Axiom Of Choice

We have the following indirect implication of form equivalence classes:

2 \(\Rightarrow\) 387 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
2 \(\Rightarrow\) 3	On successors in cardinal arithmetic, Truss, J. K. 1973c, Fund. Math.
3 \(\Rightarrow\) 9	Cardinal addition and the axiom of choice, Howard, P. 1974, Bull. Amer. Math. Soc.
9 \(\Rightarrow\) 82	clear
82 \(\Rightarrow\) 387	"Dense orderings, partitions, and weak forms of choice", Gonzalez, C. 1995a, Fund. Math.

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

Howard-Rubin Number	Statement
2:	Existence of successor cardinals: For every cardinal \(m\) there is a cardinal \(n\) such that \(m < n\) and \((\forall p < n)(p \le m)\).
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.
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.)
387:	DPO: Every infinite set has a non-trivial, dense partial order. (A partial ordering \(<\) on a set \(X\) is dense if \((\forall x, y\in X)(x \lt y \to (\exists z \in X)(x \lt z \lt y))\) and is non-trivial if \((\exists x,y\in X)(x \lt y)\)).

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