For which function are the fixed-points enumerated by $\phi_0(\alpha)$?

Question

In the Veblen function $\phi_1(\alpha)=\varepsilon_\alpha$ are fixed points of $\phi_0(\alpha)=\omega^\alpha$. It seems logical that $\phi_0(\alpha)=\omega^\alpha$ are fixed points of some other function - however I haven't been able to figure out which function has fixed-points enumerated by $\phi_0(\alpha)$.

For which function are the fixed-points enumerated by $\phi_0(\alpha)$?

Answer 1

Although $\phi_0$ doesn't enumerate the fixed points of any "elementary" function, it enumerates the so called indecomposable ordinals: an ordinal is said to be indecomposable if it can't be obtained as the sum of two smaller ordinals, and it turns out that such ordinals are precisely the powers of $\omega$ (the only exception is $0$: it is not a power of $\omega$, but it would make sense to say that it is indecomposable, since you can't construct anything from an empty set).

Anyway, if you really want, you can definitely construct a function $f$ whose fixed points are $0$ and the powers of $\omega$ (notice that, if you want $f$ to be normal like the ordinary Veblen functions, $f(1)=1$ forces $f(0)=0$). I didn't check all the details, but this should work:

• $f(0)=0$;
• $\forall \beta, \; f(\omega^\beta)=\omega^\beta$;
• if $\alpha=\omega^{\beta_1}+\omega^{\beta_2}+...+\omega^{\beta_k}$, where $k \ge 2$ and $\beta_1 \ge \beta_2 \ge ... \ge \beta_k$, then $f(\alpha)=\omega^{\beta_1}+\alpha$.