Implications

We have the following indirect implication of form equivalence classes:

378 \(\Rightarrow\) 127
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
378 \(\Rightarrow\) 11 clear
11 \(\Rightarrow\) 12 clear
12 \(\Rightarrow\) 336-n clear
336-n \(\Rightarrow\) 64 Weak choice principles, De la Cruz, O. 1998a, Proc. Amer. Math. Soc.
64 \(\Rightarrow\) 127 Amorphe Potenzen kompakter Raume, Brunner, N. 1984b, Arch. Math. Logik Grundlagenforschung

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

Howard-Rubin Number Statement
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.

11:

A Form of Restricted Choice for Families of Finite Sets: For every infinite set \(A\), \(A\) has an infinite subset \(B\) such that for every \(n\in\omega\), \(n>0\), the set of all \(n\) element subsets of \(B\) has a choice function. De la Cruz/Di Prisco [1998b]

12:

A Form of Restricted Choice for Families of Finite Sets: For every infinite set \(A\) and every \(n\in\omega\), there is an infinite subset \(B\) of \(A\) such the set of all \(n\) element subsets of \(B\) has a choice function. De la Cruz/Di Prisco} [1998b]

336-n:

(For \(n\in\omega\), \(n\ge 2\).)  For every infinite set \(X\), there is an infinite \(Y \subseteq X\) such that the set of all \(n\)-element subsets of \(Y\) has a choice function.

64:

\(E(I,Ia)\) There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.)

127:

An amorphous power of a compact \(T_2\) space, which as a set is well orderable, is well orderable.

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