Implications

We have the following indirect implication of form equivalence classes:

408 \(\Rightarrow\) 373-n
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
408 \(\Rightarrow\) 62 clear
62 \(\Rightarrow\) 10 clear
10 \(\Rightarrow\) 288-n clear
288-n \(\Rightarrow\) 373-n clear

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.

10:

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

288-n:

If \(n\in\omega-\{0,1\}\), \(C(\aleph_0,n)\): Every denumerable set of \(n\)-element sets has a choice function.

373-n:

(For \(n\in\omega\), \(n\ge 2\).) \(PC(\aleph_0,n,\infty)\): Every denumerable set of \(n\)-element sets has an infinite subset with a choice function.

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