User blog:Qlf2007/A General Scheme for Veblen Function

Let $f(x)$ be a normal function. Let $\kappa$ be the smallest cardinal, whose first ordinal (Let's call it $X$) is a fixed point of $f$. For example, if you take $f(x) = \omega^x$ then $\kappa = \aleph_0, X = \Omega$. We use $X^+$ to denote the first ordinal of $\kappa^+$.

The Cantor normal form of each ordinal $\alpha$ below $X^+$ can be uniquely written in the following way: $$\alpha = X^{\lambda_1} \times \mu_1 + X^{\lambda_2} \times \mu_2 + \cdots + X^{\lambda_n} \times \mu_n$$ where $\lambda$-values are below $X^+$, and $\mu$-values are below $X$. We will use $\alpha[\lambda]$ to denote the $\mu$-coefficient of the term $X^\lambda$ in the normal form. If the sum does not contain this term, then $\alpha[\lambda] = 0$.

For two ordinals $\alpha,\beta$ below $X^+$, we say $\alpha$ dominates $\beta$, if there exists some $\lambda$ such that $\alpha[\lambda] > \beta[\lambda]$, and for any $\lambda' < \lambda$ we have $\beta[\lambda'] = 0$, and for any $\lambda' > \lambda$ we have $\alpha[\lambda'] = \beta[\lambda']$. Note that this necessarily means $\alpha > \beta$.

We now associate each ordinal $\alpha$ below $\epsilon_{X + 1}$ with an ordinal below $X$, denoted by $\varphi(\alpha)$.

Rule 1: If $\alpha < X$, then $\varphi(\alpha) = f(\alpha)$.

Rule 2: If $\alpha \geq X$, we define $$S_0(\alpha) = \{\beta | \alpha\text{ dominates }\beta\}$$ $$S_{n + 1}(\alpha) = S_n \cup \{\sum \beta, X^\beta, \varphi(\gamma) | \beta,\gamma \in S_n,\gamma < \alpha\}$$ $$S(\alpha) = \cup S_n$$ $$\varphi(\alpha) = \sup \{\beta \in S_n : \beta < X\}$$ Here $\sum$ means finite sum of any number of (possibly different) terms.

More details on this system later, including how to define fundamental sequences, and how to make it more powerful.