Axiom Of Choice

We have the following indirect implication of form equivalence classes:

174-alpha \(\Rightarrow\) 18 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
174-alpha \(\Rightarrow\) 9	Horrors of topology without AC: A non-normal orderable space, van Douwen, E.K. 1985, Proc. Amer. Math. Soc. note-49
9 \(\Rightarrow\) 17	The independence of Ramsey's theorem, Kleinberg, E.M. 1969, J. Symbolic Logic
17 \(\Rightarrow\) 18	Ramsey's theorem in the hierarchy of choice principles, Blass, A. 1977a, J. Symbolic Logic The Axiom of Choice, Jech, 1973b, page 164 problem 11.20

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

Howard-Rubin Number	Statement
174-alpha:	\(RM1,\aleph_{\alpha }\): The representation theorem for multi-algebras with \(\aleph_{\alpha }\) unary operations: Assume \((A,F)\) is a multi-algebra with \(\aleph_{\alpha }\) unary operations (and no other operations). Then there is an algebra \((B,G)\) with \(\aleph_{\alpha }\) unary operations and an equivalence relation \(E\) on \(B\) such that \((B/E,G/E)\) and \((A,F)\) are isomorphic multi-algebras.
9:	Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.
17:	Ramsey's Theorem I: If \(A\) is an infinite set and the family of all 2 element subsets of \(A\) is partitioned into 2 sets \(X\) and \(Y\), then there is an infinite subset \(B\subseteq A\) such that all 2 element subsets of \(B\) belong to \(X\) or all 2 element subsets of \(B\) belong to \(Y\). (Also, see Form 325.), Jech [1973b], p 164 prob 11.20.
18:	\(PUT(\aleph_{0},2,\aleph_{0})\): The union of a denumerable family of pairwise disjoint pairs has a denumerable subset.

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