User blog:Edwin Shade/How do you evaluate extended Veblen notation ?

So a while back I made a blog post expressing the fact that I had learned about Veblen notation and extended Veblen functions.

To let you know where I am at the moment, I will explain what I have learned about ordinals so far; if there be any mistakes in my understanding please tell me so I can correct them.

What I Know So Far
$$\omega$$ is the first infinite ordinal, which when placed in the subscript the fast-growing hierarchy is to be replaced with the input of the fast-growing hierarchy, as the fundamental sequence of $$\omega$$ is $$1, 2, 3, 4, 5, 6, . . .$$

$$\omega$$ may be formally defined as the first ordinal $$\gamma$$ such that $$\gamma=\gamma+1$$.

$$\epsilon_0$$ is the first ordinal not expressible as a finite expression involving members of the set $$\{1,\omega \}$$ when added, multiplied, or raised to the power of another member.

The fundamental sequence of $$\epsilon_0$$ is $$\omega,\omega^{\omega},\omega^{\omega^{\omega}},\omega^{\omega^{\omega^{\omega}}},\omega^{\omega^{\omega^{\omega^{\omega}}}},. . .$$. $$\epsilon_0$$ can be formally defined as the first ordinal $$\gamma$$ such that $$\gamma =\omega^{\gamma}$$.

$$\epsilon_1$$ is the first ordinal not expressive as a finite expression involving members of the set $$\{1, \omega, \epsilon_0 \}$$ when added, multiplied, or raised to the power of another member.

It may be described as the first ordinal $$\gamma$$ such that $$\gamma =\epsilon_0^{\gamma}$$. It's fundamental sequence is $$\epsilon_0,{\epsilon_0}^{\epsilon_0},{\epsilon_0^{{\epsilon_0}^{\epsilon_0}}}, {\epsilon_0^{{\epsilon_0}^{{\epsilon_0}^{\epsilon_0}}}},. . .$$

In this way we may continue to define the epsilons, as the limit of the power tower of the preceding epsilon. $$\epsilon_{37}$$ is limit of $$\epsilon_{36}^{\epsilon_{36}^{\epsilon_{36}^{.^{.^{.}}}}}$$, for example.

We may insert any previously constructed ordinal in the subscript of $$\epsilon$$, so we can have limit ordinals such as $$\epsilon_{\epsilon_0}$$ or $$\epsilon_{{\epsilon_0}+{\omega^77}+5*{\omega^7}}$$. The limit of the epsilons is the first ordinal $$\gamma$$ such that $$\gamma =\epsilon_{\gamma}$$.

This new ordinal is known as $$\zeta_0$$ and it's fundamental sequence is $$\epsilon_0,\epsilon_{\epsilon_0},\epsilon_{\epsilon_{\epsilon_0}},\epsilon_{\epsilon_{\epsilon_{\epsilon_0}}},. . .$$

We may define a farther ordinal than zeta, (ordinals are different than cardinals so I don't feel the term "larger" is really appropriate if all countable ordinals have the same carnality; thus I say farther instead), $$\eta_0$$, as the first ordinal $$\gamma$$ such that $$\gamma =\eta_{\gamma}$$. $$\eta_0$$'s fundamental's sequence is $$\zeta_0,\zeta_{\zeta_0},\zeta_{\zeta_{\zeta_0}},\zeta_{\zeta_{\zeta_{\zeta_0}}},. . .$$

Now of course, the Greek alphabet runs out of letters and so a function to diagonalize over the letters themselves is needed. This is the Veblen function, expressed as $$\varphi_{a}(b)$$, which defines the ordinal $$L_{b}$$, where L is the ath letter after $$\omega$$, beginning with $$\epsilon$$.

The rules for inputs of the Veblen function which involve successor ordinals and limit ordinals are below. $$\alpha$$ stands for a successor ordinal, and $$\beta$$ stands for a limit ordinal.

$$\varphi_{0}(0)=1$$

$$\varphi_{\alpha}(0)[n]={{\varphi_{{\alpha}-1}}^n}(0)$$

$$\varphi_{\alpha}(a)[n]={{\varphi_{{\alpha}-1}}^n}({\varphi_{\alpha}(a-1)}+1)$$

$$\varphi_{\beta}(0)[n]=\varphi_{n}(0)$$

$$\varphi_{\beta}(a)[n]=\varphi_{n}(\varphi_{\beta}(a-1)+1)$$

Next, there is extended-Veblen notation, which utilizes multiple-argument functions.

The smallest ordinal $$\gamma$$ such that $$\varphi_{\gamma}(0)=\gamma$$ is known as $$\Gamma_0$$, and is represented as $$\varphi (1,0,0)$$ in extended-Veblen notation. It is also known as the Feferman-Schutte ordinal.

What I Don't Know So Far
This is the limit of my current comprehension on ordinals, as I cannot understand how to evaluate multiple argument Veblen-functions with a set of rules like the one I provided above for two-argument Veblen functions.

Can someone please share this general set of equations, and also explain what the Schutte Klammersymbolen is ? Deedlit11 made a blog post on it that I've read, but I can't fully grasp it because of the terminology used. I know I would be able understand it though if it were explained simply yet completely.

Leave a comment below.