Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
15 \(\Rightarrow\) 30	The Axiom of Choice, Jech, 1973b, page 53 problem 4.12
30 \(\Rightarrow\) 62	clear
62 \(\Rightarrow\) 64	Amorphe Potenzen kompakter Raume, Brunner, N. 1984b, Arch. Math. Logik Grundlagenforschung
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
15:	\(KW(\infty,\infty)\) (KW), The Kinna-Wagner Selection Principle: For every set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(\|A\|>1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 81(\(n\)).
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.
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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