Implications

We have the following indirect implication of form equivalence classes:

295 \(\Rightarrow\) 288-n
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
295 \(\Rightarrow\) 30 "Dense orderings, partitions, and weak forms of choice", Gonzalez, C. 1995a, Fund. Math.
30 \(\Rightarrow\) 10 clear
10 \(\Rightarrow\) 288-n clear

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

Howard-Rubin Number Statement
295:

DO:  Every infinite set has a dense linear ordering.

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.

288-n:

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

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