This page lists various googological functions arranged roughly by growth rate. They are grouped roughly by what theories are expected to prove them total recursive, and individual functions are also compared to the fast-growing hierarchy.
- \(\approx\) means that two functions have "comparable" growth rates in some unfixed sense.
- \(>\) 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.
Be careful that for computable functions A and B, even if the weakest theory among those which are known to prove the totality of B can prove the totality of A, B does not necessarily eventually dominate A. Similarly, the uncomputability of a function does not mean that it eventually dominates all total computable functions. (It's possible to make an uncomputable function that only outputs 0 or 1.) In other words, the order of this list does not mean the order of the growth rate.
Contents
Primitive recursive
These functions are all bounded by primitive recursive functions and it is believed that most can be proved total within the primitive recursive arithmetic (PRA).
- Successor function \(n+1 = f_0(n)\)
- Addition \(a+b = f^b_0(a)\)
- Multiplication \(a \times b > f_1(n)\) (limit)
- Exponentiation \(a^b \approx f_2(n)\) (limit)
- Factorial (and most of its extensions) \(n! \approx f_2(n)\)
- Superfactorial (Sloane and Plouffe) \(n$ \approx f_2(n)\)
- Torian \(T(x) = x!x \approx f_2(x)\)
- Hyperfactorial \(H(n) \approx f_2(n)\)
- Latin square \(L(n) \approx f_2(n)\)
- Double exponential functions \(a^{b^c} \approx f_2(f_2(n))\) (limit)
- Factorexation \(n \backslash \approx f_2(f_2(n))\)
- Googolple- \(\text{googol-}n\text{-ple-}n \approx f_2(f_2(n))\)
- Bop-counting function (not primitive recursive, but upper-bounded by primitive recursive functions, \(\theta(12)=n\) can't be proven given a natural number \(n\) in some theories including ZFC+I1)
- Exponential factorial \(a_n \approx f_3(n)\)
- Tetration \({^{b}a} \approx f_3(n)\) (limit)
- Big Ass Number \(\text{ban}(n) = n^{^nn} = {}^{n + 1}n \approx f_3(n)\)
- Expostfacto function \(\mathrm{expostfacto}(n) \approx f_3(n)\)
- Superfactorial (Pickover) \(n$ \approx f_3(f_2(n))\)
- Tetrofactorial \(n!2 \approx f_4(n)\)
- Pentation \(a \uparrow\uparrow\uparrow b\approx f_4(n)\) (limit)
- Really Big Ass Number \(\text{rban}(n) = n^{n \uparrow\uparrow\uparrow n} \approx f_4(n)\)
- Wow function \(= f_4(n)\)
- Circle notation \(Circle(n) \approx f_4(n)\)
- Pentatorial \(n[5]! \approx f_5(n)\)
- Hexation \(a \uparrow^{4} b \approx f_5(n)\) (limit)
- Heptation \(a \uparrow^{5} b \approx f_6(n)\) (limit)
- Octation \(a \uparrow^{6} b \approx f_7(n)\) (limit)
- Enneation \(a \uparrow^{7} b \approx f_8(n)\) (limit)
- Decation \(a \uparrow^{8} b \approx f_9(n)\) (limit)
- Undecation \(a \uparrow^{9} b \approx f_{10}(n)\) (limit)
- Doedecation \(a \uparrow^{10} b \approx f_{11}(n)\) (limit)
- Tredecation \(a \uparrow^{11} b \approx f_{12}(n)\) (limit)
RCA0
The totality of these functions cannot be proved in RCA0 (see second-order arithmetic) and they eventually dominate all primitive recursive functions. The fast-growing hierarchy here is given by Wainer hierarchy.
