Description:
Form 203 implies Form 94
Content:
Form 203 (Every partition of \({\cal P}(\omega )\) into non-empty subsets has a choice function) implies Form 94 (\(C(\aleph_{0},\infty,{\Bbb R})\)).
Assume \(\{A_{i}: i\in\omega\}\) is a denumerable set of sets of reals. (We identify \({\Bbb R}\) with \({\cal P}(\omega ).)\) For each \(i\in\omega\), let \(B_{i} = A_{i}\times\{ i \}\). Then the family \(\{B_{i}: i\in\omega\}\) is a disjoint family from \({\cal P}(\omega)\times\omega\) and therefore, by Form 203, has a choice function. This induces a choice function on \(\{A_{i}: i\in\omega\}\).
Howard-Rubin number:
67
Type:
Theorem
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