Restricted Tukey's Theorem
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Theorem
Weak Form
Let $X$ be a set.
Let $\AA$ be a non-empty set of subsets of $X$.
Let $'$ be a unary operation on $X$.
Let $\AA$ have finite character.
For all $A \in \AA$ and all $x \in X$, let either:
- $A \cup \set x \in \AA$
or:
- $A \cup \set {x'} \in \AA$
Then there exists a $B \in \AA$ such that for all $x \in X$, either $x \in B$ or $x' \in B$.
Strong Form
Let $X$ be a set.
Let $\AA$ be a non-empty set of subsets of $X$.
Let $'$ be a unary operation on $X$.
Let $\AA$ have finite character.
For all $A \in \AA$ and all $x \in X$, let either:
- $A \cup \set x \in \AA$
or:
- $A \cup \set {x'} \in \AA$
Then for each $A \in \AA$ there exists a $C \in \AA$ such that:
- $A \subseteq C$
and:
- for all $x \in X$, either $x \in C$ or $x' \in C$.
Variation 1
Restricted Tukey's Theorem/Variation 1
Variation 2
Restricted Tukey's Theorem/Variation 2
Source of Name
This entry was named for John Wilder Tukey.
Sources
- 2005: R.E. Hodel: Restricted versions of the Tukey-Teichmuller Theorem that are equivalent to the Boolean prime ideal theorem (Archive for Mathematical Logic no. 44: pp. 459 – 472)