Implications

We have the following indirect implication of form equivalence classes:

174-alpha \(\Rightarrow\) 17
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

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.

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