This is a page where I make a new, probably strong ordinal notation, pi notation. It relies on extending fundamental sequences to ordinal lengths.
How it works
The pi notation looks like \(\pi(\alpha,\beta)\). If \(\beta = 0\), then we could rewrite \(\pi(\alpha,\beta)\) as just \(\pi(\alpha)\).
When \(\alpha = 0\), \(\pi(0,\beta) = \omega^\beta\).
for limit \(\beta\), \(\pi(\alpha,\beta)\) is always the supremum of \(\pi(\alpha,\gamma)\) for \(\gamma < \alpha\).
For limit \(\alpha\), \(\pi(\alpha,0)\) is the supremum of all the \(\pi(\gamma,0)\) for \(\gamma < \alpha\). and \(\pi(\alpha,\beta+1)\) is the supremum of all \(\pi(\gamma,\pi(\alpha,\beta))\) for \(\gamma <\alpha\).
For successor \(\alpha\), where it is not the successor of a limit ordinal, \(\pi(\alpha+1,0)\) is the first fixed point of \(\gamma = \pi(\alpha,\gamma)\). And \(\pi(\alpha+1,\beta+1)\) is the next fixed point of that after \(\pi(\alpha+1,\beta)\).
The most interesting, but also hardest to explain, case is if \(\alpha\) is the successor of a limit ordinal. In that case, We have to extend the fundamental sequence of that ordinal. After the finite terms, These will just be written like \(\alpha[\beta]\). \(\omega\)th term is always hat ordinal itself, but they refer to different things in operations. To get to \\(\omega+1\)th term, you have to do the same thing to the \(\omega\)th term that you did to get from the nth term to the n+1-th term. So, after \(\omega^2 = \omega^2[\omega]\), there's \(\omega^2[\omega]+1\), \(\omega^2[\omega]+2, \), and the limit of that is \(\omega^2[\omega]+\omega = \omega^2[\omega+1]\). Anyway, to get the actuall definition of \(\pi(\alpha+1,\beta)\) in this case, it is the fixed points of \(\gamma = \pi([\alpha[\gamma],0)\).
Now I will analyze this notation with normal ordinal notation.
Up to \(\pi(\omega)\)
Because there are no infinites in the first argument, this is easy to understand. It's just like \(\phi(\alpha,\beta)\).
| Pi notation | Normal ordinals |
|---|---|
| \(\pi(0)\) | \(1\) |
| \(\pi(0,1)\) | \(\omega\) |
| \(\pi(0,2)\) | \(\omega^2\) |
| \(\pi(0,\omega)\) | \(\omega^\omega\) |
| \(\pi(0,\omega+1)\) | \(\omega^{\omega+1}\) |
| \(\pi(0,\omega2)\) | \(\omega^{\omega2}\) |
| \(\pi(0,\omega^2)\) | \(\omega^{\omega^2}\) |
| \(\pi(0,\omega^\omega)\) | \(\omega^{\omega^\omega}\) |
| \(\pi(0,\omega^{\omega^\omega})\) | \(\omega^{\omega^{\omega^\omega}}\) |
| \(\pi(1)\) | \(\varepsilon_0 = \psi(\Omega)\) |
| \(\pi(0,\varepsilon_0+1)\) | \(\varepsilon_0\omega = \psi(\Omega+1)\) |
| \(\pi(0,\varepsilon_02)\) | \(\varepsilon_0^2 = \psi(\Omega+\varepsilon_0)\) |
| \(\pi(0,\varepsilon_0^2)\) | \(\varepsilon_0^{\varepsilon_0} = \psi(\Omega+\varepsilon_0^2)\) |
| \(\pi(1,1)\) | \(\varepsilon_1 = \psi(\Omega2)\) |
| \(\pi(1,2)\) | \(\varepsilon_2 = \psi(\Omega3)\) |
| \(\pi(1,\omega)\) | \(\varepsilon_\omega = \psi(\Omega\omega)\) |
| \(\pi(1,\varepsilon_0)\) | \(\varepsilon_{\varepsilon_0} = \psi(\Omega\varepsilon_0)\) |
| \(\pi(2)\) | \(\zeta_0 = \psi(\Omega^2)\) |
| \(\pi(1,\pi(2)+1)\) | \(\varepsilon_{\zeta_0+1} = \psi(\Omega^2+\Omega)\) |
| \(\pi(2,1)\) | \(\zeta_1 = \psi(\Omega^22)\) |
| \(\pi(3)\) | \(\varphi(3,0) = \psi(\Omega^3)\) |
| \(\pi(4)\) | \(\varphi(4,0) = \psi(\Omega^4)\) |
Up to \(\pi(\omega+1)\)
This is the first application of infinite length fundamental sequences. Note that \(\alpha[\omega]\) can just be written as \(\alpha\) for short. To get \(\omega[\omega+1]\), we need to note how the normal fundamental sequence of \(\omega\) works. Each term is one more than the previous term. So the next ordinal in the first argument after \(\omega\) is \(\omega[\omega]+1 = \omega[\omega+1]\).
If this makes sense to you, the comparisons in this section are also quite straightforward.
| Pi notation | Normal ordinals |
|---|---|
| \(\pi(\omega)\) | \(\varphi(\omega,0) = \psi(\Omega^\omega)\) |
| \(\pi(0,\pi(\omega)+1)\) | \(\varphi(\omega,0)\omega = \psi(\Omega^\omega+1)\) |
| \(\pi(1,\pi(\omega)+1)\) | \(\varepsilon_{\varphi(\omega,0)+1} = \psi(\Omega^\omega+\Omega)\) |
| \(\pi(\omega,1)\) | \(\varphi(\omega,1) = \psi(\Omega^\omega2)\) |
| \(\pi(\omega,\pi(\omega))\) | \(\varphi(\omega,\varphi(\omega,0)) = \psi(\Omega^\omega\varphi(\omega,0))\) |
| \(\pi(\omega[\omega+1])\) | \(\varphi(\omega+1,0) = \psi(\Omega^{\omega+1})\) |
| \(\pi(\omega,\pi(\omega[\omega+1])+1)\) | \(\varphi(\omega,\varphi(\omega+1,0)+1) = \psi(\Omega^{\omega+1}+\Omega^\omega)\) |
| \(\pi(\omega[\omega+1],1)\) | \(\varphi(\omega+1,1) = \psi(\Omega^{\omega+1}2)\) |
| \(\pi(\omega[\omega+2])\) | \(\varphi(\omega+2,0) = \psi(\Omega^{\omega+2})\) |
| \(\pi(\omega[\omega2])\) | \(\varphi(\omega2,0) = \psi(\Omega^{\omega2})\) |
| \(\pi(\omega[\omega^2])\) | \(\varphi(\omega^2,0) = \psi(\Omega^{\omega^2})\) |
| \(\pi(\omega[\omega^\omega])\) | \(\varphi(\omega^\omega,0) = \psi(\Omega^{\omega^\omega})\) |
| \(\pi(\omega[\pi(1)])\) | \(\varphi(\varepsilon_0,0) = \psi(\Omega^{\varepsilon_0})\) |
| \(\pi(\omega[\pi(\omega)])\) | \(\varphi(\varphi(\omega,0),0) = \psi(\Omega^{\varphi(\omega,0)})\) |
| \(\pi(\omega[\pi(\omega[\varepsilon_0])])\) | \(\varphi(\varphi(\varepsilon_0,0),0) = \psi(\Omega^{\varphi(\varepsilon_0,0)})\) |
It seems that \(\pi(\omega[\alpha],\beta) = \varphi(\alpha,\beta)\). So this makes \(\pi(\omega+1)\) equal to \(\Gamma_0\).
