Implications

We have the following indirect implication of form equivalence classes:

408 \(\Rightarrow\) 283
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
408 \(\Rightarrow\) 62 clear
62 \(\Rightarrow\) 283 The well-ordered and well-orderable subsets of a set, Truss, J. K. 1973d, Z. Math. Logik Grundlagen Math.

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
408:

If \(\{f_i: i\in I\}\) is a family of functions such that for each \(i\in I\), \(f_i\subseteq E\times W\), where \(E\) and \(W\) are non-empty sets, and \(\cal B\) is a filter base on \(I\) such that

  1. For all \(B\in\cal B\) and all finite \(F\subseteq E\) there is an \(i\in I\) such that \(f_i\) is defined on \(F\), and
  2. For all \(B \in\cal B\) and all finite \(F\subseteq E\) there exist at most finitely many functions on \(F\) which are restrictions of the functions \(f_i\) with \(i\in I\),  
then there is a function \(f\) with domain \(E\) such that for each finite \(F\subseteq E\) and each \(B\in\cal B\) there is an \(i\in I\) such that \(f|F = f_i|F\).

62:

\(C(\infty,< \aleph_{0})\):  Every set of non-empty finite  sets  has  a choice function.

283:

Cardinality of well ordered subsets:  For all \(n\in\omega\) and for all infinite \(x\), \(|x^n| < |s(x)|\) where \(s(x)\) is the set of all well orderable subsets of \(x\).

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