List of functions

This page lists all googology functions that has an individual page. There are some definitions:


 * \(\approx\) means that two functions have comparable growth rates.
 * \(>\) means that one function significantly overgrows other.
 * \(geq\) means that it is not known exactly whether one function overgrows other or not.

From \(f_0(n)\) to \(f_\omega(n)\)

 * Addition \(> f_0(n)\)
 * Multiplication \(> f_1(n)\)
 * Exponentiation \(\approx f_2(n)\)
 * Factorial (and most of its extensions) \(\approx f_2(n)\)
 * Exponential factorial \(\approx f_3(n)\)
 * Tetration \(\approx f_3(n)\)
 * Pentation \(\approx f_4(n)\)
 * Hexation \(\approx f_5(n)\)
 * Heptation \(\approx f_6(n)\)
 * Octation \(\approx f_7(n)\)
 * Enneation \(\approx f_8(n)\)
 * Decation \(\approx f_9(n)\)

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

 * Ackermann function \(\approx f_\omega(n)\)
 * Expansion \(\approx f_{\omega+1}(n)\)
 * Multiexpansion \(\approx f_{\omega+2}(n)\)
 * Powerexpansion \(\approx f_{\omega+3}(n)\)
 * Expandotetration \(\approx f_{\omega+4}(n)\)
 * Explosion \(\approx f_{\omega 2+1}(n)\)
 * Multiexplosion \(\approx f_{\omega 2+2}(n)\)
 * Powerexplosion \(\approx f_{\omega 2+3}(n)\)
 * Explodotetration \(\approx f_{\omega 2+4}(n)\)
 * Detonation \(\approx f_{\omega 3}(n)\)
 * Pentonation \(\approx f_{\omega 4}(n)\)
 * Hexonation \(\approx f_{\omega 5}(n)\)
 * Heptonation \(\approx f_{\omega 6}(n)\)
 * Octonation \(\approx f_{\omega 7}(n)\)
 * Ennonation \(\approx f_{\omega 8}(n)\)
 * Deconation \(\approx f_{\omega 9}(n)\)
 * CG function \(\approx f_{\omega^2}(n)\)
 * Exploding Tree Function \(\approx f_{\omega^2}(n)\)
 * Hurford's C function \(\approx f_{\omega^5}(n)\)

From \(f_{\omega^\omega}(n)\) to \(f_{\omega^\text{CK}_1}(n)\)

 * n(k) function \(\approx f_{\omega^\omega}(n)\)
 * Goodstein function \(\approx f_{\varepsilon_0}(n)\)
 * tree(n) function \(\approx f_{\vartheta(\Omega^\omega)}(n)\)
 * TREE(n) function \(> f_{\vartheta(\Omega^\omega)}(n)\)
 * H(n) function \(\approx f_{\vartheta(\varepsilon_{\Omega+1})}(n)\)
 * SCG(n) function \(\geq f_{\psi_{\Omega_1}(\Omega_\omega)}(n)\)

Beyond \(f_{\omega^\text{CK}_1}(n)\)

 * Rado's sigma function (and most of its variations) \(\approx f_{\omega^\text{CK}_1}(n)\)
 * mth order sigma function \(\approx f_{\omega^\text{CK}_m}(n)\)
 * Rayo's function \(\geq f_{\omega^\text{CK}_{\omega}}(n)\)
 * Xi function \(\approx f_{\omega^\text{CK}_{\alpha}}(n)\), where \(\alpha\) is the first ordinal such that \(\omega^\text{CK}_{\alpha} = \alpha\).