Definition:Slowly Well-Ordered Class under Subset Relation
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Definition
Let $N$ be a class.
Let $N$ be well-ordered under the subset relation such that the following $3$ conditions hold:
\((\text S_1)\) | $:$ | $\O \in N$ is the smallest element of $N$ | |||||||
\((\text S_2)\) | $:$ | Each immediate successor element contains exactly $1$ more element than its immediate predecessor | |||||||
\((\text S_3)\) | $:$ | Each limit element $x$ is the union of its lower section $\bigcup x^\subset$ |
Then $N$ is slowly well-ordered under the subset relation.
Also known as
Some sources do not hyphenate, and have well ordered for well-ordered.
Some sources refer to under inclusion for under the subset relation.
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: $\S 4$ Well ordering and choice: Definition $4.1$