- Weak Goodstein function \(g(n) \approx f_\omega(n)\)
- |T[k]| function (finite ordered tree problem) \(|T[n]| \approx f_\omega(n)\)
- Ackermann function \(A(n,n) \approx f_\omega(n)\)
- Booga- \(booga(n) \approx f_\omega(n)\)
- Mythical tree problem \(\approx f_\omega(n)\) (limit)
- Friedman's vector reduction problem \(\approx f_\omega(n)\) (limit)
- Davenport-Schinzel sequence \(\approx f_\omega(n)\) (limit)
- Arrow notation (both variants) \(a \uparrow^{n} b \approx f_\omega(n)\) (limit)
- Ackermann numbers \(\approx f_\omega(n)\), the limit of the hyper operators in general
- Sudan function \(F_n(x,y) \approx f_\omega(n)\) (limit)
- Steinhaus-Moser notation \(\approx f_\omega(n)\) (limit)
- H* function \(H^*(\underbrace{n,n,\cdots n,n}_n) \approx f_\omega(n)\) (limit)
- Hyper-E notation \(E\# \approx f_\omega(n)\) (limit)
- Psi Notation \(\approx f_\omega(n)\) (limit)
- Graham's function \(g_n \approx f_{\omega+1}(n)\)
- Mag \(\text{mag}(n)\approx f_{\omega+1}(n)\)
- Trooga- \(\text{trooga}(n)\approx f_{\omega+1}(n)\)
- Exploding Tree Function \(E(n) \approx f_{\omega+1}(n)\)
- Expansion \(a \{\{1\}\} b \approx f_{\omega+1}(n)\)
- Hyperlicious function \(h_n(\underbrace{n,n,\cdots n,n}_n) \approx f_{\omega+1}(n)\) (limit)
- Multiexpansion \(a \{\{2\}\} b \approx f_{\omega+2}(n)\)
- Mixed factorial \(n^*_{(n,n)} \approx f_{\omega+2}(n)\)
- Powerexpansion \(a \{\{3\}\} b \approx f_{\omega+3}(n)\)
- Expandotetration \(a \{\{4\}\} b \approx f_{\omega+4}(n)\)
- -ag \(n\text{-ag}(n)\approx f_{\omega 2}(n)\)
- 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 \(a \{\{\{4\}\}\} b \approx f_{\omega 2+4}(n)\)
- Copy Notation \(\approx f_{\omega3}(n)\) (limit)
- Detonation \(\{a,b,1,4\} \approx f_{\omega 3+1}(n)\)
- Pentonation \(\{a,b,1,5\} \approx f_{\omega 4+1}(n)\)
- Hexonation \(\{a,b,1,6\} \approx f_{\omega 5+1}(n)\)
- Heptonation \(\{a,b,1,7\} \approx f_{\omega 6+1}(n)\)
- Octonation \(\{a,b,1,8\} \approx f_{\omega 7+1}(n)\)
- Ennonation \(\{a,b,1,9\} \approx f_{\omega 8+1}(n)\)
- Deconation \(\{a,b,1,10\} \approx f_{\omega 9+1}(n)\)
- CG function \(cg(n) \approx f_{\omega^2}(n)\)
- Extended H* Function \(H^*_n(\underbrace{n,n,\cdots n,n}_n) \approx f_{\omega^2}(n)\) (limit)
- 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 \(\{a,b,1,3,2\} \approx f_{\omega^2+\omega 2+1}(n)\)
- Megodetonation \(\{a,b,1,4,2\} \approx f_{\omega^2+\omega 3+1}(n)\)
- Gigotion \(\{a,b,1,1,3\} \approx f_{(\omega^2) 2+1}(n)\)
- Gigoexpansion \(\{a,b,1,2,3\} \approx f_{(\omega^2) 2+\omega+1}(n)\)
- Gigoexplosion \(\{a,b,1,3,3\} \approx f_{(\omega^2) 2+\omega 2+1}(n)\)
- Gigodetonation \(\{a,b,1,4,3\} \approx f_{(\omega^2) 2+\omega 3+1}(n)\)
- 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 \(\{a,b,1,1,100\} \approx f_{(\omega^2) 99+1}(n)\)
- Powiaination \(\{a,b,1,1,1,2\} \approx f_{\omega^3+1}(n)\)
- Multiaination \(\{a,b,2,1,1,2\} \approx f_{\omega^3+2}(n)\)
- Hurford's C function \(C(n) \approx f_{\omega^3 + \omega}(n)\)
- Expandaination \(\{a,b,1,2,1,2\} \approx f_{\omega^3+\omega+1}(n)\)
- Megodaination \(\{a,b,1,1,2,2\} \approx f_{\omega^3+\omega^2+1}(n)\)
- Powiairiation \(\{a,b,1,1,1,3\} \approx f_{(\omega^3) 2+1}(n)\)
- Chained array notation \(n[\underbrace{n,n,\cdots n}_n]n \approx f_{\omega^5}(n)\) (limit)
PA
The following functions eventually dominate all multirecursive functions but are still provably recursive within Peano arithmetic (PA). The fast-growing hierarchy here is given by Wainer hierarchy.
