- Class 0 and 1
- Class 2
- Class 3
- Class 4
- Class 5
- Tetration level
- Up-arrow notation level
- Linear omega level
- Quadratic omega level
- Polynomial omega level
- Exponentiated linear omega level
- Exponentiated polynomial omega level
- Double exponentiated polynomial omega level
- Triple exponentiated polynomial omega level
- Iterated Cantor normal form level
- Epsilon level
- Binary phi level
- Bachmann's collapsing level
- Higher computable level
- Uncomputable numbers
Since the comparison (or even the well-definedness) of the numbers in this level is unknown, the order of entries does not necessarily imply the order of the sizes (note that BEAF numbers beyond tetrational arrays, which are ill-defined, should not lie in this level, see also Category:Class disputed and Talk:Meameamealokkapoowa oompa#Size issue). Also, several numbers are defined by an OCF, which is uncomputable, and are not known to be computable, particularly beyond \(\psi_0(\Phi_1(0))\) level on the fast-growing hierarchy with respect to the system of fundamental sequences for Extended Buchholz's function, where \(\psi\) is Extended Buchholz's function and \(\Phi_1(0)\) is the least omega fixed point. Moreover, several approximations are using unspecified (or even ill-defined) OCFs, and hence might be mathematically meaningless. Note that various systems of fundamental sequences are used for comparisons on this level.
\(f_{\psi_0(\varepsilon_{\Omega+1})}^2(10)\) ~ \(f_{\psi_0(\Omega_\omega)}^2(10)\)
- Extremebixul, 200![1(1)[2200,200,200,200,200]]
- Kiloextremebixul, (200![1(1)[2200,200,200,200,200]])![1(1)[2200,200,200,200,200]]
- Extremetrixul, 200![1(1)[2200,200,200,200,200,200]]
- Extremequaxul, 200![1(1)[2200,200,200,200,200,200,200]]
- Bommthet, \(f_{\theta(\Omega_2,0)}(10)\)
- Gigantixul, 200![1(1)[3200,200,200]]
- Golapulusplex, {10,100} & 10 & 10 & 10
- Gigantibixul, 200![1(1)[3200,200,200,200]]
- Gigantitrixul, 200![1(1)[3200,200,200,200,200]]
- Gigantiquaxul, 200![1(1)[3200,200,200,200,200,200]]
- Trommthet, \(f_{\theta(\Omega_3,0)}(10)\)
- Quadrommthet, \(f_{\theta(\Omega_4,0)}(10)\)
- Quintommthet, \(f_{\theta(\Omega_5,0)}(10)\)
- Sextommthet, \(f_{\theta(\Omega_6,0)}(10)\)
- Septommthet, \(f_{\theta(\Omega_7,0)}(10)\)
- Octommthet, \(f_{\theta(\Omega_8,0)}(10)\)
- Dekulus / Big Mac, {10,10/2}
- Nonommthet, \(f_{\theta(\Omega_9,0)}(10)\)
- Dekommthet, \(f_{\theta(\Omega_{10},0)}(10)\)
- SCG(13) (lower bound), \(\approx f_{\psi(\Omega_\omega)}(13)\)
- Hektommthet, \(f_{\theta(\Omega_{100},0)}(10)\)
- The Whopper, {10,100/2}
- Meeting point of the three hierarchies, \(f_{\theta(\Omega_{\omega})}(100)\) = \( f_{\psi_0(\psi_{\omega}^3(0))}(100)\)[citation needed]
