Implications

We have the following indirect implication of form equivalence classes:

193 \(\Rightarrow\) 38
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 \(\Rightarrow\) 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
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.

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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