Implications

We have the following indirect implication of form equivalence classes:

49 \(\Rightarrow\) 423
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
49 \(\Rightarrow\) 30 clear
30 \(\Rightarrow\) 10 clear
10 \(\Rightarrow\) 423 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.

10:

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

423:

\(\forall n\in \omega-\{o,1\}\), \(C(\aleph_0, n)\) : For every \(n\in  \omega - \{0,1\}\), every denumerable set of \(n\) element sets has a choice function.

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