Restricted Tukey-Teichmüller Theorem/Weak Form

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Theorem

Let $X$ be a set.

Let $\mathcal A$ be a non-empty set of subsets of $X$.

Let $'$ be a unary operation on $X$.

Let $\mathcal A$ have finite character.

For all $A \in \mathcal A$ and all $x \in X$, let either:

$A \cup \set x \in \mathcal A$

or:

$A \cup \set {x'} \in \mathcal A$


Then there exists a $B \in \mathcal A$ such that for all $x \in X$, either $x \in B$ or $x' \in B$.


Proof


Source of Name

This entry was named for John Wilder Tukey and Oswald Teichmüller.


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: 459 – 472)