User blog comment:Edwin Shade/How do you evaluate extended Veblen notation ?/@comment-27516045-20171020195929

Here is a basic explanation:

First, note that what you write as \(\varphi_\alpha(\beta)\) \(\varphi(\alpha,\beta)\). You should also understand fixed points - a fixed point of an ordinal function f is an ordinal such that \(\alpha = f(\alpha)\). So the first fixed point of \(\alpha = \omega^\alpha\) is \(\varepsilon_0\).

The \(\Gamma_\alpha\) ordinals are fixed points of \(\alpha = \varphi(\alpha,0)\). And note that the limit of \(\varphi(\alpha,\Gamma_0+1)\) for \(\alpha < \Gamma_0)\) is \(\varphi(\Gamma_0,1)\).

Then, \(\Gamma_\alpha = \varphi(1,0,\alpha)\). And then \(\varphi(1,1,\alpha)\) enumerates fixed points of \(\alpha = \Gamma_\alpha\). To find fixed points of \(\alpha = \varphi(1,\alpha,0)\), use \(\varphi(2,0,\alpha)\). and so on.

In general, what you do to get fundamental sequnces is to start scanning from the ) right end, noting the last entry (call it \(\beta\)) and finding the first argument that's not 0. what you do is dependant on if that entry (let's call is \(\alpha\)) is a successor ordinal (it can be written as \(\alpha+1\) for some ordinal) or a limit ordinal (it can't). And note that if \(\beta\) itself is a limit ordinal, then the fundamental sequence is just the same thing but with the last argument replaced by the ordinals of its fundmanetal sequence.

If \(\alpha\) is a successor, then the ordinal is the \(1+\beta\)th fixed point of \(\alpha = f(x)\). To find f, what you do is decrease \(\alpha\) by 1, and set the next entry (which should be a 0) to \(\alpha\).

If \(\alpha\0 is a limit, what you do is use the members of its ffundamental sequence. If \(\beta\0 is 0, leave it there, otherwise replace the last entry by the original ordinal with the last entry decreased by, +1.

Here are some major ordinals in this notation:


 * \(\Gamma_0\)
 * \(\varphi(\Gamma_0,1)\) - limit of \(\varphi(\alpha,\gamma_0+1\) for \(\alpha < \gamma_0\)
 * \(\varphi(\Gamma_0+1,0)\) - first fixed point of \(\alpha = \varphi(\Gamma_0,\alpha)\)
 * \(\Gamma_1\) - second fixed point of \(\alpha = \varphi(\alpha,0)\) (after \(\Gamma_0\))
 * \(\Gamma_\omega\) - limit of \(\Gamma_n\)
 * \(\varphi(1,1,0)\) - first fixed point of \(\alpha = \Gamma_\alpha\)
 * \(\Gamma_{\varphi(1,1,0)+1}\) - the next fixed point of \(\alpha = \varphi(\alpha,0)\) after \(\varphi(1,1,0)
 * \(\varphi(1,2,0)\) - the first fixed point of \(\alpha = \varphi(1,1,\alpha)\)
 * \(\varphi(1,\omega,0)\) - the limit of \(\varphi(1,n,0)\)
 * \(\varphi(1,\omega,1)\) - the limit of \(\varphi(1,n,\varphi(1,\omega,0)+1)\)
 * \(\varphi(2,0,0)\) - the first fixed point of \(\alpha = \varphi(1,\alpha,0)\)
 * \(\varphi(2,0,1)\) - the second fixed point of \(\alpha = \varphi(1,\alpha,0)\)
 * \(\varphi(3,0,0)\) - the first fixed point of \(\alpha = \varphi(2,\alpha,0)\)
 * \(\varphi(\omega,0,0)\) - the limit of \(\varphi(n,0,0)\)
 * \(\varphi(\omega,0,1)\) - the limit of \(\varphi(n,0,\varphi(\omega,0,0)+1)\)
 * \(\varphi(\omega,1,0)\) - the first fixed point of \(\alpha = \varphi(\omega,0,\alpha)\)
 * \(\varphi(1,0,0,0)\) - the first fixed point of \(\alpha = \varphi(\alpha,0,0)\)
 * \(\varphi(1,1,0,0)\) - the first fixed point of \(\alpha = \varphi(1,0,\alpha,0)\)
 * \(\varphi(2,0,0,0)\) - the first fixed point of \(\alpha = \varphi(1,\alpha,0,0)\)
 * \(\varphi(1,0,0,0,0)\) - the first fixed point of \(\alpha = \varphi(\alpha,0,0,0)\)

the limit of \(\varphii(1,0,0,0,0,\cdots\) is an ordinal known as the small veblen ordinal. and this is the limit of extended veblen notation. I think there is an extension to allow infinite arguments, but it's not something I use commonly. if you want me to explain this to you, I will.

Does this make sense to you so far?