- Nucleabixul, 200![[200200]200]
- Kilommthet, \(f_{\theta(\Omega_{1000},0)}(10)\)
- Megommthet, \(f_{\theta(\Omega_{10^6},0)}(10)\)
- Gigommthet, \(f_{\theta(\Omega_{10^9},0)}(10)\)
- Terommthet, \(f_{\theta(\Omega_{10^{12}},0)}(10)\)
- Petommthet, \(f_{\theta(\Omega_{10^{15}},0)}(10)\)
- Exommthet, \(f_{\theta(\Omega_{10^{18}},0)}(10)\)
- Zettommthet, \(f_{\theta(\Omega_{10^{21}},0)}(10)\)
- Yottommthet, \(f_{\theta(\Omega_{10^{24}},0)}(10)\)
\(f_{\psi_0(\Omega_\omega)}^2(10)\) ~ \(f_{\psi_0(\Phi_1(0))+1}(10)\)
- Pair sequence number, \(\approx f_{\psi(\Omega_\omega)+1}(10)\)
- 段階配列数, \(\approx f_{\psi(\Omega_{\omega})+1}(100)\)
- Big boowa, {3,3,3 / 2}
- Great big boowa, {3,3,4 / 2}
- Grand boowa, {3,3,big boowa / 2} = {3,2,2,2 / 2}
- Gigantic big boowa, {3, 3, {3, 3, 4 / 2} / 2} = {3, 3, great big boowa / 2}
- Gorged boowa, {3, 3, {3, 3, {3, 3, 3 / 2} / 2} / 2} = {3, 3, grand boowa / 2}
- Gulp big boowa, {3, 3, {3, 3, {3, 3, 4 / 2} / 2} / 2} = {3, 3, gigantic big boowa / 2}
- Super gongulus, {10,10 (100) 2 / 2}
- Wompogulus, {10,10 (10) 2 / 100}
- Guapamonga, 10100 && (10100 & 10)
- Guapamongaplex, 10guapamonga && (10guapamonga & 10)
- Big Chunk, {10,100 [1 [1 [1 \2,2 3] 2] 2] 2}
- Bimixommwil, \(f_{\psi(\Omega_{\psi(\Omega)})}(10)\)
- Trimixommwil, \(f_{\psi(\Omega_{\psi(\Omega_{\psi(\Omega)})})}(10)\)
- Quadrimixommwil, \(f_{\psi(\Omega_{\psi(\Omega_{\psi(\Omega_{\psi(\Omega)})})})}(10)\)
- Quintimixommwil, \(f_{\psi(\Omega_{\psi(\Omega_{\psi(\Omega_{\psi(\Omega_{\psi(\Omega)})})})})}(10)\)
- Sextimixommwil, \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega)}\cdots})})}_{6\ \Omega's}}(10)\)
- Septimixommwil, \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega)}\cdots})})}_{7\ \Omega's}}(10)\)
- Octimixommwil, \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega)}\cdots})})}_{8\ \Omega's}}(10)\)
- Nonimixommwil, \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega)}\cdots})})}_{9\ \Omega's}}(10)\)
- Dekomixommwil, \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega)}\cdots})})}_{10\ \Omega's}}(10)\)
- Binommwil, \(f_{\psi(\Omega_\Omega)}(10)\)
- Hektomixommwil, \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega)}\cdots})})}_{100\ \Omega's}}(10)\)
- Kilomixommwil, \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega)}\cdots})})}_{1000\ \Omega's}}(10)\)
- Megomixommwil, \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega)}\cdots})})}_{10^6\ \Omega's}}(10)\)
- Gigomixommwil, \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega)}\cdots})})}_{10^9\ \Omega's}}(10)\)
