Definition:Finite Character

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Definition

Let $S$ be a set.

Let $A$ be a set of subsets of $S$.


Then $A$ has finite character if and only if for each $x \subseteq S$:

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


Property of Sets

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 $P$.


Class Theory

In the context of class theory, the definition follows the same lines:

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

  • Results about finite character can be found here.


Sources

  • 2005: R.E. HodelRestricted versions of the Tukey-Teichmuller Theorem that are equivalent to the Boolean prime ideal theorem (Arch. Math. Logic Vol. 44: pp. 459 – 472)