In "Regular Ordinals and Normal Forms", Theorem 2.6, David Isles claims that Bachmann proves that the sequence of normal functions generated by a Bachmann collection has property (6). Firstly, Bachmann does not prove this. Secondly, I haven't found a proof. Thirdly, it does hold for some "natural" Bachmann collections. The question is, is this statement correct?
Isles refers to p. 130 of Bachmann, where this property is numbered 2) b). For a translation, see page 20 of the translation referenced below. Bachmann in fact assumes that this property holds for the collection ${\mathfrak F}$.
I have constructed a Bachmann collection of interest in a paper referenced below. I call the property "P1", and prove that it holds for this particular Bachmann collection. I have been unable to prove that it follows from the axioms for a Bachmann collection. Section 3 contains some general lemmas of interest. The following may also be observed:
Given a built-up scheme, if $t_\xi\lt u\leq t_{\xi+1}$ where $u$ is a limit ordinal then not only does $t_\xi\leq u_0$ hold, but in fact $t_\xi\prec u_0$ holds. This may be seen from the proof of theorem 1.
Suppose (I) $t\xrightarrow\oplus t_0$ holds for all limit ordinals $t$. Then if $t_\xi\lt u\leq t_{\xi+1}$ where $u$ is a limit ordinal then $u\xrightarrow\oplus u_0\xrightarrow* t_\xi$. From this, property (6) holds.
As noted in the paper, (I) holds in various "natural" Bachmann collections.
- Self-contained statement
In response to Asaf's comment below here is a self-contained statement, in the terminology of my paper. A scheme over a regular uncountable ordinal $\Omega$ (e.g. $\aleph_1$) of length $\sigma<\Omega^+$ is a system of fundamental sequences $\alpha_\xi:\xi\lt D_\alpha$ for limit ordinals $\alpha<\sigma$, such that $D_\alpha<\Omega$ if $\alpha$ has cofinality $<\Omega$, and $D_\alpha=\Omega$ if $\alpha$ has cofinality $\Omega$. This specifies an iteration, starting with a club subset of $\Omega$ (e.g. the powers of $\omega$), applying the fixed point operation, intersection, and diagonal intersection at successor stages, limit stages of cofinality $<\Omega$, and diagonal intersection at stages of cofinality $\Omega$. Let $R_\alpha$ denote the (club) subset at stage $\alpha$.
A scheme is built-up if whenever $\alpha_\xi<\beta\leq\alpha_{\xi+1}$ and $\beta$ is a limit ordinal then $\alpha_\xi\leq\beta_0$. A scheme has property P2 if whenever $\alpha$ and $\xi$ are limit ordinals then $(\alpha_\xi)_\delta=\alpha_\delta$. (A Bachmann collection is more or less a scheme with these two properties). A scheme has property P1 if whenever $\alpha_\xi<\beta\leq\alpha_{\xi+1}$ and $\beta$ is a limit ordinal then $R_\beta\subseteq R_{\alpha_\xi+1}$.
The question is, does property P1 follow from the requirements that the scheme be built-up and have property P2?
- References:
Bachmann: H.\ Bachmann, Die Normalfunktionen und das Problem der ausgezeichnetenFolgen von Ordnungszahlen, http://www.ngzh.ch/archiv/1950_95/95_2/95_14.pdf
Bachmann translation: http://arxiv.org/abs/1903.04609
Isles: D Isles, Regular ordinals and normal forms, https://www.sciencedirect.com/science/article/pii/S0049237X08707638
My paper: https://www.researchgate.net/publication333058459_An_Ordinal_Larger_Than_the_Bachmann-Howard_Ordinal