- Teromixommwil, \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega)}\cdots})})}_{10^{12}\ \Omega's}}(10)\)
- Petomixommwil, \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega)}\cdots})})}_{10^{15}\ \Omega's}}(10)\)
- Exomixommwil, \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega)}\cdots})})}_{10^{18}\ \Omega's}}(10)\)
- Zettomixommwil, \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega)}\cdots})})}_{10^{21}\ \Omega's}}(10)\)
- Yottomixommwil, \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega)}\cdots})})}_{10^{24}\ \Omega's}}(10)\)
- Nucleatrixul, 200![[[200200]200]200]
- Trinommwil, \(f_{\psi(\Omega_{\Omega_\Omega})}(10)\)
- Nucleaquaxul, 200![[[[200200]200]200]200]
- Quadrinommwil, \(f_{\psi(\Omega_{\Omega_{\Omega_\Omega}})}(10)\)
- Quintinommwil, \(f_{\psi(\Omega_{\Omega_{\Omega_{\Omega_\Omega}}})}(10)\)
- Sextinommwil, \(f_{\psi(\Omega_{\Omega_{\Omega_{\Omega_{\Omega_\Omega}}}})}(10)\)
- Septinommwil, \(f_{\psi(\underbrace{\Omega_{\Omega_{\cdots_\Omega}}}_{7\ \Omega's})}(10)\)
- Octinommwil, \(f_{\psi(\underbrace{\Omega_{\Omega_{\cdots_\Omega}}}_{8\ \Omega's})}(10)\)
- Noninommwil, \(f_{\psi(\underbrace{\Omega_{\Omega_{\cdots_\Omega}}}_{9\ \Omega's})}(10)\)
- Dekinommwil, \(f_{\psi(\underbrace{\Omega_{\Omega_{\cdots_\Omega}}}_{10\ \Omega's})}(10)\)
- Hektinommwil, \(f_{\psi(\underbrace{\Omega_{\Omega_{\cdots_\Omega}}}_{100\ \Omega's})}(10)\)
- Big hoss, \(\lbrace 100,100 \underbrace{///\cdots ///}_{100} 2\rbrace\)
- Grand hoss, \(\lbrace 100,100 \underbrace{///\cdots ///}_{100} 100\rbrace\)
- Kilinommwil, \(f_{\psi(\underbrace{\Omega_{\Omega_{\cdots_\Omega}}}_{1000\ \Omega's})}(10)\)
- Meginommwil, \(f_{\psi(\underbrace{\Omega_{\Omega_{\cdots_\Omega}}}_{10^6\ \Omega's})}(10)\)
- Giginommwil, \(f_{\psi(\underbrace{\Omega_{\Omega_{\cdots_\Omega}}}_{10^9\ \Omega's})}(10)\)
- Terinommwil, \(f_{\psi(\underbrace{\Omega_{\Omega_{\cdots_\Omega}}}_{10^{12}\ \Omega's})}(10)\)
- Petinommwil, \(f_{\psi(\underbrace{\Omega_{\Omega_{\cdots_\Omega}}}_{10^{15}\ \Omega's})}(10)\)
- Exinommwil, \(f_{\psi(\underbrace{\Omega_{\Omega_{\cdots_\Omega}}}_{10^{18}\ \Omega's})}(10)\)
- Zettinommwil, \(f_{\psi(\underbrace{\Omega_{\Omega_{\cdots_\Omega}}}_{10^{21}\ \Omega's})}(10)\)
- Yottinommwil, \(f_{\psi(\underbrace{\Omega_{\Omega_{\cdots_\Omega}}}_{10^{24}\ \Omega's})}(10)\)
- Great big hoss, \(\lbrace \text{big hoss},\text{big hoss} \underbrace{///\cdots ///}_{\text{big hoss}} 2\rbrace\)
\(f_{\psi_0(\Phi_1(0))+1}(10)\) ~
Some of these numbers are based on functions which are neither provable nor disprovable to be total in \(\mathrm{ZFC}\).
- Unimah, \(f_{\psi(I)}(10)\)
- Bitetrotos, \(f_{\psi(I^I)}(10)\)
