Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
49 \(\Rightarrow\) 30 clear
30 \(\Rightarrow\) 62 clear
62 \(\Rightarrow\) 121 clear
121 \(\Rightarrow\) 120-K clear

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

Howard-Rubin Number Statement
49:

Order Extension Principle: Every partial ordering can be extended to a linear ordering.  Tarski [1924], p 78.

30:

Ordering Principle: Every set can be linearly ordered.

62:

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

121:

\(C(LO,<\aleph_{0})\): Every linearly ordered set of non-empty finite 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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