Madore's $$\psi$$ function is an ordinal collapsing function introduced by David Madore.

## Definition

Madore's $$\psi$$ function is defined as follows:

Let $$\omega$$ be the first transfinite ordinal and $$\Omega$$ be the first uncountable ordinal. Then,

$$C_0(\alpha) = \{0, 1, \omega, \Omega\}$$

$$C_{n+1}(\alpha) = \{\gamma + \delta, \gamma\delta, \gamma^{\delta}, \psi(\eta) | \gamma, \delta, \eta \in C_n (\alpha); \eta < \alpha\}$$

$$C(\alpha) = \bigcup_{n < \omega} C_n (\alpha)$$

$$\psi(\alpha) = \min\{\beta < \Omega|\beta \notin C(\alpha)\}$$

In other words $$\psi(\alpha)$$ is the least ordinal number less than $$\Omega$$ which cannot be generated from ordinals $$0, 1, \omega, \Omega$$ by applying of addition, multiplication, exponentiation and the function $$\psi(\eta)$$ with $$\eta < \alpha$$.

## Fundamental sequences

Now we assign a fundamental sequence for each limit ordinal below the Bachmann-Howard ordinal. The fundamental sequence for an ordinal number $$\alpha$$ with cofinality $$\text{cof}(\alpha)=\beta$$ is a strictly increasing sequence $$(\alpha[\eta])_{\eta<\beta}$$ with length $$\beta$$ and with limit $$\alpha$$, where $$\alpha[\eta]$$ is the $$\eta$$-th element of this sequence. If $$\alpha$$ is a countable limit ordinal (i.e. $$\alpha$$ is a limit ordinal less than $$\Omega$$) then $$\text{cof}(\alpha)=\omega$$. The first uncountable ordinal $$\Omega$$ is the least ordinal whose cofinality greater than $$\omega$$ since $$\text{cof}(\Omega)=\Omega$$.

At first we define the normal form for ordinals

$$\alpha=_{NF}\alpha_1+\alpha_2+\cdots+\alpha_n$$ iff $$\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n$$ and $$\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_n$$

$$\alpha=_{NF}\omega^\beta$$ iff $$\alpha=\omega^\beta$$ and $$\beta<\alpha$$

$$\alpha=_{NF}\Omega^\beta\gamma$$ iff $$\alpha=\Omega^\beta\gamma$$ and $$\gamma<\Omega$$

$$\alpha=_{NF}\psi(\beta)$$ iff $$\alpha=\psi(\beta)$$ and $$\beta\in C(\beta)$$

For limit ordinals written in normal form we assign the fundamental sequences as follows:

1) if $$\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n$$ then $$\text{cof} (\alpha)= \text{cof} (\alpha_n)$$ and $$\alpha[\eta]=\alpha_1+\alpha_2+\cdots+(\alpha_n[\eta])$$

2) if $$\alpha=\omega^\beta$$ and $$\beta$$ is a countable limit ordinal then $$\alpha[n]=\omega^{\beta[n]}$$

3) if $$\alpha=\omega^\beta$$ and $$\beta=\gamma+1$$ then $$\alpha[n]=\omega^\gamma n$$

4) if $$\alpha=\psi(0)$$ then $$\alpha[0]=1$$ and $$\alpha[n+1]=\omega^{\alpha[n]}$$

5) if $$\alpha=\psi(\beta+1)$$ then $$\alpha[0]=\psi(\beta)+1$$ and $$\alpha[n+1]=\omega^{\alpha[n]}$$

6) if $$\alpha=\Omega^{\beta}\gamma$$ and $$\text{cof} (\gamma)=\omega$$ then $$\text{cof} (\alpha)= \omega$$ and $$\alpha[\eta]=\Omega^{\beta}(\gamma[\eta])$$

7) if $$\alpha=\Omega^{\beta+1}(\gamma+1)$$ then $$\text{cof} (\alpha)=\Omega$$ and $$\alpha[\eta]=\Omega^{\beta+1}\gamma+\Omega^\beta\eta$$

8) if $$\alpha=\Omega^\beta(\gamma+1)$$ and $$\text{cof}(\beta)\geq\omega$$ then $$\text{cof}(\alpha)= \text{cof}(\beta)$$ and $$\alpha[\eta]=\Omega^\beta\gamma+\Omega^{\beta[\eta]}$$

