Implications

We have the following indirect implication of form equivalence classes:

213 \(\Rightarrow\) 314
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
213 \(\Rightarrow\) 85 clear
85 \(\Rightarrow\) 314 There can be a permutation which is not the product of two reflections, Degen, J.W. 1988, Z. Math. Logik Grundlagen Math.
note-124

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.

314:

For every set \(X\) and every permutation \(\pi\) on \(X\) there are two reflections \(\rho\) and \(\sigma\) on \(X\) such that \(\pi =\rho\circ\sigma\) and for every \(Y\subseteq X\) if \(\pi[Y]=Y\) then \(\rho[Y]=Y\) and \(\sigma[Y]=Y\).  (A reflection is a permutation \(\phi\) such that \(\phi^2\) is the identity.) \ac{Degen} \cite{1988}, \cite{2000}.

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