Description:
Form 99 (Rado's selection lemma) and Form 62 (\(C(\infty ,<\aleph _{0})\)) imply the Boolean prime ideal theorem (Form 14)
Content:
The following proof that Form 99 (Rado's selection lemma) and Form 62 (\(C(\infty ,<\aleph _{0})\)) imply the Boolean prime ideal theorem (Form 14) is due to A. Blass.
Form 99 (Rado's selection lemma) and
Form 62 (\(C(\infty ,<\aleph _{0})\)) imply the Boolean prime ideal theorem
(Form 14)
Assume Rado's lemma and \(C(\infty ,<\aleph _{0})\) and let \(B\) be a Boolean algebra. Let \(X = \left\{\{x, \bar{x}\}: x\in B\right\}\)
where \(\bar{x}\) denotes the complement of \(x\) in B. For each finite subset \(A\) of \(X\), let \(f_{A}\) be a choice function on \(A\)
chosen so that \(\{ f_{A}(Y) : Y \in A \}\) can be extended to a proper ideal in the subalgebra of \(B\) generated by \(\bigcup A\).
(Choosing \(f_{A}\) foreach finite \(A \subseteq X\) uses \(C(\infty ,<\aleph _{0})\).) By Rado's lemma there is a choice function
\(f\) for \(X\) such that for each finite \(A \subseteq X\) there is a finite \(B \subseteq X\) such that \(A\subseteq B\) and \(f_{B}\)
agrees with \(f\)on \(A\). Let \(I = \{ f(Y) : Y \in X \}\). It is reasonably easy to show that \(I\) is a maximal proper ideal in
\(B\).
Howard-Rubin number:
33
Type:
Proof
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