Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
213 \(\Rightarrow\) 85 clear
85 \(\Rightarrow\) 32 clear
32 \(\Rightarrow\) 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
213:

\(C(\infty,\aleph_{1})\): If \((\forall y\in X)(|y| = \aleph_{1})\) then \(X\) has a choice function.

85:

\(C(\infty,\aleph_{0})\):  Every family of denumerable sets has  a choice function.  Jech [1973b] p 115 prob 7.13.

32:

\(C(\aleph_0,\le\aleph_0)\): Every denumerable set of non-empty countable 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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