User:Edwin Shade/Sandbox

\(\omega\uparrow\uparrow\omega=\varepsilon_0\)

\(\omega\uparrow\uparrow(\omega+1)=\omega^{\varepsilon_0+1}\)

\(\omega\uparrow\uparrow(\omega+2)=\omega^{\omega^{\varepsilon_0+1}}\)

\(\omega\uparrow\uparrow(\omega+3)=\omega^{\omega^{\omega^{\varepsilon_0+1}}}\)

\(\omega\uparrow\uparrow(\omega+4)=\omega^{\omega^{\omega^{\omega^{\varepsilon_0+1}}}}\)

\(\omega\uparrow\uparrow\omega2=\varepsilon_1\)

\(\omega\uparrow\uparrow(\omega2+1)=\omega^{\varepsilon_1+1}\)

\(\omega\uparrow\uparrow(\omega2+2)=\omega^{\omega^{\varepsilon_1+1}}\)

\(\omega\uparrow\uparrow(\omega2+3)=\omega^{\omega^{\omega^{\varepsilon_1+1}}}\)

\(\omega\uparrow\uparrow\omega3=\varepsilon_2\)

\(\omega\uparrow\uparrow\omega4=\varepsilon_3\)

\(\omega\uparrow\uparrow\omega^2=\varepsilon_{\omega}\)

\(\omega\uparrow\uparrow(\omega^2+1)=\omega^{\varepsilon_{\omega}+1}\)

\(\omega\uparrow\uparrow(\omega^2+\omega)=\varepsilon_{\omega+1}\)

\(\omega\uparrow\uparrow(\omega^2+\omega+1)=\omega^{\varepsilon_{\omega+1}+1}\)

\(\omega\uparrow\uparrow(\omega^2+\omega2)=\varepsilon_{\omega+2}\)

\(\omega\uparrow\uparrow(\omega^2+\omega3)=\varepsilon_{\omega+3}\)

\(\omega\uparrow\uparrow(\omega^2+\omega4)=\varepsilon_{\omega+4}\)

\(\omega\uparrow\uparrow(\omega^2+\omega5)=\varepsilon_{\omega+5}\)

\(\omega\uparrow\uparrow\omega^22=\varepsilon_{\omega2}\)

\(\omega\uparrow\uparrow(\omega^22+\omega)=\varepsilon_{\omega2+1}\)

\(\omega\uparrow\uparrow(\omega^22+\omega2)=\varepsilon_{\omega2+2}\)

\(\omega\uparrow\uparrow(\omega^22+\omega3)=\varepsilon_{\omega2+3}\)

\(\omega\uparrow\uparrow(\omega^22+\omega3+1)=\omega^{\varepsilon_{\omega2+3}+1}\)

\(\omega\uparrow\uparrow(\omega^22+\omega4)=\varepsilon_{\omega2+4}\)

\(\omega\uparrow\uparrow(\omega^23)=\varepsilon_{\omega3}\)

\(\omega\uparrow\uparrow(\omega^24)=\varepsilon_{\omega4}\)

\(\omega\uparrow\uparrow(\omega^3)=\varepsilon_{\omega^2}\)

\(\omega\uparrow\uparrow(\omega\uparrow\uparrow\omega)=\varepsilon_{\varepsilon_0}\)

\(\omega\uparrow\uparrow\uparrow\omega=\zeta_0\)

Hypothesized Values
\(\omega\uparrow\uparrow\uparrow\uparrow\omega=\eta_0\)

\(\omega\uparrow^{\omega}\omega=\varphi(\omega,0)\)

\((\xi\mapsto\omega\uparrow^{\xi}\omega)=\Gamma_0\)

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\(\color{#006600}{\Rule{6ex}{44px}{0px}}\raise0pt {\moveleft13pt {\color{#FFFF00}{\Rule{3ex}{22px}{0px}}}}\)

\(\color{#006600}{\Rule{6ex}{44px}{0px}}\raise0pt {\moveleft26pt {\color{#FFFF00}{\Rule{3ex}{22px}{0px}}}}\)

\(C(0)=\{0,\Omega\}\)

\(C(\alpha)=\{\gamma\uparrow^{\eta}\delta |(\gamma\wedge\eta\wedge\delta)\in C(\beta),\beta<\alpha\}\)

\(\mathfrak{H}(\alpha)=\text{The first ordinal not in }C(\alpha)\)

Art Grids In MathJax
1 by 1 Grid

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4 by 4 Grid

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8 by 8 Grid

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