Implications

We have the following indirect implication of form equivalence classes:

392 \(\Rightarrow\) 211
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
392 \(\Rightarrow\) 394 clear
394 \(\Rightarrow\) 337 clear
337 \(\Rightarrow\) 211 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
392:

\(C(LO,LO)\): Every linearly ordered set of linearly orderable sets has a choice function.

394:

\(C(WO,LO)\): Every well ordered set of non-empty linearly orderable sets has a choice function.

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 \(X\).

211:

\(DCR\): Dependent choice for relations on \(\Bbb R\): If \(R\subseteq\Bbb R\times\Bbb R\) satisfies \((\forall x\in \Bbb R)(\exists y\in\Bbb R)(x\mathrel R y)\) then there is a sequence \(\langle x(n): n\in\omega\rangle\) of real numbers such that \((\forall n\in\omega)(x(n)\mathrel R x(n+1))\).

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