We have the following indirect implication of form equivalence classes:
| 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 \(\Rightarrow\) 124 |
Hilbertraume mit amorphen Basen (English summary), Brunner, N. 1984a, Compositio Math. |
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. |
| 124: | Every operator on a Hilbert space with an amorphous base is the direct sum of a finite matrix and a scalar operator. (A set is amorphous if it is not the union of two disjoint infinite sets.) |
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