Implications

We have the following indirect implication of form equivalence classes:

60 \(\Rightarrow\) 120-K
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
60 \(\Rightarrow\) 46-K clear
46-K \(\Rightarrow\) 120-K 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.

46-K:

If \(K\) is a finite subset of \(\omega-\{0,1\}\), \(C(\infty,K)\): For every \(n\in K\), every set of \(n\)-element sets has a choice function.

120-K:

If \(K\subseteq\omega-\{0,1\}\), \(C(LO,K)\): Every linearly ordered set of non-empty sets each of whose cardinality is in \(K\) has a choice function.

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