Implications

We have the following indirect implication of form equivalence classes:

2 \(\Rightarrow\) 57
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 \(\Rightarrow\) 57 clear

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.

57:

If \(x\) and \(y\) are Dedekind finite sets then either \(|x|\le |y|\) or \(|y|\le |x|\).
Mathias [1979], p 125.

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