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 \(a \times b > f_1(n)\)
 * Exponentiation \(a^b \approx f_2(n)\)
 * Factorial (and most of its extensions) \(n! \approx f_2(n)\)
 * Double exponential functions \(\approx a^{a^b} \approx f_2(f_2(n)) \)
 * 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
 * Sudan function \(F_n(x,y) \approx f_\omega(n)\)
 * Steinhaus-Moser notation \(\approx f_\omega(n)\)
 * Arrow notation (both variants) \(a \uparrow^{n} b \approx f_\omega(n)\)
 * Psi Notation \(\approx f_\omega(n)\)
 * |T[k]| function \(\approx f_\omega(n)\)
 * Hyper-E notation \(E\# \approx f_\omega(n)\) (limit)
 * Graham's function \(g_n \approx f_{\omega+1}(n)\)
 * Exploding Tree Function \(E(n) \approx f_{\omega+1}(n)\)
 * 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 \(cg(n) \approx f_{\omega^2}(n)\)
 * Megotion \(\{a,b,1,1,2\} \approx f_{\omega^2+1}(n)\)
 * BOX_M̃ function \(\widetilde{M}_n \approx f_{\omega^2+1}(n)\)
 * 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 \(\{\underbrace{a,b\ldots y,z}_{n}\} \approx f_{\omega^\omega}(n)\) (limit)
 * Extended hyper-E notation \(xE\# \approx f_{\omega^\omega}(n)\) (limit)
 * n(k) function \(\approx f_{\omega^\omega}(n)\)
 * Taro's multivariable Ackermann function \(\approx f_{\omega^\omega}(n)\)
 * s(n) map \(\approx f_{\omega^\omega}(n)\)
 * Planar array notation \(\{a,b (2) 2\} \approx f_{\omega^{\omega^2}}(n)\) (limit)
 * Extended array notation (dimensional) \(\{a,b (0,1) 2\} \approx f_{\omega^{\omega^\omega}}(n)\)
 * BEAF superdimensional arrays \(\{a,b (\underbrace{0,0\ldots0,0,1}_{n}) 2\} \approx f_{\omega^{\omega^{\omega^\omega}}}(n)\) (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 arrays \({^ba} \& n \approx f_{\varepsilon_0}(n)\) (limit)
 * Cascading-E notation \(E\text{^} \approx f_{\varepsilon_0}(n)\) (limit)
 * m(n) map \(\approx f_{\varepsilon_0}(n)\)
 * Goodstein function \(G(n) \approx f_{\varepsilon_0}(n)\)
 * Hydra(n) function \(\approx f_{\varepsilon_0}(n)\)
 * Worm(n) function \(\approx f_{\varepsilon_0}(n)\)
 * X-Sequence Hyper-Exponential Notation \(\approx f_{\zeta_0}(n)\)
 * m(m,n) map \(\approx f_{\zeta_0}(n)\)
 * Nested Cascading-E Notation \(\approx f_{\varphi(\omega,0)}(n)\)
 * BEAF pentational arrays \(\approx f_{\Gamma_0}(n)\) (according to Bowers )

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.


 * Extended Cascading-E Notation \(\approx f_{\vartheta(\Omega\omega)}(n)\)
 * tree(n) function \(\approx f_{\vartheta(\Omega^\omega)}(n)\)
 * TREE(n) function \(≥ f_{\vartheta(\Omega^\omega\omega)}(n)\)
 * BEAF (L-space) \(\approx f_{\vartheta(\Omega^\Omega)}(n)\) (according to Chris Bird and Bowers )
 * Bird's H(n) function \(\approx f_{\vartheta(\varepsilon_{\Omega+1})}(n)\)
 * Bird's S(n) function (original) \(\approx f_{\vartheta(\theta_1(\Omega))}(n)\)
 * Bird's U(n) function \(\approx f_{\vartheta(\Omega_\omega)}(n)\)
 * SCG(n) function \(\geq f_{\psi_{\Omega_1}(\Omega_\omega)}(n)\)
 * BH(n) function \(\approx f_{\psi_0(\varepsilon_{\Omega_\omega + 1})}(n)\)
 * Bird's array notation \(\approx f_{\vartheta(\Omega_\Omega)}(n)\) (limit)
 * Bird's S(n) function (new definition) \(\approx f_{\vartheta(\Omega_\Omega)}(n)\)
 * AH(n) function \(\approx f_{\psi(\psi_I(0))}(n)\)
 * Loader.c function \(D(n)\)
 * Friedman's finite promise games

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)
 * mth order sigma function
 * Betti number
 * Doodle function
 * Xi function
 * Infinite time Turing machine busy beaver
 * Rayo's function
 * FOOT function

Other

 * Fusible margin function
 * The growth rate of the \(m_1(x)\) function is an unsolved problem.
 * Laver tables
 * It is not yet known if corresponding function is well-defined, and only some lower bounds are known for it.
 * Friedman's finite trees
 * Although suspected to be very strong, no explicit claims on the resulting functions' rate of growth have been made.
 * Hyperfactorial array notation
 * HAN is new and still in development; its growth rate has not yet been conclusively evaluated.
 * Slow-growing hierarchy, Hardy hierarchy, Fast-growing hierarchy
 * These three hierarchies can extend indefinitely, as long as ordinals and their fundamental hierarchies can be defined.
 * BEAF
 * BEAF is not well-defined beyond tetrational arrays, therefore there are mutiple interpretations, and therefore there are also multiple growth rates possible.