Definition:Lower Section/Class Theory
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Definition
Let $A$ be a class under a total ordering $\preccurlyeq$.
Let $L$ be a subclass of $A$ such that:
- $\forall x \in L: \forall a \in A \setminus L: x \preccurlyeq a$
where $A \setminus L$ is the difference between $A$ and $L$.
Then $L$ is known as a lower section of $A$.
Also see
- Results about lower sections can be found here.
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: Exercise $1.2$
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 1$ A few preliminaries