# Diagonalization

Diagonalization is a process that helps to directly compute values of hierarchies without having to go from the bottom. Each ordinal that is not a sucsessor has a fundemental sequence that helps. When we say some ordinal diagonalized to some finite number, we use: some ordinal[number] to express. You can replace any ordinal with the (whatever number you are diagonalizing to)th of the fundemental sequence.

# Sequences

The sequence for $$\omega$$ is $$\lbrace 1,2,\cdots \rbrace$$.

The sequence for $$\omega2$$ is $$\lbrace \omega+1,\omega+2,\cdots \rbrace$$.

The sequence for $$\omega3$$ is $$\lbrace \omega2+1,\omega2+2,\cdots \rbrace$$.

$$\cdots\cdots$$

The sequence for $$\omega^2$$ is $$\lbrace \omega,\omega2,\omega3,\cdots \rbrace$$.

From now on, we just replace a loose $$\omega$$ with the number we are diagonalizing to, and replace something like $$\omega3$$ with $$\omega2+\omega$$.

The sequence for $$\omega^{\omega}$$ is $$\lbrace \omega,\omega^2,\omega^3,\cdots \rbrace$$.

The sequence for $$\omega^{\omega^{\omega}}$$ is $$\lbrace \omega^{\omega},\omega^{\omega^2},\omega^{\omega^3},\cdots \rbrace$$

$$\cdots$$

The sequence for $$\varepsilon_0$$ is $$\lbrace 1,\omega,\omega^{\omega},\omega^{\omega^{\omega}},\omega^{\omega^{\omega^{\omega}}},\cdots \rbrace$$

The sequence for $$\varepsilon_1$$ is $$\lbrace \omega^{\varepsilon_0+1},\omega^{\omega^{\varepsilon_0+1}},\omega^{\omega^{\omega^{\varepsilon_0+1}}},\cdots \rbrace$$

The sequence for $$\varepsilon_2$$ is $$\lbrace \omega^{\varepsilon_1+1},\omega^{\omega^{\varepsilon_1+1}},\omega^{\omega^{\omega^{\varepsilon_1+1}}},\cdots \rbrace$$

$$\cdots\cdots$$

The sequence for $$\varepsilon_{\omega}$$ is $$\lbrace \varepsilon_1,\varepsilon_2,\varepsilon_3,\cdots \rbrace$$

We can even replace loose ordinals with the thing it is supposed to turn into if it is alone.

The sequence for $$\varepsilon_{\varepsilon_0}$$ is $$\lbrace \varepsilon_1,\varepsilon_{\omega},\varepsilon_{\omega^{\omega}},\cdots \rbrace$$

The sequence for $$\varepsilon_{\varepsilon_{\varepsilon_0}}$$ is $$\lbrace \varepsilon_{\varepsilon_1},\varepsilon_{\varepsilon_{\omega}},\varepsilon_{\varepsilon_{\omega^{\omega}}},\cdots \rbrace$$

$$\cdots$$

The sequence for $$\zeta_0$$ is $$\lbrace \varepsilon_0,\varepsilon_{\varepsilon_0},\varepsilon_{\varepsilon_{\varepsilon_0}},\varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_0}}},\cdots \rbrace$$

The sequence for $$\eta_0$$ is $$\lbrace \zeta_0,\zeta_{\zeta_0},\zeta_{\zeta_{\zeta_0}},\zeta_{\zeta_{\zeta_{\zeta_0}}},\cdots \rbrace$$

Now we introduce the Veblen function, which is defined as follows:

1. $$\varphi_0(0)=1$$

2. If $$\alpha$$ is a succsessor, then $$\varphi_\alpha(0)[n]=\varphi_{\alpha-1}^n(0)$$ and $$\varphi_\alpha(a)[n]=\varphi_{\alpha-1}^n(\varphi_\alpha(a-1)+1)$$.

3. If $$\beta$$ is a limit ordinal, then $$\varphi_\beta(0)[n]=\varphi_n(0)$$ and $$\varphi_\beta(a)[n]=\varphi_n(\varphi_\beta(a-1)+1)$$.

The sequence for $$\varphi_4(0)$$ is $$\lbrace \eta_0,\eta_{\eta_0},\eta_{\eta_{\eta_0}},\eta_{\eta_{\eta_{\eta_0}}},\cdots \rbrace$$

The sequence for $$\varphi_\omega(0)$$ is $$\lbrace \varepsilon_0,\zeta_0,\eta_0,\varphi_4(0),\cdots \rbrace$$

$$\cdots\cdots$$

The sequence for $$\Gamma_0=\varphi(1,0,0)$$ is $$\lbrace 1,\varphi_1(0),\varphi_{\varphi_1(0)}(0),\cdots \rbrace$$

Now we encounter large countable ordinals that can only be expressed with the Extended Veblen function.

The sequence for $$\Gamma_1=\varphi(1,0,1)$$ is $$\lbrace \varphi_{\Gamma_0+1}(0),\varphi_{\varphi_{\Gamma_0+1}(0)}(0),\cdots \rbrace$$

$$\cdots\cdots$$

The sequence for $$\Gamma_{\Gamma_{\Gamma_\ddots}}=\varphi(1,1,0)$$ is $$\lbrace stuck$$

Well, that's about it. The Wainer hierarchy only lasts up to the limit of the $$\Gamma$$ function.

## ψ function sequences

In Madore's ψ function, there is a thing called $$\Omega$$.

The sequence for $$\psi(\Omega^\omega+\Omega^3\omega2+\Omega)$$ is

$$\{\psi(0),\psi(\Omega^\omega+\Omega^3\omega2+\psi(0)),\psi(\Omega^\omega+\Omega^3\omega2+\psi(\Omega^\omega+\Omega^3\omega2+\psi(0))),\psi(\Omega^\omega+\Omega^3\omega2+\psi(\Omega^\omega+\Omega^3\omega2+\psi(\Omega^\omega+\Omega^3\omega2+\psi(0)))),\cdots\}$$

Well, it's like this.