Implications

We have the following indirect implication of form equivalence classes:

231 \(\Rightarrow\) 374-n
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
231 \(\Rightarrow\) 32 clear
32 \(\Rightarrow\) 10 clear
10 \(\Rightarrow\) 423 clear
423 \(\Rightarrow\) 374-n clear

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

Howard-Rubin Number Statement
231:

\(UT(WO,WO,WO)\): The union of a well ordered collection of well orderable sets is well orderable.

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.

423:

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

374-n:

\(UT(\aleph_0,n,\aleph_0)\) for \(n\in\omega -\{0,1\}\): The union of a denumerable set of \(n\)-element sets is denumerable.

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