- Bukuwaha, {L,100100}100,100
- Tritetrotos, \(f_{\psi(I^{I^I})}(10)\)
- Quadritetrotos, \(f_{\psi(I^{I^{I^I}})}(10)\)
- Quintitetrotos, \(f_{\psi(I^{I^{I^{I^I}}})}(10)\)
- Sextitetrotos, \(f_{\psi(I\uparrow\uparrow6)}(10)\)
- Septitetrotos, \(f_{\psi(I\uparrow\uparrow7)}(10)\)
- Octitetrotos, \(f_{\psi(I\uparrow\uparrow8)}(10)\)
- Nonitetrotos, \(f_{\psi(I\uparrow\uparrow9)}(10)\)
- Dekotetrotos, \(f_{\psi(I\uparrow\uparrow10)}(10)\)
- Hektotetrotos, \(f_{\psi(I\uparrow\uparrow100)}(10)\)
- Kilotetrotos, \(f_{\psi(I\uparrow\uparrow1000)}(10)\)
- Megotetrotos, \(f_{\psi(I\uparrow\uparrow10^6)}(10)\)
- Gigotetrotos, \(f_{\psi(I\uparrow\uparrow10^9)}(10)\)
- Terotetrotos, \(f_{\psi(I\uparrow\uparrow10^{12})}(10)\)
- Petotetrotos, \(f_{\psi(I\uparrow\uparrow10^{15})}(10)\)
- Exotetrotos, \(f_{\psi(I\uparrow\uparrow10^{18})}(10)\)
- Zettotetrotos, \(f_{\psi(I\uparrow\uparrow10^{21})}(10)\)
- Yottotetrotos, \(f_{\psi(I\uparrow\uparrow10^{24})}(10)\)
- BIGG, 200? = 200![[<1(200)2>⁅200⁆1]]
- Uninotos, \(f_{\psi(I_I)}(10)\)
- Binotos, \(f_{\psi(I_{I_I})}(10)\)
- Trinotos, \(f_{\psi(I_{I_{I_I}})}(10)\)
- Sextinotos, \(f_{\psi(\underbrace{I_{I_{\cdots_I}}}_{7\ I's})}(10)\)
- Bimah, \(f_{\psi(I(2,0))}(10)\)
- Dekinotos, \(f_{\psi(\underbrace{I_{I_{\cdots_I}}}_{11\ I's})}(10)\)
- Zettinotos, \(f_{\psi(\underbrace{I_{I_{\cdots_I}}}_{10^{21}+1\ I's})}(10)\)
- Yottinotos, \(f_{\psi(\underbrace{I_{I_{\cdots_I}}}_{10^{24}+1\ I's})}(10)\)
- Trimah, \(f_{\psi(I(3,0))}(10)\)
- Quadrimah, \(f_{\psi(I(4,0))}(10)\)
- Quintimah, \(f_{\psi(I(5,0))}(10)\)
- Sextimah, \(f_{\psi(I(6,0))}(10)\)
- Septimah, \(f_{\psi(I(7,0))}(10)\)
- Octimah, \(f_{\psi(I(8,0))}(10)\)
- Nonimah, \(f_{\psi(I(9,0))}(10)\)
- Dekimah, \(f_{\psi(I(10,0))}(10)\)
- Hektimah, \(f_{\psi(I(100,0))}(10)\)
- Kilimah, \(f_{\psi(I(1000,0))}(10)\)
- Megimah, \(f_{\psi(I(10^6,0))}(10)\)
- Gigimah, \(f_{\psi(I(10^9,0))}(10)\)
- Terimah, \(f_{\psi(I(10^{12},0))}(10)\)
- Petimah, \(f_{\psi(I(10^{15},0))}(10)\)
- Eximah, \(f_{\psi(I(10^{18},0))}(10)\)
- Zettimah, \(f_{\psi(I(10^{21},0))}(10)\)
- Yottimah, \(f_{\psi(I(10^{24},0))}(10)\)
- Uninimah, \(f_{\psi(I(I(0,0),0))}(10)\)
- Trinimah, \(f_{\psi(I(I(I(I(0,0),0),0),0))}(10)\)
- Terinimah, \(f_{\psi(\underbrace{I(I(\cdots I(0,0)\cdots,0),0)}_{10^{12}+1\ I's})}(10)\)
- Zettinimah, \(f_{\psi(\underbrace{I(I(\cdots I(0,0)\cdots,0),0)}_{10^{21}+1\ I's})}(10)\)
- Yottinimah, \(f_{\psi(\underbrace{I(I(\cdots I(0,0)\cdots,0),0)}_{10^{24}+1\ I's})}(10)\)
- Tritetremar, \(f_{\psi(M^{M^M})}(10)\)
- Sextitetremar, \(f_{\psi(M\uparrow\uparrow6)}(10)\)
- Terotetremar, \(f_{\psi(M\uparrow\uparrow10^{12})}(10)\)
