Axiom Of Choice

We have the following indirect implication of form equivalence classes:

188 \(\Rightarrow\) 390 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
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
64 \(\Rightarrow\) 390	clear

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number	Statement
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.)
390:	Every infinite set can be partitioned either into two infinite sets or infinitely many sets, each of which has at least two elements. Ash [1983].

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