November 2021

The arrow OCF has some issues, so I give a fixed version. Let \(B_0(\beta)=\{0,\omega_1,\omega_2,\ldots,\omega_\omega\}\). \(B_{n+1}(\beta)=\{x,x+y,xy,x\uparrow^zy:x,y,z\in B_n(\beta)\land z<\beta\}\). \(B(\beta)=\beta\cap\bigcup_{n<\omega} B_n(\beta)\). and \(C_0(\alpha,\beta)=\alpha+1\). \(C_{\epsilon+1}(\alpha,\beta)=\{x,x+y,xy,x\uparrow^zy:x,y\in C_\epsilon(\alpha,\beta)\land z\in B(\beta)\}\). If \(\epsilon\) is a limit, \(C_\epsilon(\alpha,\beta)=S\cup\operatorname{limits}(S)\), where \(S=\bigcup_{\zeta<\epsilon}C_\zeta(\alpha,\beta)\). Then \(\alpha\uparrow^0\beta=1\) and \(\alpha\uparrow^{1+\gamma}\beta=\sup(C_\gamma(\alpha,\beta))\).

Making an OCF is easy. The idea of pre-inaccessible OCFs (OCFs without inaccessibles) are easy to grasp. Let’s define a function \(\psi(\alpha)\) as the smallest ordinal you can’t name with 0, addition, and \(\psi\) to smaller arguments. The idea is when we run out of names, we just give a name to the next ordinal

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,