User blog:Syst3ms/A sketch for an — actually — formal definition of UNOCF

'''Disclaimer : as the title implies, this is only a sketch. It is subject to changes, and I'm also open for suggestions to push it further.'''

So, recently I made a """formal""" ruleset for UNOCF, which was a complete failure. After some exchange over discord, Nish and I (especially Nish though) have come up with a definition that has the potential of being actually formal, even though it's lacking some things right now — more on that later.

The definition uses a "construction" type of function, like in most OCFs.

If this were to be fully, formally well-defined, it would work up to \(\psi(M^{M^\omega})\)

First, we need some groundwork :

\(\text{cof}(0) = 0 \\ \text{cof}(\alpha+1) = 0 \\ \text{cof}(\omega) = \omega \\ \text{cof}(\alpha+\beta) = \text{cof}(\beta) \\ \text{cof}(\alpha\beta) = \text{cof}(\beta) \\ \text{cof}(\alpha^\beta) = \text{cof}(\beta) \\ \forall i(\text{cof}(v_i)=0) : \text{cof}(C(v_n,\ldots,v_1,v_0)) = C(v_n,\ldots,v_1,v_0) \\ \exists k(k=\min\{i|\text{cof}(v_i) > 0\})  : \text{cof}(C(v_n,\ldots,v_1,v_0)) = v_k\)

\(\#\) is any succession of ordinals separated by commas.

\(\#_0\) is a succession of zeroes separated by commas

\(\#_1\) is a succession of ordinals \(>0\) separated by commas.

\(C(\alpha) = \Omega_{1+\alpha} \\ C(\#_0,\#) = C(\#)\)

Simply put, the following \(D[\kappa]\) function maps a regular cardinal to the class of all ordinals it diagonalizes over. So, for example, \(D[I]\) is the class of all ordinals of the form \(\Omega_\alpha\) with \(\alpha < I\).

\(D[C(\#,\mu+1,0,\#_0,\nu)] = \{\alpha|\exists\beta\forall\gamma((\forall\delta<\nu(\gamma>C(\#,\mu+1,0,\#_0,\delta)))\\ \Rightarrow\alpha=C(\#,\mu,\beta,\#_0,\gamma)\} \\ D[C(\#_1,\mu+1,\nu)] = \{\alpha|\exists\beta\forall\gamma((\forall\delta<\nu(\gamma>C(\#_1,\mu+1,\delta)))\\ \Rightarrow\alpha=C(\#_1,\mu,\gamma)\}\)

Now we're ready to actually define the \(\psi\) function :

\(D_0(\beta,\alpha) = \beta\cup\{0\} \\ D_{n+1}(\beta,\alpha) = D_n(\beta,\alpha) \cup\{\gamma+\delta,C(\nu_n,\ldots,\nu_1,\nu_0),\psi_\gamma(\epsilon)|  \gamma,\delta,\epsilon,\nu_i\in D_n(\beta,\alpha)\wedge\epsilon<\alpha\} \\ D(\beta,\alpha) = \bigcup\limits_{n<\omega} D_n(\beta,\alpha)\\ \psi_{C(\#,n+1,\#_0)}(0) = C(\#,n,\#_0) \\ \text{cof}(\alpha) < \kappa, \exists\xi(\kappa=C(\#,\pi+1,\#_0,\xi)): \psi_\kappa(\alpha) = \min\{\beta|\beta\in D[\kappa]\wedge\beta>\sup\{\psi_\kappa(\beta)|\beta<\alpha\}\}\\ \psi_\kappa(\alpha) = \min\{\beta|\sup(D(\beta,\alpha)\cap\kappa)=\beta\wedge\beta\in D[\kappa]\}\)

I mentioned at the beginning that this definition lacked some things to be fully formal, so here they are :
 * I actually don't understand (as in, I have no idea what the fuck I would be writing here) what's missing for this to be a formal defintion, so feel free to tell me in the comments and I'll edit this accordingly.