Implications

We have the following indirect implication of form equivalence classes:

295 \(\Rightarrow\) 178-n-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 \(\Rightarrow\) 62 clear
62 \(\Rightarrow\) 178-n-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.

62:

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

178-n-N:

If  \(n\in\omega\), \(n\ge 2\) and \(N \subseteq \{ 1, 2, \ldots , n-1 \}\), \(N \neq\emptyset\), \(MC(\infty,n, N)\):  If \(X\) is any set of \(n\)-element sets then  there is  a function \(f\) with domain \(X\) such that for all \(A\in X\), \(f(A)\subseteq A\) and \(|f(A)|\in N\).

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