Fundamental sequence

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A fundamental sequence for an ordinal $\alpha$ is a sequence $(\alpha[\xi])_{\xi\in\beta}$ of some ordinal length, satisfying these properties:

  • $\forall(\xi\in\beta)(\alpha[\xi]\in\beta)$
  • $\textrm{sup}\{\alpha[\xi]:\xi\in\beta\}=\alpha$


The $\xi$th entry of the fundamental sequence of $\alpha$ is often denoted by $\alpha[\xi]$, but some authors have used other ways to denote it, such as $\{\alpha\}(\xi)$. [2]