_{\omega+1}(n)\)
 * Expansion \(= a \{\{1\}\} b \approx f_{\omega+1}(n)\)
 * Multiexpansion \(= a \{\{2\}\} b \approx f_{\omega+2}(n)\)
 * Powerexpansion \(= a \{\{3\}\} b \approx f_{\omega+3}(n)\)
 * Expandotetration \(= a \{\{4\}\} b \approx f_{\omega+4}(n)\)
 * Explosion \(= a \{\{\{1\}\}\} b \approx f_{\omega 2+1}(n)\)
 * Multiexplosion \(= a \{\{\{2\}\}\} b \approx f_{\omega 2+2}(n)\)
 * Powerexplosion \(= a \{\{\{3\}\}\} b \approx f_{\omega 2+3}(n)\)
 * Explodotetration \(= a \{\{\{4\}\}\} b \approx f_{\omega 2+4}(n)\)
 * Detonation \(= \{a,b,1,4\} \approx f_{\omega 3}(n)\)
 * Pentonation \(= \{a,b,1,5\} \approx f_{\omega 4}(n)\)
 * Hexonation \(= \{a,b,1,6\} \approx f_{\omega 5}(n)\)
 * Heptonation \(= \{a,b,1,7\} \approx f_{\omega 6}(n)\)
 * Octonation \(= \{a,b,1,8\} \approx f_{\omega 7}(n)\)
 * Ennonation \(= \{a,b,1,9\} \approx f_{\omega 8}(n)\)
 * Deconation \(= \{a,b,1,10\} \approx f_{\omega 9}(n)\)
 * CG function \(= cg(n) \approx f_{\omega^2}(n)\)
 * Exploding Tree Function \(= E(n) \approx f_{\omega^2}(n)\)
 * Megotion \(= \{a,b,1,1,2\} \approx f_{\omega^2+1}(n)\)
 * BOX_M~ function \(= \{a,b,1,1,2\} \approx f_{\omega^2+1}(n)\)
 * Multimegotion \(= \{a,b,2,1,2\} \approx f_{\omega^2+2}(n)\)
 * Powermegotion \(= \{a,b,3,1,2\} \approx f_{\omega^2+3}(n)\)
 * Megotetration \(= \{a,b,4,1,2\} \approx f_{\omega^2+4}(n)\)
 * Megoexpansion \(= \{a,b,1,2,2\} \approx f_{\omega^2+\omega+1}(n)\)
 * Multimegoexpansion \(= \{a,b,2,2,2\} \approx f_{\omega^2+\omega+2}(n)\)
 * Powermegoexpansion \(= \{a,b,3,2,2\} \approx f_{\omega^2+\omega+3}(n)\)
 * Megoexpandotetration \(= \{a,b,4,2,2\} \approx f_{\omega^2+\omega+4}(n)\)
 * Megoexplosion \(= \{a,b,1,3,2\} \approx f_{\omega^2+\omega 2}(n)\)
 * Megodetonation \(= \{a,b,1,4,2\} \approx f_{\omega^2+\omega 3}(n)\)
 * Gigotion \(= \{a,b,1,1,3\} \approx f_{(\omega^2) 2+1}(n)\)
 * Gigoexpansion \(= \{a,b,1,2,3\} \approx f_{(\omega^2) 2+\omega}(n)\)
 * Gigoexplosion \(= \{a,b,1,3,3\} \approx f_{(\omega^2) 2+\omega 2}(n)\)
 * Gigodetonation \(= \{a,b,1,4,3\} \approx f_{(\omega^2) 2+\omega 3}(n)\)
 * Terotion \(= \{a,b,1,1,4\} \approx f_{(\omega^2) 3+1}(n)\)
 * Petotion \(= \{a,b,1,1,5\} \approx f_{(\omega^2) 4+1}(n)\)
 * Hatotion \(= \{a,b,1,1,6\} \approx f_{(\omega^2) 5+1}(n)\)
 * Hepotion \(= \{a,b,1,1,7\} \approx f_{(\omega^2) 6+1}(n)\)
 * Ocotion \(= \{a,b,1,1,8\} \approx f_{(\omega^2) 7+1}(n)\)
 * Nanotion \(= \{a,b,1,1,9\} \approx f_{(\omega^2) 8+1}(n)\)
 * Uzotion \(= \{a,b,1,1,10\} \approx f_{(\omega^2) 9+1}(n)\)
 * Uuotion \(= \{a,b,1,1,11\} \approx f_{(\omega^2) 10+1}(n)\)
 * Udotion \(= \{a,b,1,1,12\} \approx f_{(\omega^2) 11+1}(n)\)
 * Utotion \(= \{a,b,1,1,13\} \approx f_{(\omega^2) 12+1}(n)\)
 * Ueotion \(= \{a,b,1,1,14\} \approx f_{(\omega^2) 13+1}(n)\)
 * Upotion \(= \{a,b,1,1,15\} \approx f_{(\omega^2) 14+1}(n)\)
 * Uhotion \(= \{a,b,1,1,16\} \approx f_{(\omega^2) 15+1}(n)\)
 * Uaotion \(= \{a,b,1,1,17\} \approx f_{(\omega^2) 16+1}(n)\)
 * Uootion \(= \{a,b,1,1,18\} \approx f_{(\omega^2) 17+1}(n)\)
 * Unotion \(= \{a,b,1,1,19\} \approx f_{(\omega^2) 18+1}(n)\)
 * Dzotion \(= \{a,b,1,1,20\} \approx f_{(\omega^2) 19+1}(n)\)
 * Tzotion \(= \{a,b,1,1,30\} \approx f_{(\omega^2) 29+1}(n)\)
 * Uzzotion \(= \{a,b,1,1,100\} \approx f_{(\omega^2) 99+1}(n)\)
 * Hurford's C function \(= C(n) \approx f_{\omega^3 + \omega}(n)\)

