Implications

We have the following indirect implication of form equivalence classes:

337 \(\Rightarrow\) 170
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
337 \(\Rightarrow\) 92 clear
92 \(\Rightarrow\) 170 Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.

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

Howard-Rubin Number Statement
337:

\(C(WO\), uniformly linearly ordered):  If \(X\) is a well ordered collection of non-empty sets and there is a function \(f\) defined on \(X\) such that for every \(x\in X\), \(f(x)\) is a linear ordering of \(x\), then there is a choice function for \(X\).

92:

\(C(WO,{\Bbb R})\):  Every well ordered family of non-empty subsets of \({\Bbb R}\) has a choice function.

170:

\(\aleph_{1}\le 2^{\aleph_{0}}\).

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