Diagonalization

• Sequences
  • ψ function sequences

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.