Up to \(\pi(\omega^2)\)
If you can understand the previous section, this part should also be easy to understand. Each increase of \(\omega\) in the first argument increases the third-rightmost argument in the Veblen function.
| Pi notation | Normal ordinals |
|---|---|
| \(\pi(\omega+1)\) | \(\Gamma_0 = \psi(\Omega^\Omega)\) |
| \(\pi(1,\Gamma_0+1)\) | \(\varepsilon_{\Gamma_0+1} = \psi(\Omega^\Omega+\Omega)\) |
| \(\pi(2,\Gamma_0+1)\) | \(\zeta_{\Gamma_0+1} = \psi(\Omega^\Omega+\Omega^2)\) |
| \(\pi(\omega,\Gamma_0+1)\) | \(\varphi(\omega,\Gamma_0+1) = \psi(\Omega^\Omega+\Omega^\omega)\) |
| \(\pi(\omega[\varepsilon_0],\Gamma_0+1)\) | \(\varphi(\varepsilon_0,\Gamma_0+1) = \psi(\Omega^\Omega+\Omega^{\varepsilon_0})\) |
| \(\pi(\omega[\Gamma_0],1)\) | \(\varphi(\Gamma_0,1) = \psi(\Omega^\Omega+\Omega^{\Gamma_0})\) |
| \(\pi(\omega[\Gamma_0],2)\) | \(\varphi(\Gamma_0,2) = \psi(\Omega^\Omega+\Omega^{\Gamma_0}2)\) |
| \(\pi(\omega[\Gamma_0],\Gamma_0)\) | \(\varphi(\Gamma_0,\Gamma_0) = \psi(\Omega^\Omega+\Omega^{\Gamma_0}\Gamma_0)\) |
| \(\pi(\omega[\Gamma_0+1])\) | \(\varphi(\Gamma_0+1,0) = \psi(\Omega^\Omega+\Omega^{\Gamma_0+1})\) |
| \(\pi(\omega[\Gamma_02])\) | \(\varphi(\Gamma_02,0) = \psi(\Omega^\Omega+\Omega^{\Gamma_02})\) |
| \(\pi(\omega[\Gamma_0\omega])\) | \(\varphi(\Gamma_0\omega,0) = \psi(\Omega^\Omega+\Omega^{\Gamma_0\omega})\) |
| \(\pi(\omega[\varepsilon_{\Gamma_0+1}])\) | \(\varphi(\varepsilon_{\Gamma_0+1},0) = \psi(\Omega^\Omega+\Omega^{\varepsilon_{\Gamma_0+1}})\) |
| \(\pi(\omega[\varphi(\Gamma_0,1)])\) | \(\varphi(\varphi(\Gamma_0,1),0) = \psi(\Omega^\Omega+\Omega^{\varphi(\Gamma_0,1)})\) |
| \(\pi(\omega+1,1)\) | \(\Gamma_1 = \psi(\Omega^\Omega2)\) |
| \(\pi(\omega[\Gamma_1],1)\) | \(\varphi(\Gamma_1,1) = \psi(\Omega^\Omega2+\Omega^{\Gamma_1})\) |
| \(\pi(\omega+1,2)\) | \(\Gamma_2 = \psi(\Omega^\Omega3)\) |
| \(\pi(\omega+1,\omega)\) | \(\Gamma_\omega = \psi(\Omega^\Omega\omega)\) |
| \(\pi(\omega+1,\pi(\omega+1))\) | \(\Gamma_{\Gamma_0} = \psi(\Omega^\Omega\Gamma_0)\) |
| \(\pi(\omega+2)\) | \(\varphi(1,1,0) = \psi(\Omega^{\Omega+1})\) |
| \(\pi(\omega+1,\pi(\omega+2)+1)\) | \(\Gamma_{\varphi(1,1,0)+1} = \psi(\Omega^{\Omega+1}+\Omega^\Omega)\) |
| \(\pi(\omega+2,1)\) | \(\varphi(1,1,1) = \psi(\Omega^{\Omega+1}2)\) |
| \(\pi(\omega+3)\) | \(\varphi(1,2,0) = \psi(\Omega^{\Omega+2})\) |
| \(\pi(\omega2)\) | \(\varphi(1,\omega,0) = \psi(\Omega^{\Omega+\omega})\) |
| \(\pi(\omega2,1)\) | \(\varphi(1,\omega,1) = \psi(\Omega^{\Omega+\omega}2)\) |
| \(\pi(\omega2[\omega+1])\) | \(\varphi(1,\omega+1,0) = \psi(\Omega^{\Omega+\omega+1})\) |
| \(\pi(\omega2[\Gamma_0])\) | \(\varphi(1,\Gamma_0,0) = \psi(\Omega^{\Omega+\Gamma_0})\) |
| \(\pi(\omega2+1)\) | \(\varphi(2,0,0) = \psi(\omega^{\Omega2})\) |
| \(\pi(\omega+1,\pi(\omega2+1)+1)\) | \(\Gamma_{\varphi(2,0,0)+1} = \psi(\Omega^{\Omega2}+\Omega^\Omega)\) |
| \(\pi(\omega2,\pi(\omega2+1)+1)\) | \(\varphi(1,\omega,\varphi(2,0,0)+1) = \psi(\Omega^{\Omega2}+\Omega^{\Omega+\omega})\) |
| \(\pi(\omega2[\varphi(2,0,0)],1)\) | \(\varphi(1,\varphi(2,0,0),1) = \psi(\Omega^{\Omega2}+\Omega^{\Omega+\varphi(2,0,0)})\) |
| \(\pi(\omega2+1,1)\) | \(\varphi(2,0,1) = \psi(\Omega^{\Omega2}2)\) |
| \(\pi(\omega2+2,0)\) | \(\varphi(2,1,0) = \psi(\Omega^{\Omega2+1})\) |
| \(\pi(\omega3,0)\) | \(\varphi(2,\omega,0) = \psi(\Omega^{\Omega2+\omega})\) |
| \(\pi(\omega3+1)\) | \(\varphi(3,0,0) = \psi(\Omega^{\Omega3})\) |
| \(\pi(\omega4+1)\) | \(\varphi(4,0,0) = \psi(\Omega^{\Omega4})\) |
It seems that \(\pi(\omega n+m,k) = \varphi(n,m-1,k)\). And \(\pi(\omega^2) = \varphi(\omega,0,0)\).
Up to \(\pi(\omega^\omega)\)
First part of this section is to get \(\pi(\omega^2+1)\). Note here that \(\omega^2[\omega+1]\) is not the next first-argument after \(\omega^2\). That's \(\omega^2[\omega]+1\), this is because of the operation you do to get the next term of \(\omega^2\)'s fundamental sequence: Add \(\omega\).
| Pi notation | Normal ordinals |
|---|---|
| \(\pi(\omega^2)\) | \(\varphi(\omega,0,0) = \psi(\Omega^{\Omega\omega})\) |
| \(\pi(\omega+1,\pi(\omega^2)+1)\) | \(\Gamma_{\varphi(\omega,0,0)+1} = \psi(\Omega^{\Omega\omega}+\Omega^\Omega)\) |
| \(\pi(\omega^2,1)\) | \(\varphi(\omega,0,1) = \psi(\Omega^{\Omega\omega}2)\) |
| \(\pi(\omega^2[\omega]+1)\) | \(\varphi(\omega,1,0) = \psi(\Omega^{\Omega\omega+1})\) |
| \(\pi(\omega^2[\omega]+2)\) | \(\varphi(\omega,2,0) = \psi(\Omega^{\Omega\omega+2})\) |
| \(\pi(\omega^2[\omega+1])\) | \(\varphi(\omega,\omega,0) = \psi(\Omega^{\Omega\omega+\omega})\) |
| \(\pi(\omega^2[\omega+1]+1)\) | \(\varphi(\omega+1,0,0) = \psi(\Omega^{\Omega\omega+\Omega})\) |
| \(\pi(\omega^2[\omega+2]+1)\) | \(\varphi(\omega+2,0,0) = \psi(\Omega^{\Omega\omega+\Omega2})\) |
| \(\pi(\omega^2[\omega2])\) | \(\varphi(\omega2,0,0) = \psi(\Omega^{\Omega\omega2})\) |
| \(\pi(\omega^2[\omega^2])\) | \(\varphi(\omega^2,0,0) = \psi(\Omega^{\Omega\omega^2})\) |
| \(\pi(\omega^2[\Gamma_0])\) | \(\varphi(\Gamma_0,0,0) = \psi(\Omega^{\Omega\Gamma_0})\) |
| \(\pi(\omega^2[\pi(\omega^2)])\) | \(\varphi(\varphi(\omega,0,0),0,0) = \psi(\Omega^{\Omega\varphi(\omega,0,0)})\) |
| \(\pi(\omega^2+1)\) | \(\varphi(1,0,0,0) = \psi(\Omega^{\Omega^2})\) |
Then continue to \(\pi(\omega^\omega)\). This is quite easy.
