Implications

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We have the following indirect implication of form equivalence classes:

3 \(\Rightarrow\) 84 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 \(\Rightarrow\) 84	Definitions of finite, Howard, P. 1989, Fund. Math.

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.
84:	\(E(II,III)\) (Howard/Yorke [1989]): \((\forall x)(x\) is \(T\)-finite  if and only if \(\cal P(x)\) is Dedekind finite).

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