User:Username5243/Username's OCF

This is an attempt to make an OCF type notation that matches closer with UNAN. Much of what follows is mostly informal especially at higher levels, so any help here would be appreciated.

The first uncounable
Before \(\psi(\Omega)\), we can operate on three simple rules:


 * \(\psi(0) = 1\)
 * \(\psi(\alpha+1)[n] = \psi(\alpha)n\)
 * \(\psi(\alpha)[n] = \psi(\alpha[n])\)

It's easy to see that \(\psi(\alpha) = \omega^\alpha\), and \(\psi(\Omega) = \varepsilon_0\).

Now we ned to think of something called "cofinality" - basically, it works like this: Any single cardinal (\(\Omega_n\) in the early parts) has cofinality equal to itself. \(\alpha+\beta, \alpha\beta, \alpha^\beta\) etc have cofinality equal to \(\beta\), as does \(\psi_x(\beta\) for any x with cofinality greater than that of \(\beta\), otherwise it has cofinality of x.

To define FSes for ordinals with cofinality \(\Omega\), it's best to think of them as the output of some function \(f(\Omega)\). For instance, \(\Omega^{\Omega2} = f(\Omega)\), where f is defined as \(f(\alpha) = \Omega^{\Omega+\alpha}\).

We can then say that, if the ordinal has cofinality \(\Omega\):


 * \(\psi(\alpha)[1] = \psi(f(1))\)
 * \(\psi(\alpha)[n] = \psi(f(\alpha[n-1]))\)

Some examples:


 * \(\psi(\Omega) = \varepsilon_0\)
 * \(\psi(\Omega+1) = \varepsilon_0\omega\)
 * \(\psi(\Omega+\varepsilon_0) = \varepsilon_0^2\)
 * \(\psi(\Omega2) = \varepsilon_1\)
 * \(\psi(\Omega\omega) = \varepsilon_\omega\)
 * \(\psi(\Omega^2) = \zeta_0\)
 * \(\psi(\Omega^2+\Omega) = \varepsilon_{\zeta_0+1}\)
 * \(\psi(\Omega^22) = \zeta_1\)
 * \(\psi(\Omeag^3) = \varphi(3,0)\)
 * \(\psi(\Omega^\omega) = \varphi(\omega,0)\)
 * \(\psi(\Omega^\Omega) = \Gamma_0\)
 * \(\psi(\Omega^{\Omega^\Omega}\) = LVO\)

As you can see, even though it has a slower start, it does catch up to a "normal" OCF eventually.