# Difference between revisions of "Diagonalization"

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= Sequences = | = Sequences = | ||

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The sequence for \(\omega\) is \(\lbrace 1,2,\cdots \rbrace\). | The sequence for \(\omega\) is \(\lbrace 1,2,\cdots \rbrace\). | ||

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

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+ | == ψ function sequences == | ||

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+ | In [[Madore's ψ function]], there is a thing called \(\Omega\). | ||

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

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

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+ | Well, it's like this. |

## Latest revision as of 03:27, 15 April 2017

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.