User blog:Hyp cos/Monotonic FS systems

I don't know how you understand "a monotonic FS system", but in this blog post, I'll use "monotonic" for some special FS systems.

Definition 1: In a fundamental sequence system \(S:\alpha\cap\text{Lim}\times\mathbb N\rightarrow\alpha\), let \(\lambda<\alpha\) be a limit ordinal, then \(\lambda[n]_S<\lambda\) for all \(n\), and \(\sup\{\lambda[n]_S|n<\omega\}=\lambda\).

Monotone
Definition 2: \(\alpha[]_S=\left\{\begin{array}\alpha &,\ \alpha\in\text{Succ or }\alpha=0 \\ \alpha[0]_S[]_S &,\ \text{otherwise}\end{array}\right.\)

So In FS system \(S\), \(\alpha[]_S\) repeatedly takes the 0-th term of FS of \(\alpha\) until a successor ordinal or 0.

Definition 3: A FS system \(S\) is monotonic if for all limit \(\lambda\) and natural number \(n\), \(\lambda[n+1]_S[]_S=\lambda[n]_S+1\).