Madore's function

Madore's \(\psi\) function is an ordinal collapsing function introduced by David Madore (under "Gro-Tsen" on Wikipedia).

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.