Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
388 \(\Rightarrow\) 106
106 \(\Rightarrow\) 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
388:

Every infinite branching poset (a partially ordered set in which each element has at least two lower bounds) has either an infinite chain or an infinite antichain.

106:

Baire Category Theorem for Compact Hausdorff Spaces: Every compact Hausdorff space is Baire.

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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