- 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 \(\approx f_{\omega^\omega}(n)\)
- The Q-supersystem \(Q_{\underbrace{1,0,0,\ldots,0,0}_n}(n) \approx f_{\omega^\omega}(n)\) (limit)
- Taro's multivariable Ackermann function \(\approx f_{\omega^\omega}(n)\) (limit)
- SAN linear array notation \(s(n,n,\ldots,n,n) \approx f_{\omega^\omega}(n)\) (limit)
- s(n) map \(\approx f_{\omega^\omega}(n)\)
- Hyper-Moser notation \(M(n,\underbrace{0,0,\cdots0,}_nn) \approx f_{\omega^\omega}(n)\) (limit)
- Aarex's Graham Generator \(\text{Forcal}_{\underbrace{1,1,\cdots1,}_n2}(1) \approx f_{\omega^\omega}(n)\) (limit)
- Nested factorial notation \(n![\underbrace{n,n,\cdots n,n}_n] \approx f_{\omega^\omega}(n)\) (limit)
- Matthew's Function \(n\uparrow\rightarrow\uparrow n \approx f_{\omega^{\omega^2}}(n)\)
- Planar array notation \(\{a,b (2) 2\} \approx f_{\omega^{\omega^2}}(n)\) (limit)
- Extended array notation (dimensional) \(\{a,b (0,1) 2\} \approx f_{\omega^{\omega^\omega}}(n)\) (limit)
- BEAF superdimensional arrays \(\{a,b (\underbrace{0,0\ldots0,0,1}_{n}) 2\} \approx f_{\omega^{\omega^{\omega^\omega}}}(n)\) (limit)
ATR0
Starting from here, the totality of these functions is not provable in PA. The fast-growing hierarchy here is given by Veblen hierarchy.
- Goodstein function \(G(n) \approx f_{\varepsilon_0}(n)\)
- Bracket notation \(n\$[[0]_2] \approx f_{\varepsilon_0}(n)\)
- BEAF tetrational arrays \({^ba} \& n \approx f_{\varepsilon_0}(n)\) (limit)
- Notation Array Notation \(\approx f_{\varepsilon_0}(n)\) (limit)
- Cascading-E notation \(E\text{^} \approx f_{\varepsilon_0}(n)\) (limit)
- Pound-Star Notation \(\approx f_{\varepsilon_0}(n)\) (limit)
- SAN extended array notation \(s(n,n\{1\{1\ldots\{1\{1,2\}2\}\ldots2\}2\}2) \approx f_{\varepsilon_0}(n)\) (limit)
- Circle(n) function \(\approx f_{\varepsilon_0}(n)\)
- m(n) map \(\approx f_{\varepsilon_0}(n)\)
- Hydra(n) function \(\approx f_{\varepsilon_0}(n)\)
- Worm(n) function \(\approx f_{\varepsilon_0}(n)\)
- Primitive sequence system
- Marxen.c function \(h(g(n),n)\)
- X-Sequence Hyper-Exponential Notation \(\approx f_{\zeta_0}(n)\)
- m(m,n) map \(\approx f_{\zeta_0}(n)\)
- Nested Cascading-E Notation \(\approx f_{\varphi(\omega,0)}(n)\) (limit)
- Three Bracket NaN \(\approx f_{\varphi(\omega,0)}(n)\)
ZFC
These functions cannot be proved total in arithmetical transfinite induction but are believed to be provably total in \(\textrm{ZFC}\). It does not mean that the totality is actually verified, and actually the list contains functions whose totality or even computability is not known in the current googology community.
