Implications

We have the following indirect implication of form equivalence classes:

211 \(\Rightarrow\) 38
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
211 \(\Rightarrow\) 94 Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.
94 \(\Rightarrow\) 5 clear
5 \(\Rightarrow\) 38 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
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 \((\forall n\in\omega)(x(n)\mathrel R x(n+1))\).

94:

\(C(\aleph_{0},\infty,{\Bbb R})\): Every denumerable family of non-empty sets of reals  has a choice function. Jech [1973b], p 148 prob 10.1.

5:

\(C(\aleph_0,\aleph_0,\Bbb R)\): Every denumerable set of non-empty denumerable subsets of \({\Bbb R}\) has a choice function.

38:

\({\Bbb R}\) is not the union of a countable family of countable sets.

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