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${\displaystyle ()=1}$

${\displaystyle ()()=2}$

${\displaystyle ()()()=3}$

${\displaystyle (())=\omega }$

${\displaystyle (())()=\omega +1}$

${\displaystyle (())()()=\omega +2}$

${\displaystyle (())()()()=\omega +3}$

${\displaystyle (())(())=\omega 2}$

${\displaystyle (())(())(())=\omega 3}$

${\displaystyle (()())=\omega ^{2}}$

${\displaystyle (()()())=\omega ^{3}}$

${\displaystyle ((()))=\omega ^{\omega }}$

${\displaystyle ((()()))=\omega ^{\omega ^{2}}}$

${\displaystyle ((()()()))=\omega ^{\omega ^{3}}}$

${\displaystyle (((())))=\omega ^{\omega ^{\omega }}}$

${\displaystyle ([])=\varepsilon _{0}}$

${\displaystyle ([])([])=\varepsilon _{0}2}$

${\displaystyle ([]())=\varepsilon _{0}\omega }$

${\displaystyle ([]([]))=\varepsilon _{0}^{2}}$

${\displaystyle ([]([])([]))=\varepsilon _{0}^{3}}$

${\displaystyle ([]([]()))=\varepsilon _{0}^{\omega }}$

${\displaystyle ([]([]([])))=\varepsilon _{0}^{\varepsilon _{0}}}$

${\displaystyle ([]([]([]())))=\varepsilon _{0}^{\varepsilon _{0}^{\omega }}}$

${\displaystyle ([]([]([]([]))))=\varepsilon _{0}^{\varepsilon _{0}^{\varepsilon _{0}}}}$

${\displaystyle ([][])=\varepsilon _{1}}$

${\displaystyle ([][][])=\varepsilon _{2}}$

${\displaystyle ([()])=\varepsilon _{\omega }}$

${\displaystyle ([([])])=\varepsilon _{\varepsilon _{0}}}$

${\displaystyle ([([([])])])=\varepsilon _{\varepsilon _{\varepsilon _{0}}}}$

${\displaystyle ([[]])=\zeta _{0}}$

${\displaystyle ([[]][])=\varepsilon _{\zeta _{0}+1}}$

${\displaystyle ([[]][([[]])])=\varepsilon _{\zeta _{0}2}}$

${\displaystyle ([[]][([[]][])])=\varepsilon _{\varepsilon _{\zeta _{0}+1}}}$

${\displaystyle ([[]][[]])=\zeta _{1}}$

${\displaystyle ([[]()])=\zeta _{\omega }}$

${\displaystyle ([[]([[]])])=\zeta _{\zeta _{0}}}$

${\displaystyle ([[][]])=\eta _{0}}$

${\displaystyle ([[][][]])=\varphi (4,0)}$

${\displaystyle ([[()]])=\varphi (\omega ,0)}$

${\displaystyle ([[([])]])=\varphi (\varepsilon _{0},0)}$

${\displaystyle ([[([[()]])]])=\varphi (\varphi (\omega ,0),0)}$

${\displaystyle ([[([[([])]])]])=\varphi (\varphi (\varepsilon _{0},0),0)}$

${\displaystyle ([[[]]])=\Gamma _{0}}$

${\displaystyle ([[[]]][[([[[]]])]])=\varphi (\Gamma _{0},1)}$

${\displaystyle ([[[]]][[[]]])=\Gamma _{1}}$

${\displaystyle ([[[]]()])=\Gamma _{\omega }}$

${\displaystyle ([[[]][]])=\varphi (1,1,0)}$

${\displaystyle ([[[]][][]])=\varphi (1,2,0)}$

${\displaystyle ([[[]][()]])=\varphi (1,\omega ,0)}$

${\displaystyle ([[[]][[]]])=\varphi (2,0,0)}$

${\displaystyle ([[[]][[]][[]]])=\varphi (3,0,0)}$

${\displaystyle ([[[]()]])=\varphi (\omega ,0,0)}$

${\displaystyle ([[[][]]])=\varphi (1,0,0,0)}$

${\displaystyle ([[[][][]]])=\varphi (1,0,0,0,0)}$

${\displaystyle ([[[()]]])=\psi (\Omega ^{\Omega ^{\omega }})}$

${\displaystyle ([[[([[[]]])]]])=\psi (\Omega ^{\Omega ^{\psi (\Omega ^{\Omega ^{\omega }})}})}$

