Definition:Finite Character/Property of Sets
< Definition:Finite Character(Redirected from Definition:Finite Character (Property of Sets))
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Definition
Let $P$ be a property of sets.
Then $P$ has finite character if and only if for every set $x$:
- $x$ has property $P$ if and only if every finite subset of $x$ has property $P$.
Also known as
To say that:
- $A$ has finite character
is the same as saying that:
- $A$ is of finite character.
Also see
- Tukey's Lemma, an equivalent of the Axiom of Choice
- Restricted Tukey's Theorem, an equivalent of the Boolean Prime Ideal Theorem.
- Results about finite character 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 $\text {II}$ -- Maximal principles: $\S 5$ Maximal principles