Implications

We have the following indirect implication of form equivalence classes:

49 \(\Rightarrow\) 46-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 \(\Rightarrow\) 61 clear
61 \(\Rightarrow\) 46-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.

61:

\((\forall n\in\omega, n\ge 2\))\((C(\infty,n))\): For each \(n\in\omega\), \(n\ge 2\), every set of \(n\) element  sets has a choice function.

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.

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