Axiom Of Choice

We have the following indirect implication of form equivalence classes:

3 \(\Rightarrow\) 378 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
3 \(\Rightarrow\) 9	Cardinal addition and the axiom of choice, Howard, P. 1974, Bull. Amer. Math. Soc.
9 \(\Rightarrow\) 376	clear
376 \(\Rightarrow\) 377	Weak choice principles, De la Cruz, O. 1998a, Proc. Amer. Math. Soc.
377 \(\Rightarrow\) 378	clear

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

Howard-Rubin Number	Statement
3:	\(2m = m\): For all infinite cardinals \(m\), \(2m = m\).
9:	Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.
376:	Restricted Kinna Wagner Principle: For every infinite set \(X\) there is an infinite subset \(Y\) of \(X\) and a function \(f\) such that for every \(z\subseteq Y\), if \(\|z\| \ge 2\) then \(f(z)\) is a non-empty proper subset of \(z\).
377:	Restricted Ordering Principle: For every infinite set \(X\) there is an infinite subset \(Y\) of \(X\) such that \(Y\) can be linearly ordered.
378:	Restricted Choice for Families of Well Ordered Sets: For every infinite set \(X\) there is an infinite subset \(Y\) of \(X\) such that the family of non-empty well orderable subsets of \(Y\) has a choice function.

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