Implications

We have the following indirect implication of form equivalence classes:

60 \(\Rightarrow\) 390
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
60 \(\Rightarrow\) 45-n clear
45-n \(\Rightarrow\) 64 Classes of Dedekind finite cardinals, Truss, J. K. 1974a, Fund. Math.
64 \(\Rightarrow\) 390 clear

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

Howard-Rubin Number Statement
60:

\(C(\infty,WO)\): Every set of non-empty, well orderable sets has a choice function.
Moore, G. [1982], p 125.

45-n:

If \(n\in\omega-\{0,1\}\), \(C(\infty,n)\): Every set of \(n\)-element sets has a choice function.

64:

\(E(I,Ia)\) There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.)

390:

Every infinite set can be partitioned either into two infinite sets or infinitely many sets, each of which has at least two elements. Ash [1983].

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