The fast-growing hierarchy here is given by Denis' fundamental sequences associated to Buchholz's function and Extended Buchholz's function for expressions including those functions, and is given by the fundamental sequence associated to the tautological norm on the ordinal notation associated to Rathjen's psi for expressions including the function. On the other hand, \(\vartheta\) with inputs larger than \(\Omega_2\) and \(\psi\) with inputs larger than \(\varepsilon_{\Omega_{\omega}+1}\) in estimations are unspecified (and possibly ill-defined) OCFs, and hence those estimations are meaningless. Also, these estimations are not necessarily verified, and might be just expectations, i.e. conjectural one without proofs.
- Fusible margin function \(m_1(x)\) (A non-trivial description of the conjectural growth rate is unknown.)
- Username5243's Array Notation \(\approx f_{\Gamma_0}(n)\) (limit)
- Extended Cascading-E Notation \(\approx f_{\vartheta(\Omega^{2}\omega)}(n)\) (limit)
- Hyper-Extended Cascading-E Notation \(\approx f_{\vartheta(\Omega^{3})}(n)\) (limit)
- Extended Q-supersystem \(xQ_{\underbrace{1,0,0,\ldots,0,0}_{n}}(n) \approx f_{\vartheta(\Omega^{\omega})}(n)\) (limit)
- TREE(n) function \(>\) tree(n) function \(> f_{\psi_0(\Omega^\omega)}(n)\) (More precisely, the strict lowerbound is proved for tree(1.00001n+2) rather than tree(n).)
- Bird's H(n) function \(\approx f_{\vartheta(\varepsilon_{\Omega+1})}(n)\)
- SAN expanding array notation \(s(n,n\{1\{1\{1\ldots\{1\{1,2`\}2`\}\ldots2`\}2`\}2\}2) \approx f_{\vartheta(\varepsilon_{\Omega+1})}(n)\) (limit)
- Bird's S(n) function (original) \(\approx f_{\vartheta(\theta_1(\Omega))}(n)\)
- Bird's U(n) function \(\approx f_{\vartheta(\Omega_\omega)}(n)\)
- SAN multiple expanding array notation \(s(n,n \{1\underbrace{``\ldots`}_{n}2\} 2) = s(n,n\{1,,1,2\}2) \approx f_{\vartheta(\Omega_\omega)}(n)\) (limit)
- Pair sequence system \(\approx f_{\psi_0(\Omega_\omega)}(n)\) with respect to Buchholz's function
- Hyper primitive sequence system \(\approx f_{\psi_0(\Omega_{\omega})}(n)\)
- 段階配列表記 \(\approx f_{\psi_0(\Omega_{\omega})}(n)\)
- SCG(n) function \(\geq f_{\psi_0(\Omega_\omega)}(n)\) with respect to Buchholz's function
- BH(n) function \(\approx f_{\psi_0(\varepsilon_{\Omega_\omega + 1})}(n)\) with respect to Buchholz's function
- Bird's array notation \(\approx f_{\vartheta(\Omega_\Omega)}(n)\) (limit)
- Bird's S(n) function (new definition) \(\approx f_{\vartheta(\Omega_\Omega)}(n)\)
- Hyperfactorial array notation \(\geq f_{\psi_0(\varepsilon_{\Omega_\omega+1})}(n)\) with respect to Buchholz's function
- 降下段階配列表記 \(\approx f_{\psi_{\chi_0(0)}(\Phi_1(0))}(n)\) with respect to Rathjen's ψ = \(f_{\psi_0(\Omega_{\Omega_{...} })}(n)\) with respect to Extended Buchholz's function
- KumaKuma ψ function \(\approx f_{\psi_{\chi_0(0)}(\psi_{\chi_{3}(0)}(0))}(n)\)
- 多変数段階配列表記 \(\approx f_{\psi_{\chi_0(0)}(\psi_{\chi_{\omega}(0)}(0))}(n)\)
- ε function \(\approx f_{\psi_{\chi_0(0)}(\chi_{M+1}(0))}(n)\)
- Primary dropping array notation (strong array notation) (A conjectural growth rate corresponds to C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω2,C(Ω22,0))),C(Ω2,C(Ω22,0))),0),0),0)[1] in Taranovsky's ordinal notation using Hyp_cos's fundamental sequences, but there is no actual proof.)
