Proposition
187
[26J]This model of \(β\) is a well-ordered set with the ordering
\[ nβ€ m \iff nβ m\quad . \]
Moreover in this model we have
\begin{equation} β n,mβ β, n β m \iff (n β m β§ nβ m)\quad . \label{eq:n_ in_ m_ iff} \end{equation}
188
so, defining (as usual)
\[ n{\lt}m β (nβ€ m β§ nβ m) \]
we can write
\[ nβ m \iff n{\lt}m\quad . \]
This is proven in the following exercises, see in particular [269].