Description:
Form 214 implies Form 152
Content:
In this note we give a proof that
Form 214 (For every family \(A\) of infinite sets, there is a function \(f\) such that \(\forall y\in A\), \(f(y)\subseteq y\) and \(|f(y)|=\aleph_{0}\).) implies Form 152 (Every non-well orderable set is the union of a pairwise disjoint, well orderable family of denumerable sets.)
Let \(X\) be a set which cannot be well ordered and let \(Y\) be the setof all infinite subsets of \(X\). Form 214 implies that there is a function \(f\) such that for each \(y\in Y\), \(f(y)\subseteq y\) and\(|f(y)|=\aleph_0\). We define a function \(g\) on the ordinals so that \(g(0)= f(X)\) and \(g(\alpha)=f(X\setminus \bigcup_{\beta<\alpha}g(\beta))\). Since \(X\) is a set, there is an ordinal \(\gamma\) such that \(X\setminus\bigcup_{\beta < \gamma}g(\beta)\) is finite. We can use this function \(g\) to cover \(X\) with a well ordered family of pairwise disjoint denumerable sets.
Howard-Rubin number:
140
Type:
Proof
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