| Pi notation | Normal ordinals |
|---|---|
| \(\pi(\omega^2+1)\) | \(\varphi(1,0,0,0) = \psi(\Omega^{\Omega^2})\) |
| \(\pi(\omega^2[\pi(\omega^2+1)],1)\) | \(\varphi(\varphi(1,0,0,0),0,1) = \psi(\Omega^{\Omega^2}+\Omega^{\Omega\varphi(1,0,0,0)})\) |
| \(\pi(\omega^2+1,1)\) | \(\varphi(1,0,0,1) = \psi(\Omega^{\Omega^2}2)\) |
| \(\pi(\omega^2+2)\) | \(\varphi(1,0,1,0) = \psi(\Omega^{\Omega^2+1})\) |
| \(\pi(\omega^2+3)\) | \(\varphi(1,0,2,0) = \psi(\Omega^{\Omega^2+2})\) |
| \(\pi(\omega^2+\omega)\) | \(\varphi(1,0,\omega,0) = \psi(\Omega^{\Omega^2+\omega})\) |
| \(\pi(\omega^2+\omega+1)\) | \(\varphi(1,1,0,0) = \psi(\Omega^{\Omega^2+\Omega})\) |
| \(\pi(\omega^2+\omega2)\) | \(\varphi(1,1,\omega,0) = \psi(\Omega^{\Omega^2+\Omega+\omega})\) |
| \(\pi(\omega^2+\omega2+1)\) | \(\varphi(1,2,0,0) = \psi(\Omega^{\Omega^2+\Omega2})\) |
| \(\pi(\omega^22)\) | \(\varphi(1,\omega,0,0) = \psi(\Omega^{\Omega^2+\Omega\omega})\) |
| \(\pi(\omega^22+1)\) | \(\varphi(2,0,0,0) = \psi(\Omega^{\Omega^22})\) |
| \(\pi(\omega^22+\omega)\) | \(\varphi(2,0,\omega,0) = \psi(\Omega^{\Omega^22+\Omega\omega})\) |
| \(\pi(\omega^23)\) | \(\varphi(2,\omega,0,0) = \psi(\Omega^{\Omega^22+\Omega\omega})\) |
| \(\pi(\omega^23+1)\) | \(\varphi(3,0,0,0) = \psi(\Omega^{\Omega^23})\) |
| \(\pi(\omega^3)\) | \(\varphi(\omega,0,0,0) = \psi(\Omega^{\Omega^2\omega})\) |
| \(\pi(\omega^3[\omega]+1)\) | \(\varphi(\omega,0,1,0) = \psi(\Omega^{\Omega^2\omega+1})\) |
| \(\pi(\omega^3[\omega]+\omega)\) | \(\varphi(\omega,0,\omega,0) = \psi(\Omega^{\Omega^2\omega+\omega})\) |
| \(\varphi(\omega^3[\omega+1])\) | \(\varphi(\omega,\omega,0,0) = \psi(\Omega^{\Omega^2\omega+\Omega\omega})\) |
| \(\pi(\omega^3[\omega2])\) | \(\varphi(\omega2,0,0,0) = \psi(\Omega^{\Omega^2\omega2})\) |
| \(\pi(\omega^3+1)\) | \(\varphi(1,0,0,0,0) = \psi(\Omega^{\Omega^3})\) |
| \(\pi(\omega^3+\omega)\) | \(\varphi(1,0,0,\omega,0) = \psi(\Omega^{\Omega^3+\omega})\) |
| \(\pi(\omega^3+\omega+1)\) | \(\varphi(1,0,1,0,0) = \psi(\Omega^{\Omega^3+\Omega})\) |
| \(\pi(\omega^3+\omega^2)\) | \(\varphi(1,0,\omega,0,0) = \psi(\Omega^{\Omega^3+\Omega\omega})\) |
| \(\pi(\omega^3+\omega^2+1)\) | \(\varphi(1,1,0,0,0) = \psi(\Omega^{\Omega^3+\Omega^2})\) |
| \(\pi(\omega^32)\) | \(\varphi(1,\omega,0,0,0) = \psi(\Omega^{\Omega^3+\Omega^2\omega})\) |
| \(\pi(\omega^32+1)\) | \(\varphi(2,0,0,0,0) = \psi(\Omega^{\Omega^32})\) |
| \(\pi(\omega^4)\) | \(\varphi(\omega,0,0,0,0) = \psi(\Omega^{\Omega^3\omega})\) |
| \(\pi(\omega^4+1)\) | \(\varphi(1,0,0,0,0,0) = \psi(\Omega^{\Omega^4})\) |
| \(\pi(\omega^5+1)\) | \(\varphi(1,0,0,0,0,0,0) = \psi(\Omega^{\Omega^5})\) |
So \(\pi(\omega^\omega)=\psi(\Omega^{\Omega^\omega})\).
Up to \(\pi(\varepsilon_0)\)
First we need to get \(\pi(\omega^\omega+1)\):
| Pi notation | Normal ordinals |
|---|---|
| \(\pi(\omega^\omega)\) | \(\psi(\Omega^{\Omega^\omega})\) |
| \(\pi(\omega^\omega,1)\) | \(\psi(\Omega^{\Omega^\omega}2)\) |
| \(\pi(\omega^\omega[\omega]+1)\) | \(\psi(\Omega^{\Omega^\omega+1})\) |
| \(\pi(\omega^\omega[\omega]+\omega)\) | \(\psi(\Omega^{\Omega^\omega+\omega})\) |
| \(\pi(\omega^\omega[\omega]+\omega+1)\) | \(\psi(\Omega^{\Omega^\omega+\Omega})\) |
| \(\pi(\omega^\omega[\omega]+\omega2+1)\) | \(\psi(\Omega^{\Omega^\omega+\Omega2})\) |
| \(\pi(\omega^\omega[\omega]+\omega^2)\) | \(\psi(\Omega^{\Omega^\omega+\Omega\omega})\) |
| \(\pi(\omega^\omega[\omega]+\omega^2+1)\) | \(\psi(\Omega^{\Omega^\omega+\Omega^2})\) |
| \(\pi(\omega^\omega[\omega]+\omega^3+1)\) | \(\psi(\Omega^{\Omega^\omega+\Omega^3})\) |
| \(\pi(\omega^\omega[\omega]2)\) | \(\psi(\Omega^{\Omega^\omega2})\) |
| \(\pi(\omega^\omega[\omega]3)\) | \(\psi(\Omega^{\Omega^\omega3})\) |
| \(\pi(\omega^\omega[\omega+1])\) | \(\psi(\Omega^{\Omega^\omega\omega})\) |
| \(\pi(\omega^\omega[\omega+1]+1)\) | \(\psi(\Omega^{\Omega^{\omega+1}})\) |
| \(\pi(\omega^\omega[\omega+2]+1)\) | \(\psi(\Omega^{\Omega^{\omega+2}})\) |
| \(\pi(\omega^\omega[\omega2])\) | \(\psi(\Omega^{\Omega^{\omega2}})\) |
| \(\pi(\omega^\omega[\varepsilon_0])\) | \(\psi(\Omega^{\Omega^{\varepsilon_0}})\) |
| \(\pi(\omega^\omega[\pi(\omega^\omega)])\) | \(\psi(\Omega^{\Omega^{\psi(\Omega^{\Omega^\omega})}})\) |
So \(\pi(\omega^\omega+1) = \psi(\Omega^{\Omega^\Omega})\).
One thing to note before contuing: it is possible to have these kinds of ordinals in the exponent of \(\omega\) too. So the next power of \(\omega\) after \(\omega^{\omega^2}[\omega]\) is \(\omega^{\omega^2[\omega]+1}\).
