Minimally Closed Class under Progressing Mapping
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Theorem
Let $N$ be a class which is closed under a progressing mapping $g$.
Let $b$ be an element of $N$ such that $N$ is minimally closed under $g$ with respect to $b$.
Then the following results hold:
Minimally Closed Class under Progressing Mapping induces Nest
For all $x, y \in N$:
- either $\map g x \subseteq y$ or $y \subseteq x$
and $N$ forms a nest:
- $\forall x, y \in N: x \subseteq y$ or $y \subseteq x$
Bounded Subset of Minimally Closed Class under Progressing Mapping has Greatest Element
Every bounded subset of $N$ has a greatest element.
Fixed Point of Progressing Mapping on Minimally Closed Class is Greatest Element
$g$ has no fixed point, unless possibly the greatest element, if there is one.
Minimally Closed Class under Progressing Mapping is Well-Ordered
$N$ is well-ordered under the subset relation.
Smallest Element of Minimally Closed Class under Progressing Mapping
$b$ is the smallest element of $N$.
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 4$ A double induction principle and its applications: Theorem $4.17$