${\displaystyle ([[[[]]]])=\psi (\Omega ^{\Omega ^{\Omega }})}$

${\displaystyle ([\{\}])=\psi (\Omega _{2})}$

${\displaystyle ([\{\}][[\{\}]])=\psi (\Omega _{2}+\varepsilon _{\Omega +1})}$

${\displaystyle ([\{\}][\{\}])=\psi (\Omega _{2}2)}$

${\displaystyle ([\{\}][\{\}][\{\}])=\psi (\Omega _{2}3)}$

${\displaystyle ([\{\}()])=\psi (\Omega _{2}\omega )}$

${\displaystyle ([\{\}[]])=\psi (\Omega _{2}\Omega )}$

${\displaystyle ([\{\}[[\{\}]]])=\psi (\Omega _{2}\varepsilon _{\Omega +1})}$

${\displaystyle ([\{\}[\{\}]])=\psi (\Omega _{2}^{2})}$

${\displaystyle ([\{\}[\{\}][\{\}]])=\psi (\Omega _{2}^{3})}$

${\displaystyle ([\{\}[\{\}()]])=\psi (\Omega _{2}^{\omega })}$

${\displaystyle ([\{\}[\{\}[]]])=\psi (\Omega _{2}^{\Omega })}$

${\displaystyle ([\{\}[\{\}[\{\}]]])=\psi (\Omega _{2}^{\Omega _{2}})}$

${\displaystyle ([\{\}[\{\}[\{\}][\{\}]]])=\psi (\Omega _{2}^{\Omega _{2}^{2}})}$

${\displaystyle ([\{\}[\{\}[\{\}()]]])=\psi (\Omega _{2}^{\Omega _{2}^{\omega }})}$

${\displaystyle ([\{\}[\{\}[\{\}[]]]])=\psi (\Omega _{2}^{\Omega _{2}^{\Omega }})}$

${\displaystyle ([\{\}[\{\}[\{\}[\{\}]]]])=\psi (\Omega _{2}^{\Omega _{2}^{\Omega _{2}}})}$

${\displaystyle ([\{\}\{\}])=\psi (\Omega _{3})}$

${\displaystyle ([\{\}\{\}\{\}])=\psi (\Omega _{4})}$

${\displaystyle ([\{()\}])=\psi (\Omega _{\omega })}$

${\displaystyle ([\{([\{()\}])\}])=\psi (\Omega _{\psi (\Omega _{\omega })})}$

${\displaystyle ([\{[]\}])=\psi (\Omega _{\Omega })}$

${\displaystyle ([\{[\{\}]\}])=\psi (\Omega _{\Omega _{2}})}$

${\displaystyle ([\{[\{()\}]\}])=\psi (\Omega _{\Omega _{\omega }})}$

${\displaystyle ([\{[\{[]\}]\}])=\psi (\Omega _{\Omega _{\Omega }})}$

${\displaystyle ([\{\{\}\}])=\psi (\Omega _{\Omega _{\Omega _{\ldots }}})}$

${\displaystyle [\{[\{\{\}\}\{\}]\}]=I}$

${\displaystyle [\{[\{\{\}\}\{\}][\{\{\}\}\{\}]\}]=I(2,0)}$

${\displaystyle [\{[\{\{\}\}\{\}[\{\{\}\}\{\}]]\}]=M}$

${\displaystyle [\{[\{\{\}\}\{\}[\{\{\}\}\{\}][\{\{\}\}\{\}]]\}]=\Xi (3,0)}$

${\displaystyle [\{[\{\{\}\}\{\}[\{\{\}\}\{\}()]]\}]=\Xi (\omega ,0)}$

${\displaystyle [\{[\{\{\}\}\{\}[\{\{\}\}\{\}()][\{\{\}\}\{\}]]\}]}$ is ${\displaystyle n}$-Mahlo for all finite ${\displaystyle n}$

${\displaystyle [\{[\{\{\}\}\{\}[\{\{\}\}\{\}[\{\{\}\}]]]\}]}$ is larger than the least ${\displaystyle \alpha }$-Mahlo cardinal for all ${\displaystyle \alpha <[\{[\{\{\}\}\{\}[\{\{\}\}\{\}[\{\{\}\}]]]\}]}$

${\displaystyle [\{[\{\{\}\}\{\}[\{\{\}\}\{\}[\{\{\}\}]][\{\{\}\}\{\}]]\}]}$ is ${\displaystyle \alpha }$-Mahlo for all ${\displaystyle \alpha <[\{[\{\{\}\}\{\}[\{\{\}\}\{\}[\{\{\}\}]][\{\{\}\}\{\}]]\}]}$

${\displaystyle [\{[\{\{\}\}\{\}[\{\{\}\}\{\}[\{\{\}\}\{\}]]]\}]}$