Now continue to \(\pi(\varepsilon_0)\):
| Pi notation | Normal ordinals |
|---|---|
| \(\pi(\omega^\omega+1)\) | \(\psi(\Omega^{\Omega^\Omega})\) |
| \(\pi(\omega^\omega+1,1)\) | \(\psi(\Omega^{\Omega^\Omega}2)\) |
| \(\pi(\omega^\omega+2)\) | \(\psi(\Omega^{\Omega^\Omega+1})\) |
| \(\pi(\omega^\omega+3)\) | \(\psi(\Omega^{\Omega^\Omega+2})\) |
| \(\pi(\omega^\omega+\omega)\) | \(\psi(\Omega^{\Omega^\Omega+\omega})\) |
| \(\pi(\omega^\omega+\omega+1)\) | \(\psi(\Omega^{\Omega^\Omega+\Omega})\) |
| \(\pi(\omega^\omega+\omega2+1)\) | \(\psi(\Omega^{\Omega^\Omega+\Omega2})\) |
| \(\pi(\omega^\omega+\omega^2)\) | \(\psi(\Omega^{\Omega^\Omega+\Omega\omega})\) |
| \(\pi(\omega^\omega+\omega^2+1)\) | \(\psi(\Omega^{\Omega^\Omega+\Omega^2})\) |
| \(\pi(\omega^\omega+\omega^3+1)\) | \(\psi(\Omega^{\Omega^\Omega+\Omega^3})\) |
| \(\pi(\omega^\omega2)\) | \(\psi(\Omega^{\Omega^\Omega+\Omega^\omega})\) |
| \(\pi(\omega^\omega2+1)\) | \(\psi(\Omega^{\Omega^\Omega2})\) |
| \(\pi(\omega^\omega3+1)\) | \(\psi(\Omega^{\Omega^\Omega3})\) |
| \(\pi(\omega^{\omega+1})\) | \(\psi(\Omega^{\Omega^\Omega\omega})\) |
| \(\pi(\omega^{\omega+1}+1)\) | \(\psi(\Omega^{\Omega^{\Omega+1}})\) |
| \(\pi(\omega^{\omega+1}+\omega)\) | \(\psi(\Omega^{\Omega^{\Omega+1}\omega})\) |
| \(\pi(\omega^{\omega+1}+\omega^\omega)\) | \(\psi(\Omega^{\Omega^{\Omega+1}+\Omega^\omega})\) |
| \(\pi(\omega^{\omega+1}2)\) | \(\psi(\Omega^{\Omega^{\Omega+1}+\Omega^\Omega\omega})\) |
| \(\pi(\omega^{\omega+1}2+1)\) | \(\psi(\Omega^{\Omega^{\Omega+1}2})\) |
| \(\pi(\omega^{\omega+2})\) | \(\psi(\Omega^{\Omega^{\Omega+1}\omega})\) |
| \(\pi(\omega^{\omega+2}+1)\) | \(\psi(\Omega^{\Omega^{\Omega+2}})\) |
| \(\pi(\omega^{\omega+3}+1)\) | \(\psi(\Omega^{\Omega^{\Omega+3}})\) |
| \(\pi(\omega^{\omega2})\) | \(\psi(\Omega^{\Omega^{\Omega+\omega}})\) |
| \(\pi(\omega^{\omega2}[\omega+1])\) | \(\psi(\Omega^{\Omega^{\Omega+\omega}\omega})\) |
| \(\pi(\omega^{\omega2}[\omega+1]+1)\) | \(\psi(\Omega^{\Omega^{\Omega+\omega+1}})\) |
| \(\pi(\omega^{\omega2}+1)\) | \(\psi(\Omega^{\Omega^{\Omega2}})\) |
| \(\pi(\omega^{\omega2}+\omega^\omega)\) | \(\psi(\Omega^{\Omega^{\Omega2}+\Omega^\omega})\) |
| \(\pi(\omega^{\omega2}2)\) | \(\psi(\Omega^{\Omega^{\Omega2}+\Omega^{\Omega+\omega}})\) |
| \(\pi(\omega^{\omega2}2+1)\) | \(\psi(\Omega^{\Omega^{\Omega2}2})\) |
| \(\pi(\omega^{\omega2+1})\) | \(\psi(\Omega^{\Omega^{\Omega2}\omega})\) |
| \(\pi(\omega^{\omega2+1}+1)\) | \(\psi(\Omega^{\Omega^{\Omega2+1}})\) |
| \(\pi(\omega^{\omega3})\) | \(\psi(\Omega^{\Omega^{\Omega2+\omega}})\) |
| \(\pi(\omega^{\omega3}+1)\) | \(\psi(\Omega^{\Omega^{\Omega3}})\) |
| \(\pi(\omega^{\omega4}+1)\) | \(\psi(\Omega^{\Omega^{\Omega4}})\) |
| \(\pi(\omega^{\omega^2})\) | \(\psi(\Omega^{\Omega^{\Omega\omega}})\) |
| \(\pi(\omega^{\omega^2}[\omega]2)\) | \(\psi(\Omega^{\Omega^{\Omega\omega}2})\) |
| \(\pi(\omega^{\omega^2[\omega]+1})\) | \(\psi(\Omega^{\Omega^{\Omega\omega}\omega})\) |
| \(\pi(\omega^{\omega^2[\omega]+1}+1)\) | \(\psi(\Omega^{\Omega^{\Omega\omega+1}})\) |
| \(\pi(\omega^{\omega^2}[\omega+1])\) | \(\psi(\Omega^{\Omega^{\Omega\omega+\omega}})\) |
| \(\pi(\omega^{\omega^2}[\omega2])\) | \(\psi(\Omega^{\Omega^{\Omega\omega2}})\) |
| \(\pi(\omega^{\omega^2}+1)\) | \(\psi(\Omega^{\Omega^{\Omega^2}})\) |
| \(\pi(\omega^{\omega^2}2+1)\) | \(\psi(\Omega^{\Omega^{\Omega^2}2})\) |
| \(\pi(\omega^{\omega^2+1})\) | \(\psi(\Omega^{\Omega^{\Omega^2}\omega})\) |
| \(\pi(\omega^{\omega^2+1}+1)\) | \(\psi(\Omega^{\Omega^{\Omega^2+1}})\) |
| \(\pi(\omega^{\omega^2+\omega})\) | \(\psi(\Omega^{\Omega^{\Omega^2+\omega}})\) |
| \(\pi(\omega^{\omega^22})\) | \(\psi(\Omega^{\Omega^{\Omega^2+\Omega\omega}})\) |
| \(\pi(\omega^{\omega^22}+1)\) | \(\psi(\Omega^{\Omega^{\Omega^22}})\) |
| \(\pi(\omega^{\omega^3})\) | \(\psi(\Omega^{\Omega^{\Omega^2\omega}})\) |
| \(\pi(\omega^{\omega^3}+1)\) | \(\psi(\Omega^{\Omega^{\Omega^3}})\) |
| \(\pi(\omega^{\omega^4}+1)\) | \(\psi(\Omega^{\Omega^{\Omega^4}})\) |
| \(\pi(\omega^{\omega^\omega})\) | \(\psi(\Omega^{\Omega^{\Omega^\omega}})\) |
| \(\pi(\omega^{\omega^\omega}[\omega]2)\) | \(\psi(\Omega^{\Omega^{\Omega^\omega}2})\) |
| \(\pi(\omega^{\omega^\omega[\omega]+1})\) | \(\psi(\Omega^{\Omega^{\Omega^\omega+1}})\) |
| \(\pi(\omega^{\omega^\omega[\omega]2})\) | \(\psi(\Omega^{\Omega^{\Omega^\omega2}})\) |
| \(\pi(\omega^{\omega^\omega}[\omega+1])\) | \(\psi(\Omega^{\Omega^{\Omega^\omega\omega}})\) |
| \(\pi(\omega^{\omega^\omega}[\omega2])\) | \(\psi(\Omega^{\Omega^{\Omega^{\omega2}}})\) |
| \(\pi(\omega^{\omega^\omega}+1)\) | \(\psi(\Omega^{\Omega^{\Omega^\Omega}})\) |
| \(\pi(\omega^{\omega^\omega}2+1)\) | \(\psi(\Omega^{\Omega^{\Omega^\Omega}2})\) |
| \(\pi(\omega^{\omega^\omega+1})\) | \(\psi(\Omega^{\Omega^{\Omega^\Omega}\omega})\) |
| \(\pi(\omega^{\omega^\omega+1}+1)\) | \(\psi(\Omega^{\Omega^{\Omega^\Omega+1}})\) |
| \(\pi(\omega^{\omega^\omega+\omega})\) | \(\psi(\Omega^{\Omega^{\Omega^\Omega+\omega}})\) |
| \(\pi(\omega^{\omega^\omega2})\) | \(\psi(\Omega^{\Omega^{\Omega^\Omega+\Omega^\omega}})\) |
| \(\pi(\omega^{\omega^\omega2}+1)\) | \(\psi(\Omega^{\Omega^{\Omega^\Omega2}})\) |
| \(\pi(\omega^{\omega^{\omega+1}})\) | \(\psi(\Omega^{\Omega^{\Omega^\Omega\omega}})\) |
| \(\pi(\omega^{\omega^{\omega+1}}+1)\) | \(\psi(\Omega^{\Omega^{\Omega^{\Omega+1}}})\) |
| \(\pi(\omega^{\omega^{\omega+2}}+1)\) | \(\psi(\Omega^{\Omega^{\Omega^{\Omega+2}}})\) |
| \(\pi(\omega^{\omega^{\omega2}})\) | \(\psi(\Omega^{\Omega^{\Omega^{\Omega+\omega}}})\) |
| \(\pi(\omega^{\omega^{\omega2}}+1)\) | \(\psi(\Omega^{\Omega^{\Omega^{\Omega2}}})\) |
| \(\pi(\omega^{\omega^{\omega^2}})\) | \(\psi(\Omega^{\Omega^{\Omega^{\Omega\omega}}})\) |
| \(\pi(\omega^{\omega^{\omega^2}}+1)\) | \(\psi(\Omega^{\Omega^{\Omega^{\Omega^2}}})\) |
| \(\pi(\omega^{\omega^{\omega^3}}+1)\) | \(\psi(\Omega^{\Omega^{\Omega^{\Omega^3}}})\) |
| \(\pi(\omega^{\omega^{\omega^\omega}})\) | \(\psi(\Omega^{\Omega^{\Omega^{\Omega^\omega}}})\) |
| \(\pi(\omega^{\omega^{\omega^\omega}}+1)\) | \(\psi(\Omega^{\Omega^{\Omega^{\Omega^\Omega}}})\) |
| \(\pi(\omega^{\omega^{\omega^{\omega^2}}}+1)\) | \(\psi(\Omega^{\Omega^{\Omega^{\Omega^{\Omega^2}}}})\) |
| \(\pi(\omega^{\omega^{\omega^{\omega^\omega}}}+1)\) | \(\psi(\Omega^{\Omega^{\Omega^{\Omega^{\Omega^\Omega}}}})\) |
| \(\pi(\omega^{\omega^{\omega^{\omega^{\omega^\omega}}}}+1)\) | \(\psi(\Omega^{\Omega^{\Omega^{\Omega^{\Omega^{\Omega^\Omega}}}}})\) |
So \(\pi(\varepsilon_0) = \pi(\pi(1)) = \psi(\Omega_2)\).
