Condition for Woset to be Isomorphic to Ordinal/Mistake

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Source Work

1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.): $\S 1$: Naive Set Theory:

$\S 1.7$: Well-Orderings and Ordinals:
Theorem $1.7.11$


Also, since:
\(\ds X_x\) \(=\) \(\ds \set {z \in X \mid z \le x}\)
\(\ds \) \(=\) \(\ds \set {z \in X \mid z \le y \land z \le x}\)
\(\ds \) \(=\) \(\ds \set {z \in X_y \mid z \le x}\)
\(\ds \) \(=\) \(\ds \paren {X_y}_x,\)

we have

$(2): \quad \paren {g_y X_x}: X_x \cong \paren {\map Z y}_{\map {g_y} x}$.

Now, $\map Z y$ is an ordinal, so by Theorem $1.7.6$, $\paren {\map Z y}_{\map {g_y} x}$ is an ordinal.



Statement $(2)$ has a misprint.

It should read:

$(2): \quad \paren {g_y \restriction X_x}: X_x \cong \paren {\map Z y}_{\map {g_y} x}$

thereby providing the crucial information that the mapping under consideration is the restriction of $g_y$ to $X_x$.