Axiom Of Choice

We have the following indirect implication of form equivalence classes:

133 \(\Rightarrow\) 327 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
133 \(\Rightarrow\) 231	note-123
231 \(\Rightarrow\) 151	clear
151 \(\Rightarrow\) 122	Russell's alternative to the axiom of choice, Howard, P. 1992, Z. Math. Logik Grundlagen Math. note-27
122 \(\Rightarrow\) 327	clear

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

Howard-Rubin Number	Statement
133:	Every set is either well orderable or has an infinite amorphous subset.
231:	\(UT(WO,WO,WO)\): The union of a well ordered collection of well orderable sets is well orderable.
151:	\(UT(WO,\aleph_{0},WO)\) (\(U_{\aleph_{1}}\)): The union of a well ordered set of denumerable sets is well orderable. (If \(\kappa\) is a well ordered cardinal, see note 27 for \(UT(WO,\kappa,WO)\).)
122:	\(C(WO,<\aleph_{0})\): Every well ordered set of non-empty finite sets has a choice function.
327:	\(KW(WO,<\aleph_0)\), The Kinna-Wagner Selection Principle for a well ordered family of finite sets: For every well ordered set \(M\) of finite sets there is a function \(f\) such that for all \(A\in M\), if \(\|A\|>1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 15.)

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