Up to \(\pi(\psi(\Omega_2))\)
In the last section, you probably noticed how \(\omega\)'s in pi notation corresponds to \(\Omega\)'s in psi notation. Now we can continue. In this part, you'll notice a correspondence between \(\Omega\)'s in \(\pi(\psi())\) and \(\Omega_2\)'s in \(\psi\).
THe first steps is to figure out the fundamental sequence of \(\varepsilon_0\) for transfinite values. for this I will use \(\varepsilon_0[\omega+1] = \omega^{\varepsilon_0[\omega]+1}\). Then \(\varepsilon_0[\omega2] = \varepsilon_0[\omega]\uparrow\uparrow\omega\).
So let's find \(\pi(\varepsilon_0+1)\):
| Pi notation | Normal ordinals |
|---|---|
| \(\pi(\varepsilon_0)\) | \(\psi(\varepsilon_{\Omega+1}) = \psi(\Omega_2)\) |
| \(\pi(\varepsilon_0,1)\) | \(\psi(\varepsilon_{\Omega+1}2) = \psi(\Omega_2+\psi_1(\Omega_2))\) |
| \(\pi(\varepsilon_0[\omega]+1)\) | \(\psi(\varepsilon_{\Omega+1}\Omega) = \psi(\Omega_2+\psi_1(\Omega_2+\Omega))\) |
| \(\pi(\varepsilon_0[\omega]+2)\) | \(\psi(\varepsilon_{\Omega+1}\Omega^2) = \psi(\Omega_2+\psi_1(\Omega_2+\Omega2))\) |
| \(\pi(\varepsilon_0[\omega]+\omega)\) | \(\psi(\varepsilon_{\Omega+1}\Omega^\omega) = \psi(\Omega_2+\psi_1(\Omega_2+\Omega\omega))\) |
| \(\pi(\varepsilon_0[\omega]+\omega+1)\) | \(\psi(\varepsilon_{\Omega+1}\Omega^\Omega) = \psi(\Omega_2+\psi_1(\Omega_2+\Omega^2))\) |
| \(\pi(\varepsilon_0[\omega]+\omega^2+1)\) | \(\psi(\varepsilon_{\Omega+1}\Omega^{\Omega^2}) = \psi(\Omega_2+\psi_1(\Omega_2+\Omega^3))\) |
| \(\pi(\varepsilon_0[\omega]+\omega^\omega+1)\) | \(\psi(\varepsilon_{\Omega+1}\Omega^{\Omega^\Omega}) = \psi(\Omega_2+\psi_1(\Omega_2+\Omega^\Omega))\) |
| \(\pi(\varepsilon_0[\omega]+\omega^{\omega^\omega}+1)\) | \(\psi(\varepsilon_{\Omega+1}\Omega^{\Omega^{\Omega^\Omega}}) = \psi(\Omega_2+\psi_1(\Omega_2+\Omega^{\Omega^\Omega}))\) |
| \(\pi(\varepsilon_0[\omega]2)\) | \(\psi(\varepsilon_{\Omega+1}^2) = \psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)))\) |
| \(\pi(\varepsilon_0[\omega]2+1)\) | \(\psi(\varepsilon_{\Omega+1}^2\Omega) = \psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)+\Omega))\) |
| \(\pi(\varepsilon_0[\omega]3)\) | \(\psi(\varepsilon_{\Omega+1}^3) = \psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)2))\) |
| \(\pi(\varepsilon_0[\omega]\omega) = \pi(\varepsilon_0[\omega+1])\) | \(\psi(\varepsilon_{\Omega+1}^\omega) = \psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+1)))\) |
| \(\pi(\varepsilon_0[\omega]\omega+1)\) | \(\psi(\varepsilon_{\Omega+1}^\Omega) = \psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\Omega)))\) |
| \(\pi(\varepsilon_0[\omega]\omega2+1)\) | \(\psi(\varepsilon_{\Omega+1}^{\Omega2}) = \psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+1)2))\) |
| \(\pi(\varepsilon_0[\omega]\omega^2+1)\) | \(\psi(\varepsilon_{\Omega+1}^{\Omega^2}) = \psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\Omega2)))\) |
| \(\pi(\varepsilon_0[\omega]\omega^\omega)\) | \(\psi(\varepsilon_{\Omega+1}^{\Omega^\omega}) = \psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\Omega\omega)))\) |
| \(\pi(\varepsilon_0[\omega]\omega^{\omega^\omega})\) | \(\psi(\varepsilon_{\Omega+1}^{\Omega^{\Omega^\omega}}) = \psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\Omega^\omega)))\) |
| \(\pi(\varepsilon_0[\omega]^2)\) | \(\psi(\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}}) = \psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2))))\) |
| \(\pi(\varepsilon_0[\omega]^2\omega)\) | \(\psi(\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}\omega}) = \psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)+\Omega)))\) |
| \(\pi(\varepsilon_0[\omega]^3)\) | \(\psi(\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}^2}) = \psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)2)))\) |
| \(\pi(\varepsilon_0[\omega]^\omega)\) | \(\psi(\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}^\omega}) = \psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+1))))\) |
| \(\pi(\varepsilon_0[\omega]^\omega+1)\) | \(\psi(\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}^\Omega}) = \psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\Omega))))\) |
| \(\pi(\varepsilon_0[\omega]^{\omega^\omega})\) | \(\psi(\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}^{\Omega^\omega}}) = \psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\Omega\omega))))\) |
| \(\pi(\varepsilon_0[\omega]^{\varepsilon_0[\omega]})\) | \(\psi(\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}}}) = \psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)))))\) |
| \(\pi(\varepsilon_0[\omega]^{\varepsilon_0[\omega]^{\varepsilon_0[\omega]}})\) | \(\psi(\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}}}}) = \psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2))))))\) |
| \(\pi(\varepsilon_0[\omega2])\) | \(\psi(\varepsilon_{\Omega+2}) = \psi(\Omega_22)\) |
| \(\pi(\varepsilon_0[\omega2]\omega)\) | \(\psi(\varepsilon_{\Omega+2}^\omega) = \psi(\Omega_22+\psi_1(\Omega_22+1))\) |
| \(\pi(\varepsilon_0[\omega3])\) | \(\psi(\varepsilon_{\Omega+3}) = \psi(\Omega_23)\) |
| \(\pi(\varepsilon_0[\omega^2])\) | \(\psi(\varepsilon_{\Omega+\omega}) = \psi(\Omega_2\omega)\) |
| \(\pi(\varepsilon_0[\omega^\omega])\) | \(\psi(\varepsilon_{\Omega+\omega^\omega}) = \psi(\Omega_2\omega^\omega)\) |
| \(\pi(\varepsilon_0[\varepsilon_0])\) | \(\psi(\varepsilon_{\Omega+\varepsilon_0}) = \psi(\Omega_2\psi(\Omega))\) |
| \(\pi(\varepsilon_0[\pi(\varepsilon_0)])\) | \(\psi(\varepsilon_{\Omega+\psi(\varepsilon_{\Omega+1})}) = \psi(\Omega_2\psi(\Omega_2))\) |
so \(\pi(\varepsilon_0+1) = \psi(\varepsilon_{\Omega2})\).
