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 $1.7.11$
Mistake
- 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.
...
Correction
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 $X_x$.
Sources
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.7$: Well-Orderings and Ordinals: Theorem $1.7.11$