Axiom Of Choice

We have the following indirect implication of form equivalence classes:

193 \(\Rightarrow\) 104 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 \(\Rightarrow\) 126	Injectivity, projectivity and the axiom of choice, Blass, A. 1979, Trans. Amer. Math. Soc.
126 \(\Rightarrow\) 94	Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.
94 \(\Rightarrow\) 34	Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.
34 \(\Rightarrow\) 104	clear

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.
126:	\(MC(\aleph_0,\infty)\), Countable axiom of multiple choice: For every denumerable set \(X\) of non-empty sets there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\).
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.
34:	\(\aleph_{1}\) is regular.
104:	There is a regular uncountable aleph. Jech [1966b], p 165 prob 11.26.

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