Book:Raymond M. Smullyan/Set Theory and the Continuum Problem/Revised Edition/Errata

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Errata for 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.)

Exercise $5.6$: $B - \paren {A - B} = \O$

Chapter $2$: Some Basics of Class-Set Theory:

Show that for any classes $A$, $B$, $\ldots{}$
$B - \paren {A - B} = \O$


Class which has Injection to Subclass of Well-Orderable Class is Well-Orderable

Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering: Proposition $1.2$

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 $\le$).


Condition on Proper Lower Sections for Total Ordering to be Well-Ordering

Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering: Exercise $1.2$

Prove that a sufficient condition for a linear ordering $\le$ of $A$ to be a well ordering of $A$ is that, for every proper lower section $L$ of $A$, there is a least element $x$ of $A$ not in $L$.