Now to continue. This is relatively simple.
| Pi notation | Normal ordinals |
|---|---|
| \(\pi(\varepsilon_0+1)\) | \(\psi(\varepsilon_{\Omega2}) = \psi(\Omega_2\Omega)\) |
| \(\pi(\varepsilon_0+1,1)\) | \(\psi(\varepsilon_{\Omega2}2) = \psi(\Omega_2\Omega+\psi_1(\Omega_2\Omega))\) |
| \(\pi(\varepsilon_0+2)\) | \(\psi(\varepsilon_{\Omega2}\Omega) = \psi(\Omega_22+\psi_1(\Omega_22+\Omega))\) |
| \(\pi(\varepsilon_0+\omega)\) | \(\psi(\varepsilon_{\Omega2}\Omega^\omega) = \psi(\Omega_22+\psi_1(\Omega_22+\Omega\omega))\) |
| \(\pi(\varepsilon_0+\omega+1)\) | \(\psi(\varepsilon_{\Omega2}\Omega^\Omega) = \psi(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\Omega^2))\) |
| \(\pi(\varepsilon_0+\omega^\omega+1)\) | \(\psi(\varepsilon_{\Omega2}\Omega^{\Omega^\Omega}) = \psi(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\Omega^\Omega))\) |
| \(\pi(\varepsilon_02)\) | \(\psi(\varepsilon_{\Omega2}\varepsilon_{\Omega+1}) = \psi(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\psi_1(\Omega_2)))\) |
| \(\pi(\varepsilon_02+1)\) | \(\psi(\varepsilon_{\Omega2}^2) = \psi(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\psi_1(\Omega_2\Omega)))\) |
| \(\pi(\varepsilon_03+1)\) | \(\psi(\varepsilon_{\Omega2}^3) = \psi(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\psi_1(\Omega_2\Omega)2))\) |
| \(\pi(\varepsilon_0\omega)\) | \(\psi(\varepsilon_{\Omega2}^\omega) = \psi(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\psi_1(\Omega_2\Omega+1)))\) |
| \(\pi(\varepsilon_0\omega+1)\) | \(\psi(\varepsilon_{\Omega2}^\Omega) = \psi(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\Omega)))\) |
| \(\pi(\varepsilon_0\omega^2+1)\) | \(\psi(\varepsilon_{\Omega2}^{\Omega^2}) = \psi(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\psi_1(\Omega_2+\Omega2)))\) |
| \(\pi(\varepsilon_0\omega^\omega+1)\) | \(\psi(\varepsilon_{\Omega2}^{\Omega^\Omega}) = \psi(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\Omega^2)))\) |
| \(\pi(\varepsilon_0^2)\) | \(\psi(\varepsilon_{\Omega2}^{\varepsilon_{\Omega+1}}) = \psi(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\psi_1(\Omega_2))))\) |
| \(\pi(\varepsilon_0^2+1)\) | \(\psi(\varepsilon_{\Omega2}^{\varepsilon_{\Omega2}}) = \psi(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\psi_1(\Omega_2\Omega))))\) |
| \(\pi(\varepsilon_0^3+1)\) | \(\psi(\varepsilon_{\Omega2}^{\varepsilon_{\Omega2}^2}) = \psi(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\psi_1(\Omega_2\Omega)2)))\) |
| \(\pi(\varepsilon_0^\omega+1)\) | \(\psi(\varepsilon_{\Omega2}^{\varepsilon_{\Omega2}^\Omega}) = \psi(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\Omega))))\) |
| \(\pi(\varepsilon_0^{\varepsilon_0})\) | \(\psi(\varepsilon_{\Omega2}^{\varepsilon_{\Omega2}^{\varepsilon_{\Omega+1}}}) = \psi(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\psi_1(\Omega_2\Omega+\psi_1(\Omega_2)))))\) |
| \(\pi(\varepsilon_0^{\varepsilon_0^{\varepsilon_0}})\) | \(\psi(\varepsilon_{\Omega2}^{\varepsilon_{\Omega2}^{\varepsilon_{\Omega2}^{\varepsilon_{\Omega+1}}}})\) |
| \(\pi(\varepsilon_1)\) | \(\psi(\varepsilon_{\Omega2+1}) = \psi(\Omega_2\Omega+\Omega_2)\) |
| \(\pi(\varepsilon_1[\omega2])\) | \(\psi(\varepsilon_{\Omega2+2}) = \psi(\Omega_2\Omega+\Omega_22)\) |
| \(\pi(\varepsilon_1+1)\) | \(\psi(\varepsilon_{\Omega3}) = \psi(\Omega_2\Omega2)\) |
| \(\pi(\varepsilon_12+1)\) | \(\psi(\varepsilon_{\Omega3}^2) = \psi(\Omega_2\Omega2+\psi_1(\Omega_2\Omega2+\psi_1(\Omega_2\Omega2)))\) |
| \(\pi(\varepsilon_1^2+1)\) | \(\psi(\varepsilon_{\Omega3}^{\varepsilon_{\Omega3}}) = \psi(\Omega_2\Omega2)+\psi_1(\Omega_2+\Omega2+\psi_1(\Omega_2\Omega2+\psi_1(\Omega_2\Omega2))))\) |
| \(\pi(\varepsilon_2)\) | \(\psi(\varepsilon_{\Omega3+1}) = \psi(\Omega_2\Omega2+\Omega_2)\) |
| \(\pi(\varepsilon_3)\) | \(\psi(\varepsilon_{\Omega4+1}) = \psi(\Omega_2\Omega3+\Omega_2)\) |
| \(\pi(\varepsilon_\omega)\) | \(\psi(\varepsilon_{\Omega\omega}) = \psi(\Omega_2\Omega\omega)\) |
| \(\pi(\varepsilon_\omega+1)\) | \(\psi(\varepsilon_{\Omega^2}) = \psi(\Omega_2\Omega^2)\) |
| \(\pi(\varepsilon_{\omega+1})\) | \(\psi(\varepsilon_{\Omega^2+1}) = \psi(\Omega_2\Omega^2+\Omega_2)\) |
| \(\pi(\varepsilon_{\omega2}+1)\) | \(\psi(\varepsilon_{\Omega^22}) = \psi(\Omega_2\Omega^22)\) |
| \(\pi(\varepsilon_{\omega^2})\) | \(\psi(\varepsilon_{\Omega^2\omega}) = \psi(\Omega_2\Omega^2\omega)\) |
| \(\pi(\varepsilon_{\omega^2}+1)\) | \(\psi(\varepsilon_{\Omega^3}) = \psi(\Omega_2\Omega^3)\) |
| \(\pi(\varepsilon_{\omega^3}+1)\) | \(\psi(\varepsilon_{\Omega^4}) = \psi(\Omega_2\Omega^4)\) |
| \(\pi(\varepsilon_{\omega^\omega})\) | \(\psi(\varepsilon_{\Omega^\omega}) = \psi(\Omega_2\Omega^\omega)\) |
| \(\pi(\varepsilon_{\omega^\omega}+1)\) | \(\psi(\varepsilon_{\Omega^\Omega}) = \psi(\Omega_2\Omega^\Omega)\) |
| \(\pi(\varepsilon_{\omega^{\omega^\omega}}+1)\) | \(\psi(\varepsilon_{\Omega^{\Omega^\Omega}}) = \psi(\Omega_2\Omega^{\Omega^\Omega})\) |
| \(\pi(\varepsilon_{\varepsilon_0})\) | \(\psi(\varepsilon_{\varepsilon_{\Omega+1}}) = \psi(\Omega_2\psi_1(\Omega_2))\) |
| \(\pi(\varepsilon_{\varepsilon_0}+1)\) | \(\psi(\varepsilon_{\varepsilon_{\Omega2}}) = \psi(\Omega_2\psi_1(\Omega_2\Omega))\) |
| \(\pi(\varepsilon_{\varepsilon_1})\) | \(\psi(\varepsilon_{\varepsilon_{\Omega2+1}}) = \psi(\omega_2\psi_1(\Omega_2\Omega+\Omega_2))\) |
| \(\pi(\varepsilon_{\varepsilon_{\varepsilon_0}})\) | \(\psi(\varepsilon_{\varepsilon_{\varepsilon_{\Omega+1}}}) = \psi(\Omega_2\psi_1(\Omega_2\psi_1(\Omega_2)))\) |
| \(\pi(\zeta_0) = \pi(\psi(\Omega^2))\) | \(\psi(\zeta_{\Omega+1}) = \psi(\Omega_2^2)\) |
