User blog:Simply Beautiful Art/Intuition behind the Mahlo OCF

The Mahlo OCF, as it would turn out, is not too bad to try and understand. The idea lies behind what properties of a regular ordinal we want from \(\Omega\) for Madore's OCF and the like. \(\Omega\) satisfies the following property:

\(\forall f:\Omega\mapsto\Omega,\exists\alpha\in\Omega,\forall\eta\in\alpha,f(\eta)\in\alpha\)

which, written out in English, translates to something like:

Every function \(f\) mapping ordinals less than \(\Omega\) to ordinals less than \(\Omega\) has a point less than \(\Omega\) where \(f\) will get "stuck".

It should be obvious how this property plays into an ordinal collapsing function. In fact, the above defines the uncountable regular ordinals, which are used in extensions to the basic ordinal collapsing functions.

Let \(\Omega^\star\) denote the class of uncountable regular ordinals.

The Mahlo ordinals can then be defined as:

\(\forall f:M\mapsto M,\exists\alpha\in M\cap\Omega^\star,\forall\eta\in\alpha,f(\eta)\in\alpha\)

which, written out in English, translates to something like:

Every function \(f\) mapping ordinals less than \(M\) to ordinals less than \(M\) has a point less than \(M\) that is regular and \(f\) gets "stuck".

The change is fairly minor, but it plays a major role in an OCF which outputs regular ordinals. For example, we can define:

\(\displaystyle B(\alpha,\xi)_0=\xi\cup\{0,M\}\)

\(\displaystyle B(\alpha,\xi)_{n+1}=\{\gamma+\delta,\omega^\gamma~|~\gamma,\delta\in B(\alpha,\xi)_n\}\)

\(\displaystyle\hphantom{B(\alpha,\xi)_{n+1}={}}{}\cup\{\chi(\eta)~|~\eta\in\alpha\cap B(\alpha,\xi)_n\}\)

\(\displaystyle B(\alpha,\xi)=\bigcup_{n\in\mathbb N}B(\alpha,\xi)_n\)

\(\displaystyle\chi(\alpha)=\min\{\xi\in\Omega^\star~|~\xi=M\cap B(\alpha,\xi)\}\)

Note the simplicity in definition and the overall familiarity to smaller ordinal collapsing functions.

Some basic values of this include:

\(\chi(\alpha)\) is the \(\alpha\)th regular ordinal, up to \(\alpha=\chi(M)\).

\(\chi(M)\) is the least regular ordinal that \(\chi\) gets stuck at, which is the first inaccessible. Note that it doesn't get stuck at \(\Omega_{\text{fp}}=\sup\{\Omega,\Omega_\Omega,\Omega_{\Omega_\Omega},\dots\}\) since this point is not regular.

From there \(\chi(M+1)\) would simply be the regular ordinal after \(\chi(M)\) and so on.

Then \(\chi(M\cdot2)\) would be the second inaccessible, and so on.

One can then define an OCF using this, such as:

\(\displaystyle C(\alpha,\xi)_0=\xi\cup\{0,M\}\)

\(\displaystyle C(\alpha,\xi)_{n+1}=\{\gamma+\delta,\omega^\gamma,\chi(\gamma)~|~\gamma,\delta\in C(\alpha,\xi)_n\}\)

\(\displaystyle\hphantom{C(\alpha,\xi)_{n+1}={}}{}\cup\{\psi_\pi(\eta)~|~\pi\in\Omega^\star\cap C(\alpha,\xi)_n\land\eta\in\alpha\cap C(\alpha,\xi)_n\}\)

\(\displaystyle C(\alpha,\xi)=\bigcup_{n\in\mathbb N}C(\alpha,\xi)_n\)

\(\displaystyle\psi_\pi(\alpha)=\min\{\xi~|~\xi=\pi\cap C(\alpha,\xi)\},~\pi\in\Omega^\star\)

which has things such as \(\psi_{\chi(M)}(0)=\Omega_{\text{fp}}\). In general, \(\psi_{\chi(\alpha)}\) produces epsilon numbers less than \(\chi(\alpha)\) if \(\operatorname{cof}(\alpha)<M\) and \(\alpha\in B(\alpha,\chi(\alpha))\) (to avoid things such as \(\chi(\chi(M))\)), while \(\psi_{\chi(\alpha)}\) produces limits of expressions given by iterating over an \(M\) in \(\alpha\) for \(\operatorname{cof}(\alpha)=M\) and \(\alpha\in B(\alpha,\chi(\alpha))\), e.g. \(\psi_{\chi(M)}(0)=\sup\{0,\chi(0),\chi(\chi(0)),\dots\}\).