- Stegert's \(f\) functions (limit)
- Strong array notation (It has not been formalised. A non-trivial description of the conjectural growth rate is unknown for a fixed formalisation, but it is believed to be strong.)
- Bashicu matrix system (A non-trivial description of the conjectural growth rate is unknown, but it is believed to be strong.)
- N primitive (A non-trivial description of the conjectural growth rate is unknown, but it is believed to be strong.)
- Y sequence (A non-trivial description of the conjectural growth rate of the desired behaviour is unknown, but it is believed to be strong.)
- ω-Y sequence (A non-trivial description of the conjectural growth rate of the desired behaviour is unknown, but it is believed to be strong.)
- Loader.c function \(D(n)\) (A non-trivial description of the conjectural growth rate is unknown, but it is believed to be strong.)
Stronger set theories
These functions cannot be proven total in \(\textrm{ZFC}\), but are known to be provably total in stronger set theories.
- Friedman's finite promise games functions \(FPLCI(a)\), \(FPCI(a)\), and \(FLCI(a)\) defined in \(\textrm{SMAH}+\)
- Greedy clique sequence functions \(USGCS(k)\), \(USGDCS(k)\), \(USGDCS_2(k)\), and (\(USGDCS_2(n)\) defined in \(\textrm{HUGE}+\)
- Friedman's finite trees function \(F(k,p)\) defined in \(\textrm{ZFC}\) augmented by the axiom "for any \(k \in \mathbb{N}\), there exists a \(k\)-subtle cardinal"
- Laver's function \(q(n)\) defined in \(\textrm{ZFC}+\textrm{I}3\)
- I0 function \(\textrm{I}0(n)\) defined in \(\textrm{ZFC}+\textrm{I}0\) augumented by the \(\Sigma_1\)-soundness of \(\textrm{ZFC}+\textrm{I}0\)
Uncomputable functions
These functions are uncomputable, and cannot be evaluated by computer programs in finite time.
- Busy beaver function
- Frantic frog function
- Placid platypus function (slow-growing)
- Weary wombat function (slow-growing)
- mth order busy beaver function
- \(b_k(N)\) function using Betti numbers.
- Doodle function
- Xi function
- Infinite time Turing machine busy beaver function
- Rayo's function (A problem on axioms has been pointed out.)
- FOOT (A problem on its definition has been pointed out.)
Other
The following are functions which are not single maps:
- Slow-growing hierarchy, Hardy hierarchy, Fast-growing hierarchy. These three hierarchies can be extended indefinitely, as long as ordinals and their fundamental hierarchies can be defined. Although they are literally uncomputable with transfinite indices, their segments can be interpreted into computable functions if ordinals are replaced by terms in an ordinal notation and fundamental sequences are given by an algorithm on expressions.
- BEAF. BEAF is not formalized and well-defined beyond tetrational arrays, so there are multiple interpretations.
- Lossy channel systems and priority channel systems. The complexity classes of some decision problems are googologically large, but no single fast-growing function or number has been extracted from these.
- TR function. It is a hierarchy of functions indexed by formal theories, or a function on the pair of a formal theory and a natural number.
