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So I made this Bump Funk, and the first version was basically defined as follows:

(all ordinals in Cantor Normal Form)

B(k,\delta)=k\forall k<\omega

B(\omega,\delta)=\delta

B(\alpha+\beta,\delta)=B(\alpha,\delta)+B(\beta,\delta)

B(\alpha\beta,\delta)=B(\alpha,\delta)B(\beta,\delta)

B(\alpha^\beta,\delta)=B(\alpha,\delta)^{B(\beta,\delta)}

If your ordinal is not one of the above, then

B(\alpha,\delta)[\eta]=B(\alpha[\eta],\delta)

Where \alpha[\eta] represents the \eta-th ordinal in the fundamental sequence of \alpha.

Let's look at an example:

B(\varepsilon_0,\varepsilon_0)

=\sup\{B(1,\varepsilon_0),B(\omega,\varepsilon_0),B(\omega^\omega,\varepsilon_0),\dots\}

=\sup\{1,\varepsilon_0,\varepsilon_0^{\varepsilon_0},\dots\}

=\varepsilon_1

Pretty basic right? Now we have my Bump Ordinal Collapsing Funk, BOCF or \psi_B.

C(\alpha)_0=\{\Omega\}

C(\alpha)_{n+1}=C(\alpha)_n\cup\{\gamma+\delta,\gamma\delta,\gamma^\delta,B(\gamma,\delta),\psi_B(\eta)|\gamma,\delta,\eta\in C(\alpha)_n,\eta<\alpha\}

C(\alpha)=\bigcup_{n<\omega}C(\alpha)_n

\psi_B(\alpha)=\min\{\beta|\beta\notin C(\alpha)\}

And so we have...

\psi_B(0)=0

\psi_B(1)=\omega

\psi_B(2)=\varepsilon_0

\psi_B(3)=\varepsilon_\omega

\psi_B(2+\alpha)=\varepsilon_{\omega^\alpha}

\psi_B(\Omega)=\zeta_0

\psi_B(\Omega+1)=\varepsilon_{\zeta_0\omega}

\psi_B(\Omega+2)=\varepsilon_{\zeta_0\omega^2}

\psi_B(\Omega+\alpha)=\varepsilon_{\zeta_0\omega^\alpha}

\psi_B(\Omega+\varepsilon_0)=\varepsilon_{\zeta_0\varepsilon_0}

\psi_B(\Omega2)=\zeta_1

In general, for \alpha\le\varepsilon_{\Omega+1} and \alpha being a perfect multiple/power of \Omega, we have that this is equal to Madore's psi function.

We can then go further that Madore's psi function since

B(\psi(2),\Omega)=\varepsilon_{\Omega+1}

In general, we have

B(\varepsilon_\alpha,\varepsilon_\beta)=\varepsilon_{\alpha+1+\beta}

So now we can do neat little things like

\psi_B(\varepsilon_{\Omega+\psi_B(\varepsilon_{\dots})})

And furthermore, since B(\Omega,\Omega)=\varepsilon_{\Omega2}, we can even go further...

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