Implications

We have the following indirect implication of form equivalence classes:

346 \(\Rightarrow\) 185
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
346 \(\Rightarrow\) 126 The vector space Kinna-Wagner Principle is equivalent to the axiom of choice, Keremedis, K. 2001a, Math. Logic Quart.
126 \(\Rightarrow\) 185 note-76

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.

126:

\(MC(\aleph_0,\infty)\), Countable axiom of multiple choice: For every denumerable set \(X\) of non-empty sets there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\).

185:

Every linearly ordered Dedekind finite set is finite.

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