- Petotetremar, \(f_{\psi(M\uparrow\uparrow10^{15})}(10)\)
- Exotetremar, \(f_{\psi(M\uparrow\uparrow10^{18})}(10)\)
- Zettotetremar, \(f_{\psi(M\uparrow\uparrow10^{21})}(10)\)
- Yottotetremar, \(f_{\psi(M\uparrow\uparrow10^{24})}(10)\)
- Uninemar, \(f_{\psi(M_M)}(10)\)
- Trinemar, \(f_{\psi(M_{M_{M_M}})}(10)\)
- Quadrinemar, \(f_{\psi(M_{M_{M_{M_M}}})}(10)\)
- Quintinemar, \(f_{\psi(\underbrace{M_{M_{\cdots_M}}}_{6\ M's})}(10)\)
- Sextinemar, \(f_{\psi(\underbrace{M_{M_{\cdots_M}}}_{7\ M's})}(10)\)
- Kilinemar, \(f_{\psi(\underbrace{M_{M_{\cdots_M}}}_{1001\ M's})}(10)\)
- Meginemar, \(f_{\psi(\underbrace{M_{M_{\cdots_M}}}_{10^6+1\ M's})}(10)\)
- Giginemar, \(f_{\psi(\underbrace{M_{M_{\cdots_M}}}_{10^9+1\ M's})}(10)\)
- Terinemar, \(f_{\psi(\underbrace{M_{M_{\cdots_M}}}_{10^{12}+1\ M's})}(10)\)
- Petinemar, \(f_{\psi(\underbrace{M_{M_{\cdots_M}}}_{10^{15}+1\ M's})}(10)\)
- Exinemar, \(f_{\psi(\underbrace{M_{M_{\cdots_M}}}_{10^{18}+1\ M's})}(10)\)
- Zettinemar, \(f_{\psi(\underbrace{M_{M_{\cdots_M}}}_{10^{21}+1\ M's})}(10)\)
- Yottinemar, \(f_{\psi(\underbrace{M_{M_{\cdots_M}}}_{10^{24}+1\ M's})}(10)\)
- Uninamus, \(f_{\psi(M(M(0;0);0))}(10)\)
- Zettinamus, \(f_{\psi(\underbrace{M(M(\cdots M(0;0)\cdots;0);0)}_{10^{21}+1\ M's})}(10)\)
- Yottinamus, \(f_{\psi(\underbrace{M(M(\cdots M(0;0)\cdots;0);0)}_{10^{24}+1\ M's})}(10)\)
- Kumakuma 3 variables ψ number, F10100(10100)
- グラハム数ver ε.0.1.0, G64(4)
- Tritar, Tar(3)
- Quadritar, Tar(4)
- Quintitar, Tar(5)
- Sextitar, Tar(6)
- Septitar, Tar(7)
- Octitar, Tar(8)
- Nonitar, Tar(9)
- Dekotar, Tar(10)
- Hektotar, Tar(100)
- Kilotar, Tar(1000)
- Megotar, Tar(106)
- Gigotar, Tar(109)
- Terotar, Tar(1012)
- Petotar, Tar(1015)
- Exotar, Tar(1018)
- Zettotar, Tar(1021)
- Yottotar, Tar(1024)
- Unintar, Tar(Dekotar) = Tar(Tar)
- Bintar, Tar(Tar(Dekotar)) = Tar(Tar(Tar))
- Trintar, Tar(Tar(Tar(Tar)))
- Quadrintar, Tar4(Tar)
- Quintintar, Tar5(Tar)
- Sextintar, Tar6(Tar)
- Septintar, Tar7(Tar)
- Octintar, Tar8(Tar)
- Nonintar, Tar9(Tar)
- Dekintar, Tar10(Tar)
- Hektintar, Tar100(Tar)
- Kilintar, Tar1000(Tar)
- Megintar, Tar106(Tar)
- Gigintar, Tar109(Tar)
- Terintar, Tar1012(Tar)
- Petintar, Tar1015(Tar)
- Exintar, Tar1018(Tar)
- Zettintar, Tar1021(Tar)
- Yottintar, Tar1024(Tar)
- Tarintar, TarTar(Tar)
- Loader's number (output of loader.c), \(D^5(99)\)
- Bashicu matrix number with respect to Bashicu matrix system version 2.3
- 6 (N primitive)
- Y sequence number, f2000(1)
Stronger set theory level
These numbers are based on functions which are not provably total in \(\mathrm{ZFC}\), but based on functions that are known to be provably total in stronger set theories.