User blog:Syst3ms/YAUD

YAUD = Yet Another UNOCF Definition

Ok, you know the drill by now.

So I and Nish have decided to define UNOCF as an ordinal notation without an OCF; on Discord. It works in an, er, unusual way, to say the least. This definition (hopefully) works up to \(\psi(\varepsilon_{M+1})\). It could be extended to stage cardinal, metastage cardinal and so on, but the definitions gets significantly more redundant when we do that, and we run into a nesting problem. Nish was thinking about a solution, but this is irrelevant here.

Maybe this lacks a couple of things to be fully well-defined, but I'm confident it's not that big of deal this time. But I've been wrong twice already, so fuck anything I say. Let's just get to the point.

First, some misc stuff :

\(\psi(\alpha) = \psi_{\psi_M(0)}(\alpha) \\ \alpha+0=\alpha \\ \alpha*0 = 0 \\ \alpha^0 = 1 \\ \alpha+(\beta+1) = (\alpha+\beta)+1 \\ \alpha{\beta+1} = \alpha\beta+\alpha \\ \alpha^{\beta+1} = \alpha^\beta*\alpha \\ \text{cof}(\kappa) = \kappa: \kappa[\alpha] = \alpha \\ 0^0 = 1 \\ \alpha > 0: 0^\alpha = 0 \\ 1^\alpha = 1\)

Second, the recursive definition of cofinality :

\(\text{cof}(0) = 0 \\ \text{cof}(\alpha+1) = 0 \\ \text{cof}(\omega) = \omega \\ \text{cof}(M) = M \\ \text{cof}(\alpha) = 0 \vee \text{cof}(\alpha) = M: \text{cof}(\psi_M(\alpha)) = \psi_M(\alpha)  \\ \text{cof}(\alpha) > 0: \text{cof}(\psi_M(\alpha)) = \text{cof}(\alpha)  \\ \text{cof}(\alpha+\beta) = \text{cof}(\beta)  \\ \text{cof}(\beta) = 0: \text{cof}(\alpha*\beta), \text{cof}(\alpha^\beta) = \text{cof}(\alpha)  \\ \text{cof}(\beta) > 0, \alpha > 0 = \text{cof}(\alpha*\beta) = \text{cof}(\beta)  \\ \text{cof}(\beta) > 0, \alpha > 1 = \text{cof}(\alpha^\beta) = \text{cof}(\beta)  \\ \text{cof}(0*\beta) = 0 \\ \text{cof}(\beta) > 0 \wedge \alpha <= 1: \text{cof}(\alpha^\beta) = 0  \\ \text{cof}(\psi(0)) = 0 \\ \text{cof}(\alpha) >= \text{cof}(\kappa) : \text{cof}(\psi_\kappa(\alpha)) = \omega \\ \text{cof}(\kappa) < M \wedge \text{cof}(\alpha) = 0: \text{cof}(\psi_{\psi_M(\kappa)}(\alpha)) = \omega \\ 0 < \text{cof}(\alpha) < \text{cof}(\kappa) : \text{cof}(\psi_\kappa(\alpha)) = \text{cof}(\alpha)\)

Last but not least, the actual definition of \(\psi\) :

\(\psi_\psi_M(0)(0) = 1 \\ \psi_\psi_M(\alpha+1)(0) = \psi_M(\alpha)  \\ \psi_\kappa(\alpha+1)[0] = 0 \\ \psi_\kappa(\alpha+1)[n+1] = \psi_\kappa(\alpha+1)[n] + \psi_\kappa(\alpha) \\ \text{cof}(\kappa) > \text{cof}(\alpha) > 0: \psi_\kappa(\alpha)[\beta] = \psi_\kappa(\alpha[\beta])  \\ \text{cof}(\alpha) > \text{cof}(\kappa): \psi_\kappa(\alpha) = \psi_\kappa(\psi_{\text{cof}}(\alpha)(\alpha)) \\ \text{cof}(\alpha) = \text{cof}(\kappa): \psi_\kappa(\alpha)[0] = 0 \\ \text{cof}(\alpha) = \text{cof}(\kappa): \psi_\kappa(\alpha)[n+1] = \psi_\kappa(\alpha[\psi_\kappa(\alpha)[n]]) \\ \text{cof}(\kappa) = M : \psi_{\psi_M(\kappa)}(\alpha) = \psi_M(\kappa[\alpha])\)

It should be noted that whereas a lot of "regular" UNOCF expressions are invalid in this ruleset, all expressions permitted by this ruleset are valid "regular" UNOCF expressions.

If you have any questions/remarks, ask them in a comment, yaddi yaddi yadda.