I try to understand the Löb-Wainer-hierarchy and one definition just doesn't open. I hope someone could clarify this to me.
A fundamental sequence to limit ordinal $\alpha$ is $\omega$-sequence $\{\alpha_i\}_{i\in\mathbb{N}}$ of ordinals such that for each $i\in\mathbb{N}$ we have $\alpha_i<\alpha_{i+1}<\alpha$ and $\lim_{i\in\mathbb{N}}\alpha_i=\alpha$.
For each countable limit ordinal $\beta$, let $\{\beta\}(i)$,$i\in\mathbb{N}$ denote a arbitrarily chosen fixed fundamental sequence to $\beta$. Define now $F_\alpha^n$ such that
- $F_0^n(x)=(n+1)(x+1)$
- $F_{\alpha+1}^0(x)=F_{\alpha}^x(x)$
- $F_\beta^0(x)=F^0_{\{\beta\}(x)}(\varrho_\beta(x))$ where $\beta$ is a limit ordinal
- $F_\gamma^{n+1}(x)=F_\gamma^0(F_\gamma^n(x)),\gamma\neq 0.$
Here for limit ordinal $\beta$ we have:
- $\varrho_\beta(0)=0$,
- $\varrho_\beta(m+1)=\mu_z(z>\varrho_\beta(m)\ \&\ (i)_{\leq m}(F_{\{\beta\}(m+1)}^0(z)>F_{\{\beta\}(i)}(z)))$.
I don't understand the definition of $F_\beta^0(x)$. It seems a sort of diagonalisation process (does it?) but the notation with a weak understanding of ordinal numbers (I'm working on the latter at the very moment) are giving me hard time. I guess $\mu_z$ refers to smallest $z$ such this-and-that, but I don't quite understand the overall construction.
What the definition says?