Implications

We have the following indirect implication of form equivalence classes:

85 \(\Rightarrow\) 283
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
85 \(\Rightarrow\) 62 clear
62 \(\Rightarrow\) 283 The well-ordered and well-orderable subsets of a set, Truss, J. K. 1973d, Z. Math. Logik Grundlagen Math.

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

Howard-Rubin Number Statement
85:

\(C(\infty,\aleph_{0})\):  Every family of denumerable sets has  a choice function.  Jech [1973b] p 115 prob 7.13.

62:

\(C(\infty,< \aleph_{0})\):  Every set of non-empty finite  sets  has  a choice function.

283:

Cardinality of well ordered subsets:  For all \(n\in\omega\) and for all infinite \(x\), \(|x^n| < |s(x)|\) where \(s(x)\) is the set of all well orderable subsets of \(x\).

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