# Extended Veblen function

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.

## Definition

\(\varphi(1,0,0)=\varphi_{\varphi_{\ddots_{\varphi_0(0)}}(0)}(0) \\ \varphi(1,0,1)=\varphi_{\varphi_{\ddots_{\varphi(1,0,0)+1}}(0)}(0) \\ \varphi(1,0,n+1)=\varphi_{\varphi_{\ddots_{\varphi(1,0,n)+1}}(0)}(0) \\ \varphi(1,1,0)=\varphi(1,0,\varphi(1,0,\varphi(1,0,\cdots))) \\ \varphi(1,1,1)=\varphi_{\varphi_{\ddots_{\varphi(1,1,0)+1}}(0)}(0) \\ \varphi(1,2,0)=\varphi(1,1,\varphi(1,1,\varphi(1,1,\cdots))) \\ \varphi(1,n+1,0)=\varphi(1,n,\varphi(1,n,\varphi(1,n,\cdots))) \\ \varphi(2,0,0)=\varphi(1,\varphi(1,\varphi(1,\cdots,0),0),0) \\ \varphi(2,0,1)=\varphi_{\varphi_{\ddots_{\varphi(2,0,0)+1}}(0)}(0) \\ \varphi(2,1,0)=\varphi(2,0,\varphi(2,0,\varphi(2,0,\cdots))) \\ \varphi(3,0,0)=\varphi(2,\varphi(2,\varphi(2,\cdots,0),0),0) \\ \varphi(n+1,0,0)=\varphi(n,\varphi(n,\varphi(n,\cdots,0),0),0) \\ \varphi(1,0,0,0)=\varphi(\varphi(\varphi(\cdots,0,0),0,0),0,0)\)