| \(\pi(\zeta_0+1)\) | \(\psi(\zeta_{\Omega2}) = \psi(\Omega_2^2\Omega)\) |
| \(\pi(\zeta_02+1)\) | \(\psi(\zeta_{\Omega2}^2) = \psi(\Omega_2^2\Omega+\psi_1(\Omega_2^2\Omega+\psi_1(\Omega_2^2\Omega)))\) |
| \(\pi(\varepsilon_{\zeta_0+1})\) | \(\psi(\varepsilon_{\zeta_{\Omega2}+1}) = \psi(\Omega_2^2\Omega+\Omega_2)\) |
| \(\pi(\varepsilon_{\zeta_0+1}+1)\) | \(\psi(\varepsilon_{\zeta_{\Omega2}+\Omega}) = \psi(\Omega_2^2\Omega+\Omega_2\Omega)\) |
| \(\pi(\varepsilon_{\zeta_02}+1)\) | \(\psi(\varepsilon_{\zeta_{\Omega2}2}) = \psi(\Omega_2^2\Omega+\Omega_2\psi_1(\Omega_2^2\Omega))\) |
| \(\pi(\zeta_1)\) | \(\psi(\zeta_{\Omega2+1}) = \psi(\Omega_2^2\Omega+\Omega_2^2)\) |
| \(\pi(\zeta_1+1)\) | \(\psi(\zeta_{\Omega3}) = \psi(\Omega_2^2\Omega2)\) |
| \(\pi(\zeta_2+1)\) | \(\psi(\zeta_{\Omega4}) = \psi(\Omega_2^2\Omega3)\) |
| \(\pi(\zeta_\omega)\) | \(\psi(\zeta_{\Omega\omega}) = \psi(\Omega_2^2\Omega\omega)\) |
| \(\pi(\zeta_\omega+1)\) | \(\psi(\zeta_{\Omega^2}) = \psi(\Omega_2^2\Omega^2)\) |
| \(\pi(\zeta_{\omega^\omega}+1)\) | \(\psi(\zeta_{\Omega^\Omega}) = \psi(\Omega_2^2\Omega^\omega)\) |
| \(\pi(\zeta_{\varepsilon_0})\) | \(\psi(\zeta_{\varepsilon_{\Omega+1}}) = \psi(\Omega_2^2\psi_1(\Omega_2))\) |
| \(\pi(\zeta_{\zeta_0})\) | \(\psi(\zeta_{\zeta_{\Omega+1}}) = \psi_1(\Omega_2^2\psi_1(\Omega_2^2))\) |
| \(\pi(\varphi(3,0))\) | \(\psi(\varphi(3,\Omega+1)) = \psi(\Omega_2^3)\) |
| \(\pi(\varphi(3,0)+1)\) | \(\psi(\varphi(3,\Omega2)) = \psi(\Omega_2^3\Omega)\) |
| \(\pi(\varphi(3,1))\) | \(\psi(\varphi(3,\Omega2+1)) = \psi(\Omega_2^3\Omega+\Omega_2^3)\) |
| \(\pi(\varphi(4,0))\) | \(\psi(\varphi(4,\Omega+1)) = \psi(\Omega_2^4)\) |
| \(\pi(\varphi(\omega,0))\) | \(\psi(\varphi(\omega,\Omega+1)) = \psi(\Omega_2^\omega)\) |
| \(\pi(\varphi(\omega[\omega],1))\) | \(\psi(\varphi(\omega,\Omega+2)) = \psi(\Omega_2^\omega2)\) |
| \(\pi(\varphi(\omega,0)[\omega+1])\) | \(\psi(\varphi(\omega,\Omega2)) = \psi(\Omega_2^\omega\Omega)\) |
| \(\pi(\varphi(\omega[\omega],\varphi(\omega,0)[\omega+1]+1)\) | \(\psi(\varphi(\omega,\Omega2+1)) = \psi(\Omega_2^\omega\Omega+\Omega_2^\omega)\) |
| \(\pi(\varphi(\omega[\omega+1],1))\) | \(\psi(\varphi(\omega,\Omega3)) = \psi(\Omega_2^\omega\Omega2)\) |
| \(\pi(\varphi(\omega,0)[\omega+2])\) | \(\psi(\varphi(\omega+1,\Omega+1)) = \psi(\Omega_2^{\omega+1})\) |
| \(\pi(\varphi(\omega,0)[\omega2])\) | \(\psi(\varphi(\omega2,\Omega+1)) = \psi(\Omega_2^{\omega2})\) |
| \(\pi(\varphi(\omega,0)+1)\) | \(\psi(\varphi(\Omega,1)) = \psi(\Omega_2^\Omega)\) |
| \(\pi(\varphi(\omega,1))\) | \(\psi(\varphi(\omega,\varphi(\Omega,1)+1)) = \psi(\Omega_2^\Omega+\Omega_2^\omega)\) |
| \(\pi(\varphi(\omega,1)+1)\) | \(\psi(\varphi(\Omega,2)) = \psi(\Omega_2^\Omega2)\) |
| \(\pi(\varphi(\omega,\omega)+1)\) | \(\psi(\varphi(\Omega,\Omega)) = \psi(\Omega_2^\Omega\Omega)\) |
| \(\pi(\varphi(\omega,\varphi(\omega,0))+1)\) | \(\psi(\varphi(\Omega,\varphi(\Omega,1))) = \psi(\Omega_2^\Omega\psi_1(\Omega_2^\omega))\) |
| \(\pi(\varphi(\omega+1,0))\) | \(\psi(\varphi(\Omega+1,0)) = \psi(\Omega_2^{\Omega+1})\) |
| \(\pi(\varphi(\omega+2,0))\) | \(\psi(\varphi(\Omega+2,0)) = \psi(\Omega_2^{\Omega+2})\) |
| \(\pi(\varphi(\omega2,0))\) | \(\psi(\varphi(\Omega+\omega,0)) = \psi(\Omega_2^{\Omega+\omega})\) |
| \(\pi(\varphi(\omega2,0)+1)\) | \(\psi(\varphi(\Omega2,0)) = \psi(\Omega_2^{\Omega2})\) |
| \(\pi(\varphi(\omega^2,0))\) | \(\psi(\varphi(\Omega\omega,0)) = \psi(\Omega_2^{\Omega\omega})\) |
| \(\pi(\varphi(\omega^2,0)+1)\) | \(\psi(\varphi(\Omega^2,0)) = \psi(\Omega_2^{\Omega^2})\) |
| \(\pi(\varphi(\omega^\omega,0))\) | \(\psi(\varphi(\Omega^\omega,0)) = \psi(\Omega_2^{\Omega^\omega})\) |
| \(\pi(\varphi(\varepsilon_0,0))\) | \(\psi(\varphi(\varepsilon_{\Omega+1},0)) = \psi(\Omega_2^{\psi_1(\Omega_2)})\) |
| \(\pi(\varphi(\zeta_0,0))\) | \(\psi(\varphi(\zeta_{\Omega+1},0)) = \psi(\Omega_2^{\psi_1(\Omega_2^2)})\) |
| \(\pi(\varphi(\varphi(\omega,0),0))\) | \(\psi(\varphi(\varphi(\omega,\Omega+1),0)) = \psi(\Omega_2^{\psi_1(\Omega_2^\omega)})\) |
| \(\pi(\Gamma_0) = \pi(\psi(\Omega^\Omega))\) | \(\psi(\Omega_2^{\Omega_2})\) |
| \(\pi(\psi(\Omega^\Omega)+1)\) | \(\psi(\Omega_2^{\Omega_2}\Omega)\) |
| \(\pi(\psi(\Omega^\Omega+\Omega^\omega))\) | \(\psi(\Omega_2^{\Omega_2}\Omega+\Omega_2^\omega)\) |
| \(\pi(\psi(\Omega^\Omega+\Omega^{\Gamma_0}))\) | \(\psi(\Omega_2^{\Omega_2}+\Omega_2^{\psi_1(\Omega_2^{\Omega_2)}})\) |
| \(\pi(\psi(\Omega^\Omega2))\) | \(\psi(\Omega_2^{\Omega_2}\Omega+\Omega_2^{\Omega_2})\) |
| \(\pi(\psi(\Omega^\Omega\omega))\) | \(\psi(\Omega_2^{\Omega_2}\Omega\omega)\) |
| \(\pi(\psi(\Omega^{\Omega+1}))\) | \(\psi(\Omega_2^{\Omega_2+1})\) |
| \(\pi(\psi(\Omega^{\Omega+2}))\) | \(\psi(\Omega_2^{\Omega_2+2})\) |
| \(\pi(\psi(\Omega^{\Omega+\omega}))\) | \(\psi(\Omega_2^{\Omega_2+\omega})\) |
| \(\pi(\psi(\Omega^{\Omega+\omega})+1)\) | \(\psi(\Omega_2^{\Omega_2+\Omega})\) |
| \(\pi(\psi(\Omega^{\Omega2}))\) | \(\psi(\Omega_2^{\Omega_22})\) |
| \(\pi(\psi(\Omega^{\Omega3}))\) | \(\psi(\Omega_2^{\Omega_23})\) |
| \(\pi(\psi(\Omega^{\Omega\omega}))\) | \(\psi(\Omega_2^{\Omega_2\omega})\) |
| \(\pi(\psi(\Omega^{\Omega^2}))\) | \(\psi(\Omega_2^{\Omega_2^2})\) |
| \(\pi(\psi(\Omega^{\Omega^3}))\) | \(\psi(\Omega_2^{\Omega_2^3})\) |
| \(\pi(\psi(\Omega^{\Omega^\omega}))\) | \(\psi(\Omega_2^{\Omega_2^\omega})\) |
| \(\pi(\psi(\Omega^{\Omega^\Omega}))\) | \(\psi(\Omega_2^{\Omega_2^{\Omega_2}})\) |
| \(\pi(\psi(\Omega^{\Omega^{\Omega^\Omega}}))\) | \(\psi(\Omega_2^{\Omega_2^{\Omega_2^{\Omega_2}}})\) |
So \(\pi(\psi(\Omega_2)) = \psi(\Omega_3)\).