9) if $$\alpha=\varepsilon_{\Omega+1}$$ then $$\text{cof} (\alpha)=\omega$$ and $$\alpha[0]=1$$ and $$\alpha[n+1]=\Omega^{\alpha[n]}$$

10) if $$\alpha=\psi(\beta)$$ and $$\text{cof}(\beta)=\omega$$ then $$\text{cof} (\alpha)=\omega$$ and $$\alpha[n]=\psi(\beta[n])$$

11) if $$\alpha=\psi(\beta)$$ and $$\text{cof}(\beta)=\Omega$$ then $$\text{cof} (\alpha)=\omega$$ and $$\alpha[0]=1$$ and $$\alpha[n+1]=\psi(\beta[\alpha[n]])$$

For example, for ordinal $$\psi(\Omega^{\Omega^2+\Omega3})$$ we have the following fundamental sequence (using rules 1, 7, 8, 10)

$$\psi(\Omega^{\Omega^2+\Omega3})[0]=1$$

$$\psi(\Omega^{\Omega^2+\Omega3})[1]=\psi(\Omega^{\Omega^2+\Omega2+1})$$

$$\psi(\Omega^{\Omega^2+\Omega3})[2]=\psi(\Omega^{\Omega^2+\Omega2+\psi(\Omega^{\Omega^2+\Omega2+1})})$$

and so on.

Assignation of fundamental sequences is vital for definition of the fast-growing hierarchy, slow-growing hierarchy and Hardy hierarchy.

## Values

\begin{eqnarray*} \psi(0) &=& \varepsilon_0 \\ \psi(1) &=& \varepsilon_1 \\ \psi(2) &=& \varepsilon_2 \\ \psi(n) &=& \varepsilon_n \\ \psi(\zeta_0) &=& \zeta_0 \\ \psi(\zeta_0+1) &=& \zeta_0 \end{eqnarray*}

That seems strange. Shouldn't $$\psi(\zeta_0+1)=\varepsilon_{\zeta_0+1}$$? No. Look at $$C(\zeta_0+1)$$. It has all the things from $$C(\zeta_0)$$ and $$+\times\text{^}$$ $$\psi$$ of it. But in order to get $$\zeta_0$$, you have to have $$\zeta_0$$in your list. So you are never going to have $$\zeta_0$$ in your list. Or are you $$\cdots\cdots$$

\begin{eqnarray*} \psi(\Omega) &=& \zeta_0 \\ \psi(\Omega+1) &=& \varepsilon_{\zeta_0+1} \\ \psi(\Omega+n) &=& \varepsilon_{\zeta_0+n} \\ \psi(\Omega+\zeta_1) &=& \varepsilon_{\zeta_0+\zeta_1} &=& \zeta_1 \\ \psi(\Omega+\zeta_1+1) &=& \zeta_1 \end{eqnarray*}

We see that this $$\psi$$ function got stuck at $$\zeta_0$$. But when we arrive at $$\Omega+1$$, then we are allowed to use the $$\Omega$$ in $$C(\Omega)$$ to create bigger ordinals. We then arrive at $$\psi(\Omega+\zeta_1)$$ which is signaling that this function is stuck again until $$\Omega2$$.

\begin{eqnarray*} \psi(\Omega2) &=& \zeta_1 \\ \psi(\Omega2+1) &=& \varepsilon_{\zeta_1+1} \\ \psi(\Omega2+n) &=& \varepsilon_{\zeta_1+n} \\ \psi(\Omega2+\zeta_2) &=& \varepsilon_{\zeta_1+\zeta_2} &=& \zeta_2 \\ \psi(\Omega2+\zeta_2+1) &=& \zeta_2 \end{eqnarray*}

Stuck again. We are going to fast forward now.

\begin{eqnarray*} \psi(\Omega3) &=& \zeta_2 \ \psi(\Omega n) &=& \zeta_{n-1} \\ \psi(\Omega \eta_0) &=& \eta_0 \\ \psi(\Omega \eta_0+1) &=& \eta_0 \end{eqnarray*}

