Definition:Finite Character/Class Theory

Definition

Let $A$ be a class.

Then $A$ has finite character if and only if, for all $x$:

$x \in A$ if and only if every finite subset of $x$ is in $A$.

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