Up to \(\alpha=\pi(\alpha)\)
This part should be fairly simple if you understood everything so far.
| Pi notation | Normal ordinals |
|---|---|
| \(\pi(\psi(\varepsilon_{\Omega+1})) = \psi(\Omega_2)\) | \(\psi(\varepsilon_{\Omega_2+1}) = \psi(\Omega_3)\) |
| \(\pi(\psi(\varepsilon_{\Omega+1})[\omega2])\) | \(\psi(\varepsilon_{\Omega_2+2}) = \psi(\Omega_32)\) |
| \(\pi(\psi(\varepsilon_{\Omega+1})+1)\) | \(\psi(\varepsilon_{\Omega_2+\Omega}) = \psi(\Omega_3\Omega)\) |
| \(\pi(\psi(\varepsilon_{\Omega+1})+2)\) | \(\psi(\varepsilon_{\Omega_2+\Omega}\Omega) = \psi(\Omega_3\Omega+\psi_1(\Omega_3\Omega+\Omega))\) |
| \(\pi(\psi(\varepsilon_{\Omega+1})2+1)\) | \(\psi(\varepsilon_{\Omega_2+\Omega}^2) = \psi(\Omega_3\Omega+\psi_1(\Omega_3\Omega+\psi_1(\Omega_3\Omega)))\) |
| \(\pi(\psi(\varepsilon_{\Omega+1}+\Omega)) = \pi(\psi(\Omega_2+\Omega))\) | \(\psi(\varepsilon_{\Omega_2+\Omega}+\Omega_2) = \psi(\Omega_3\Omega+\Omega_2)\) |
| \(\pi(\psi(\varepsilon_{\Omega+1}+\Omega^\Omega)) = \pi(\psi(\Omega_2+\Omega^\Omega))\) | \(\psi(\varepsilon_{\Omega_2+\Omega}+\Omega_2^{\Omega_2}) = \psi(\Omega_3\Omega+\Omega_2^{\Omega_2})\) |
| \(\pi(\psi(\varepsilon_{\Omega+1}2)) = \psi(\Omega_2+\psi_1(\Omega_2))\) | \(\psi(\varepsilon_{\Omega_2+\Omega}+\varepsilon_{\Omega_2+1}) = \psi(\Omega_3\Omega+\psi_2(\Omega_3))\) |
| \(\pi(\psi(\varepsilon_{\Omega+1}2)+1)\) | \(\psi(\varepsilon_{\Omega_2+\Omega}2) = \psi(\Omega_3\Omega+\psi_2(\Omega_3\Omega))\) |
| \(\pi(\psi(\varepsilon_{\Omega+1}^2)+1)\) | \(\psi(\varepsilon_{\Omega_2+\Omega}^2) = \psi(\Omega_3\Omega+\psi_2(\Omega_3\Omega+\psi_2(\Omega_3\Omega)))\) |
| \(\pi(\psi(\varepsilon_{\Omega+2})) = \pi(\psi(\Omega_22))\) | \(\psi(\varepsilon_{\Omega_2+\Omega+1}) = \psi(\Omega_3\Omega+\Omega_3)\) |
| \(\pi(\psi(\varepsilon_{\Omega+2})+1)\) | \(\psi(\varepsilon_{\Omega_2+\Omega2}) = \psi(\Omega_3\Omega2)\) |
| \(\pi(\psi(\varepsilon_{\Omega+3})) = \pi(\psi(\Omega_23))\) | \(\psi(\varepsilon_{\Omega_2+\Omega2+1}) = \psi(\Omega_3\Omega2+\Omega_3)\) |
| \(\pi(\psi(\varepsilon_{\Omega+\omega})) = \pi(\psi(\Omega_2\omega))\) | \(\psi(\varepsilon_{\Omega_2+\Omega\omega}) = \psi(\Omega_3\Omega\omega)\) |
| \(\pi(\psi(\varepsilon_{\Omega2})) = \pi(\psi(\Omega_2\Omega))\) | \(\psi(\varepsilon_{\Omega_22}) = \psi(\Omega_3\Omega_2)\) |
| \(\pi(\psi(\varepsilon_{\Omega2})+1)\) | \(\psi(\varepsilon_{\Omega_22+\Omega}) = \psi(\Omega_3\Omega_2+\Omega_3\Omega)\) |
| \(\pi(\psi(\varepsilon_{\Omega3})) = \pi(\psi(\Omega_2\Omega2))\) | \(\psi(\varepsilon_{\Omega_23}) = \psi(\Omega_3\Omega_22)\) |
| \(\pi(\psi(\varepsilon_{\Omega\omega})) = \pi(\psi(\Omega_2\Omega\omega))\) | \(\psi(\varepsilon_{\Omega_2\omega}) = \psi(\Omega_3\Omega_2\omega)\) |
| \(\pi(\psi(\varepsilon_{\Omega^2})) = \pi(\psi(\Omega_2\Omega^2))\) | \(\psi(\varepsilon_{\Omega_2^2}) = \psi(\Omega_3\Omega_2^2)\) |
| \(\pi(\psi(\varepsilon_{\Omega^\Omega})) = \pi(\psi(\Omega_2\Omega^\Omega))\) | \(\psi(\varepsilon_{\Omega_2^{\Omega_2}}) = \psi(\Omega_3\Omega_2^{\Omega_2})\) |
| \(\pi(\psi(\varepsilon_{\varepsilon_{\Omega+1}})) = \pi(\psi(\Omega_2\psi_1(\Omega_2)))\) | \(\psi(\varepsilon_{\varepsilon_{\Omega_2+1}}) = \psi(\Omega_3\psi_2(\Omega_3))\) |
| \(\pi(\psi(\zeta_{\Omega+1})) = \pi(\psi(\Omega_2^2))\) | \(\psi(\zeta_{\Omega_2+1}) = \psi(\Omega_3^2)\) |
| \(\pi(\psi(\Omega_2^2)+1)\) | \(\psi(\Omega_3^2\Omega)\) |
| \(\pi(\psi(\Omega_2^2+\Omega_2))\) | \(\psi(\Omega_3^2\Omega+\Omega_3)\) |
| \(\pi(\psi(\Omega_2^22))\) | \(\psi(\Omega_3^2\Omega+\Omega_3^2)\) |
| \(\pi(\psi(\Omega_2^2\Omega))\) | \(\psi(\Omega_3^2\Omega_2)\) |
| \(\pi(\psi(\Omega_2^3))\) | \(\psi(\Omega_3^3)\) |
| \(\pi(\psi(\Omega_2^\omega))\) | \(\psi(\Omega_3^\omega)\) |
| \(\pi(\psi(\Omega_2^\Omega))\) | \(\psi(\Omega_3^{\Omega_2})\) |
| \(\pi(\psi(\Omega_2^{\Omega_2}))\) | \(\psi(\Omega_3^{\Omega_3})\) |
| \(\pi(\psi(\Omega_2^{\Omega_2+1}))\) | \(\psi(\Omega_3^{\Omega_3+1})\) |
| \(\pi(\psi(\Omega_2^{\Omega_22}))\) | \(\psi(\Omega_3^{\Omega_32})\) |
| \(\pi(\psi(\Omega_2^{\Omega_2^2}))\) | \(\psi(\Omega_3^{\Omega_3^2})\) |
| \(\pi(\psi(\Omega_2^{\Omega_2^{\Omega_2}}))\) | \(\psi(\Omega_3^{\Omega_3^{\Omega_3}})\) |
| \(\pi(\psi(\varepsilon_{\Omega_2+1})) = \pi(\psi(\Omega_3))\) | \(\psi(\varepsilon_{\Omega_3+1}) = \psi(\Omega_4)\) |
| \(\pi(\psi(\Omega_3\Omega_2))\) | \(\psi(\Omega_4\Omega_3))\) |
| \(\pi(\psi(\Omega_3^2))\) | \(\psi(\Omega_4^2)\) |
| \(\pi(\psi(\Omega_3^3))\) | \(\psi(\Omega_4^3)\) |
| \(\pi(\psi(\Omega_3^{\Omega_3}))\) | \(\psi(\Omega_4^{\Omega_4})\) |
| \(\pi(\psi(\varepsilon_{\Omega_3+1})) = \pi(\psi(\Omega_4))\) | \(\psi(\varepsilon_{\Omega_4+1}) = \psi(\Omega_5)\) |
| \(\pi(\psi(\Omega_5))\) | \(\psi(\Omega_6)\) |
The pattern is now obvious: an \(\Omega_n\) in pi notation corresponds to a \(\Omega_{n+1}\) in normal notation. And the limit of the first part of pi notation is \(\psi(\Omega_\omega)\).
But pi notation can be extended easily to produce much higher ordinals...