Ordinal numbers:- An ordinal number is a set $\alpha$ with the following properties:
(1) If $x,y \in \alpha$, then either $x\in y, y\in x, $ or $y=x$
(2)If $y \in \alpha$ and $x\in y$, then $x\in \alpha$.
Theorem H.10 There is an uncountable well-ordered set $\Omega$ of ordinal numbers with maximal element $\omega_1$ having the property if $x\in \Omega$ and $x\neq \omega_1$, then $\{y\in \Omega:y\leq x\}$ is countable.
Definition of interval in the book is given by:-
Doubt:- By the definition of the interval $\{x \in \mathcal X:x>a \} \cap \{x \in \mathcal X:x<b+1 \}=(a,b+1)$. Am I correct? If there is no element between $b$ and $b+1$. I can agree with the notation in the textbook. Please help me.
It is important here that we're talking about sets of ordinals, so we know there is no possible element strictly between $b$ and $b+1$.
(Recall that $b+1$ is by definition the set $b\cup\{b\}$, and the standard ordering of ordinals is the same as ordering by $\subseteq$, so there can't be any ordinal strictly between $b$ and $b+1$ because $b+1$ has only one additional element).
Therefore the condition $x<b+1$ is the same as $x\le b$, so $(a,b+1)$ and $(a,b]$ are two ways of writing the same set of ordinals.