The arrow OCF

Not-so-recently the Knuth arrow notation was extended to infinite ordinals. The definition is here. It’s pretty straightforward, and the precise details are unimportant for this post.

Anyways, we have an aordinal function. (Conjecturally on par with the two-argument Veblen function.) So, we can pull a Solomon Feferman and define an OCF which extends it. Conjecturally, the strength of the OCF is on par with Buchholz’s and Feferman’s. Also I’m making this post partially because it’s quite different from other OCFs and might be of intrest.

First we define the sets \(C^\epsilon(\alpha,\beta)\):

• $C^0(\alpha,\beta)=\{\alpha,\omega_1,\omega_2,\ldots,\omega_\omega\}$.
• $C^{\epsilon+1}(\alpha,\beta) = C^\epsilon (\alpha,\beta)$
  $\begin{align*}\hspace{4.5em} & \cup \{\alpha+\beta:\alpha,\beta\in C^\epsilon (\alpha,\beta)\}\\ & \cup \{\alpha\beta:\alpha,\beta\in C^\epsilon (\alpha,\beta)\}\\ & \cup \{\alpha\uparrow^\beta\gamma: \alpha,\beta\in C^\epsilon (\alpha,\beta)\land\gamma<\alpha\} \end{align*}$
• If $\epsilon$ is a limit, $C^\epsilon(\alpha,\beta)=S\cup \operatorname{limits}(S)$, where $S=\bigcup_{\zeta<\epsilon}C^\zeta(\alpha,\beta)$

Then \(\alpha\uparrow^\beta 0=1\) and \(\alpha\uparrow^\beta(1+\gamma)=\sup C^\gamma(\alpha,\beta)\cap N\), where \(N\) is the smallest regular cardinal strictly greater than both \(\alpha\) and \(\beta\), or zero if \(\alpha=0\).

Properties

• If \(\beta\le\Gamma_0\) then probably \(\alpha\uparrow^\beta\gamma=\alpha\downarrow^\beta\gamma\).
• For all \(\Gamma_0\le\beta\le\Omega\), probably \(\alpha\uparrow^\beta\gamma=\alpha\downarrow^{\Gamma_0}\gamma\).
• \(\omega\uparrow^{\varepsilon_{\Omega_\omega+1}}\omega\) is probably the Takeuti-Feferman-Buchholz ordinal.