Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.
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The Extended Veblen function is a function with more than 2 arguments to express ordinals from \(\Gamma_0\) to \(\psi(\Omega^{\Omega^\omega})\), the small Veblen ordinal.
\(\varphi(1,0,0)=\varphi_{\varphi_{\ddots_{\varphi_0(0)}}(0)}(0)
\
\varphi(1,0,1)=\varphi_{\varphi_{\ddots_{\varphi(1,0,0)+1}}(0)}(0)
\
\varphi(1,0,n+1)=\varphi_{\varphi_{\ddots_{\varphi(1,0,n)+1}}(0)}(0)
\\ \varphi(1,1,0)=\varphi(1,0,\varphi(1,0,\varphi(1,0,\cdots)))
\
\varphi(1,1,1)=\varphi_{\varphi_{\ddots_{\varphi(1,1,0)+1}}(0)}(0)
\\ \varphi(1,2,0)=\varphi(1,1,\varphi(1,1,\varphi(1,1,\cdots)))
\\ \varphi(1,n+1,0)=\varphi(1,n,\varphi(1,n,\varphi(1,n,\cdots)))
\\ \varphi(2,0,0)=\varphi(1,\varphi(1,\varphi(1,\cdots,0),0),0)
\
\varphi(2,0,1)=\varphi_{\varphi_{\ddots_{\varphi(2,0,0)+1}}(0)}(0)
\\ \varphi(2,1,0)=\varphi(2,0,\varphi(2,0,\varphi(2,0,\cdots)))
\\ \varphi(3,0,0)=\varphi(2,\varphi(2,\varphi(2,\cdots,0),0),0)
\\ \varphi(n+1,0,0)=\varphi(n,\varphi(n,\varphi(n,\cdots,0),0),0)
\
\varphi(1,0,0,0)=\varphi(\varphi(\varphi(\cdots,0,0),0,0),0,0)\)