Class which has Injection to Subclass of Well-Orderable Class is Well-Orderable/Mistake
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Source Work
2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.):
- Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering:
- $\S 1$ Introduction to well ordering:
- Proposition $1.2$: Proof
- $\S 1$ Introduction to well ordering:
Mistake
- Finally, suppose $C$ is any non-empty subclass of $A$. Let $C'$ be the class of all elements $x'$ such that $x \in C$. Then $C'$ contains a least element (with respect to $R$), and this element is $b'$ for some $b \in C$. Then for any $x \in C$, $x' \in C'$, and so $x' R b'$, and therefore $x \le b$. Thus $b$ is the least element of $C$ (with respect to $\le$).
Correction
As $b'$ is the least element of $C'$, it follows that for arbitrary $x' \in C'$ that $b' R x'$, not $x' R b'$.
Hence similarly $b \le x$, not $x \le b$.
However, the conclusion is correct.
Also see
- Class which has Injection to Subclass of Well-Orderable Class is Well-Orderable: $\mathsf{Pr} \infty \mathsf{fWiki}$'s rendition of this result
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering: Proposition $1.2$