User:Vel!/pu/Big Phi

The big phi function is an OCF that I made. It is very similar to Madore's Psi, but it is less powerful, enumerating powers of omega instead of fixed points of $$\alpha \rightarrow \omega^{\alpha}$$ that the Psi function does.

This is the definition of the psi function:

Let $$C(\alpha)$$ be the set of all ordinals constructible from $$0,1,\omega,\Omega$$ and finite applications of the operations addition, multiplication, exponentiation, and $$\kappa\rightarrow\psi(\kappa)$$, the latter only if $$\psi(\kappa)$$ has been defined yet. $$\psi(\alpha)$$ is the first ordinal not in $$C(\alpha)$$

This is the definition of the big phi function:

Let $$P(\alpha)$$ be the set of all ordinals constructible from $$0,\Omega$$ and finite applications of the operations addition and $$\kappa\rightarrow\Phi(\kappa)$$, the latter only if $$\Phi(\kappa)$$ has been defined yet. $$\Phi(\alpha)$$ is the first ordinal not in $$P(\alpha)$$

If it works how I think it does, the limit, $$\Phi(\Omega^{\Omega^{\Omega^{...}}})$$, is the BHO. You could also write it as $$\Phi(\varepsilon_{\Omega+1})$$. The big phi function first catches up with the psi function at $$\Phi(\Omega^{\omega}) = \psi(\Omega^{\omega}) = \varphi(\omega,0)$$, and does again at every subsequent power of big omega.

So that is the big phi function!