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Smaller 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.

\(f_{\psi_0(\varepsilon_{\Omega+1})}^2(10)\) ~ \(f_{\psi_0(\Omega_\omega)}^2(10)\)

\(f_{\psi_0(\Omega_\omega)}^2(10)\) ~ \(f_{\psi_0(\Phi_1(0))+1}(10)\)

  • 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 \(\mathrm{ZFC}\).

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.

Larger numbers

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