From \(f_{\omega^\omega}(n)\) to \(f_{\varepsilon_0}(n)\)

 * Linear array notation \(= \{a,b (1) 2\} \approx f_{\omega^\omega}(n)\) (limit)
 * Extended hyper-E notation \(xE\# \approx f_{\omega^\omega}(n)\) (limit)
 * n(k) function \(\approx f_{\omega^\omega}(n)\)
 * Aarex's array notation (2 row) \(= \{a,2 (2) 2\} \approx f_{\omega^{\omega 2}}(n)\)
 * Planar array notation \(= \{a,b (2) 2\} \approx f_{\omega^{\omega^2}}(n)\) (limit)
 * Aarex's array notation (planer) \(= \{a,a (2) 2\} \approx f_{\omega^{\omega^2}}(n)\)
 * Aarex's array notation (cubic) \(= \{a,a (3) 2\} \approx f_{\omega^{\omega^3}}(n)\)
 * Extended array notation (dimensional) \(= \{a,b (0,1) 2\} \approx f_{\omega^{\omega^\omega}}(n)\)
 * BEAF (superdimensional) \(= \{a,b (\underbrace{0,0\ldots0,0,1}_{n}) 2\} \approx f_{\omega^{\omega^{\omega^\omega}}}(n)\) (limit)
 * Aarex's array notation (superdimensional) \(= \{a,b [\underbrace{1,1\ldots1,1,2}_{n}] 2\} \approx f_{\omega^{\omega^{\omega^\omega}}}(n)\) (limit)

From \(f_{\varepsilon_0}(n)\) to \(f_{\Gamma_0}(n)\)
Starting from here, the totality of these functions is not provable in Peano arithmetic.


 * BEAF (tetrational) \(= {^ba} \& n \approx f_{\varepsilon_0}(n)\) (limit)
 * Cascading-E notation \(E\text{^} \approx f_{\varepsilon_0}(n)\) (limit)
 * Goodstein function \(= G(n) \approx f_{\varepsilon_0}(n)\)
 * Hydra(n) function \(\approx f_{\varepsilon_0}(n)\)
 * Nested Cascading-E Notation \(\approx f_{\varphi(\omega,0)}(n)\)

From \(f_{\Gamma_0}(n)\) to \(f_{\omega^\text{CK}_1}(n)\)
These functions and all those that follow cannot be proved total in arithmetical transfinite induction.


 * tree(n) function \(\approx f_{\theta(\Omega^\omega)}(n)\)
 * TREE(n) function \(> f_{\theta(\Omega^\omega)}(n)\)
 * BEAF (L-space) \(= \{a,b / 2\} \approx f_{\theta(\Omega^\Omega)}(n)\) (limit)
 * BEAF \(\approx f_{\theta((\Omega^\Omega)\omega)}(n)\) (estimated limit)
 * Bird's H(n) function \(\approx f_{\theta(\epsilon_{\Omega+1})}(n)\)
 * Bird's S(n) function \(\approx f_{\theta(\theta_1(\Omega))}(n)\)
 * Bird's U(n) function \(\approx f_{\theta(\Omega_\omega)}(n)\)
 * Bird's array notation \(\approx f_{\theta(\Omega_\omega)}(n)\) (limit)
 * SCG(n) function \(\geq f_{\psi_{\Omega_1}(\Omega_\omega)}(n)\)
 * BH(n) function \(\geq f_{\psi_0(\epsilon_{\Omega_\omega + 1})}(n)\)
 * Loader.c \(= D(n) \gg f_{\theta(\Omega_{\Omega_{\Omega_\cdots}})}(n)\)
 * No notation has yet been devised to describe the ordinal for this notation.

Beyond \(f_{\omega^\text{CK}_1}(n)\)
These functions are uncomputable, and cannot be evaluated by computer programs in finite time.


 * Rado's sigma function (and most of its variations) \(= \Sigma(n) \approx f_{\omega^\text{CK}_1}(n)\)
 * mth order sigma function \(= \Sigma_m(n) \approx f_{\omega^\text{CK}_m}(n)\)
 * Xi function \(= \Xi(n) \approx f_{\alpha}(n)\), where \(\alpha\) is \(\epsilon_0^{CK}\), the first fixed point of \(\alpha \mapsto \omega^\text{CK}_{\alpha}\).
 * 1st Aarex function \(\approx f_{\alpha+\omega}(n)\)
 * 2nd Aarex function \(\approx f_{\alpha+\omega^2}(n)\)
 * 3rd Aarex function \(\approx f_{\alpha+\omega^{\omega^2}}(n)\)
 * 4th Aarex function \(\approx f_{\alpha+\omega^{\omega^\omega}}(n)\)
 * 5th Aarex function \(\approx f_{\beta+\omega^{\omega^\omega}}(n)\), where \(\beta\) is \(\phi(\omega,1)^{CK}\).
 * Rayo's function \(\gg f_{\omega_\alpha^\text{CK}}(n)\)
 * The ordinal describing Rayo(n) is large enough that no notation is yet known for it.

Other

 * Fusible margin function
 * The growth rate of the \(m_1(x)\) function is an unsolved problem.
 * Hyperfactorial array notation
 * HAN is new and still in development; its growth rate has not yet been conclusively evaluated.
 * Slow-growing hierarchy (virtually unlimited; \(\omega_1\))
 * Hardy hierarchy (\(\omega_1\))
 * Fast-growing hierarchy (\(\omega_1\))
 * These three hierarchies can extend indefinitely, as long as ordinals and their fundamental hierarchies can be defined.