Implications

We have the following indirect implication of form equivalence classes:

3 \(\Rightarrow\) 64
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 \(\Rightarrow\) 64 The independence of various definitions of finiteness, Levy, A. 1958, Fund. Math.
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.

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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