Implications

We have the following indirect implication of form equivalence classes:

346 \(\Rightarrow\) 18
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
346 \(\Rightarrow\) 18 The vector space Kinna-Wagner Principle is equivalent to the axiom of choice, Keremedis, K. 2001a, Math. Logic Quart.

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

Howard-Rubin Number Statement
346:

If \(V\) is a vector space without a finite basis then \(V\) contains an infinite, well ordered, linearly independent subset.

18:

\(PUT(\aleph_{0},2,\aleph_{0})\):  The union of a denumerable family of pairwise disjoint pairs has a denumerable subset.

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