Implications

We have the following indirect implication of form equivalence classes:

193 \(\Rightarrow\) 64
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 \(\Rightarrow\) 82 note-76
82 \(\Rightarrow\) 83 Definitions of finite, Howard, P. 1989, Fund. Math.
83 \(\Rightarrow\) 64 The Axiom of Choice, Jech, 1973b, page 52 problem 4.10

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 \(y\).

82:

\(E(I,III)\) (Howard/Yorke [1989]): If \(X\) is infinite then \(\cal P(X)\) is Dedekind infinite. (\(X\) is finite \(\Leftrightarrow {\cal P}(X)\) is Dedekind finite.)

83:

\(E(I,II)\) Howard/Yorke [1989]: \(T\)-finite is equivalent to finite.

64:

\(E(I,Ia)\) There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.)

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