This function is stuck again until $$\psi(\Omega^2)$$ because $$C(\Omega \eta_0)$$ contains all countable ordinals up to but not incliding $$\eta_0$$, but there is no ordinal called $$\Omega \eta_0$$ in $$C(\Omega \eta_0)$$. So if you want to have $$\eta_0$$ in your list, you need $$\Omega \eta_0$$, and therefore, $$\eta_0$$.

\begin{eqnarray*} \psi(\Omega^2) &=& \eta_0 \\ \psi(\Omega^2+1) &=& \varepsilon_{\eta_0+1} \\ \psi(\Omega^2+n) &=& \varepsilon_{\eta_0+n} \\ \psi(\Omega^2+\Omega) &=& \zeta_{\eta_0+1} \\ \psi(\Omega^2+\Omega2) &=& \zeta_{\eta_0+2} \\ \psi(\Omega^2+\Omega n) &=& \zeta_{\eta_0+n} \\ \psi(\Omega^2+\Omega\eta_1) &=& \eta_1 \\ \psi(\Omega^2 2) &=& \eta_1 \\ \psi(\Omega^2 n) &=& \eta_{n-1} \\ \psi(\Omega^2 \varphi_4(0)) &=& \varphi_4(0) \\ \psi(\Omega^3) &=& \varphi_4(0) \end{eqnarray*}

Now we are introducing the Veblen function, which is explained in Diagonalization, and also the Extended Veblen function.

\begin{eqnarray*} \psi(\Omega^3 \varphi_5(0)) &=& \varphi_5(0) \\ \psi(\Omega^4) &=& \varphi_5(0) \\ \psi(\Omega^n) &=& \varphi_{1+n}(0) \\ \psi(\Omega^{\Gamma_0}) &=& \Gamma_0 \\ \psi(\Omega^\Omega) &=& \Gamma_0 \\ \psi(\Omega^\Omega+1) &=& \varepsilon_{\Gamma_0+1} \\ \psi(\Omega^\Omega+\Omega) &=& \zeta_{\Gamma_0+1} \\ \psi(\Omega^\Omega+\Omega^n) &=& \varphi_{1+n}(\Gamma_0+1) \\ \psi(\Omega^\Omega+\Omega^{\Gamma_1}) &=& \Gamma_1 \\ \psi(\Omega^\Omega2) &=& \Gamma_1 \\ \psi(\Omega^\Omega n) &=& \Gamma_{n-1} \\ \psi(\Omega^{\Omega+1}) &=& \varphi(1,1,0) \\ \psi(\Omega^{\Omega+1}2) &=& \varphi(1,1,1) \\ \psi(\Omega^{\Omega+2}) &=& \varphi(1,2,0) \\ \psi(\Omega^{\Omega2}) &=& \varphi(2,0,0) \\ \psi(\Omega^{\Omega2+1}) &=& \varphi(2,1,0) \\ \psi(\Omega^{\Omega3}) &=& \varphi(3,0,0) \\ \psi(\Omega^{\Omega n}) &=& \varphi(n,0,0) \\ \psi(\Omega^{\Omega^2}) &=& \varphi(1,0,0,0) \\ \psi(\Omega^{\Omega^3}) &=& \varphi(1,0,0,0,0) \end{eqnarray*}

## Small Veblen ordinal

The small veblen ordinal is defined as $$\psi(\Omega^{\Omega^\omega}) = \varphi(1,\underbrace{0,\cdots,0}_\omega)$$. But it's only small compared to...

## Large Veblen ordinal

The large veblen ordinal is defined as $$\psi(\Omega^{\Omega^\Omega}) = \psi(\Omega^{\Omega^{\psi(\Omega^{\Omega^{\psi (\cdots)}})}}) = \varphi(1,\underbrace{0,\cdots,0}_{\varphi(1,\underbrace{0,\cdots,0}_{\varphi(1,\underbrace{0,\cdots,0}_{\varphi(\cdots)})})})$$. But even that's nothing compared to...

## Bachmann-Howard ordinal

$$BHO = \psi(\varepsilon_{\Omega+1}) = \psi(\underbrace{\Omega^{\Omega^{\cdots^\Omega}}}_\omega)$$

Madore's $$\psi$$ function is one of the simpliest collapsing functions. There are much stronger functions of such kind: