Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
193 \(\Rightarrow\) 188	Injectivity, projectivity and the axiom of choice, Blass, A. 1979, Trans. Amer. Math. Soc.
188 \(\Rightarrow\) 106	Injectivity, projectivity and the axiom of choice, Blass, A. 1979, Trans. Amer. Math. Soc.
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
193:	\(EFP\ Ab\): Every Abelian group is a homomorphic image of a free projective Abelian group.
188:	\(EP\ Ab\): For every Abelian group \(A\) there is a projective Abelian group \(G\) and a homomorphism from \(G\) onto \(A\).
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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