Implications

We have the following indirect implication of form equivalence classes:

377 \(\Rightarrow\) 374-n
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
377 \(\Rightarrow\) 378 clear
378 \(\Rightarrow\) 132 Weak choice principles, De la Cruz, O. 1998a, Proc. Amer. Math. Soc.
132 \(\Rightarrow\) 10 Amorphe Potenzen kompakter Raume, Brunner, N. 1984b, Arch. Math. Logik Grundlagenforschung
10 \(\Rightarrow\) 423 clear
423 \(\Rightarrow\) 374-n clear

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

Howard-Rubin Number Statement
377:

Restricted Ordering Principle:  For every infinite set \(X\) there is an infinite subset \(Y\) of \(X\) such that \(Y\) can be linearly ordered.

378:

Restricted Choice for Families of Well Ordered Sets:  For every infinite set \(X\) there is an infinite subset \(Y\) of \(X\) such that the family of non-empty well orderable subsets of \(Y\) has a choice function.

132:

\(PC(\infty, <\aleph_0,\infty)\):  Every infinite family of finite  sets has an infinite subfamily with a choice function.

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.

374-n:

\(UT(\aleph_0,n,\aleph_0)\) for \(n\in\omega -\{0,1\}\): The union of a denumerable set of \(n\)-element sets is denumerable.

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