We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 43 \(\Rightarrow\) 8 | clear |
| 8 \(\Rightarrow\) 9 | Was sind und was sollen die Zollen?, Dedekind, [1888] |
| 9 \(\Rightarrow\) 17 |
The independence of Ramsey's theorem, Kleinberg, E.M. 1969, J. Symbolic Logic |
| 17 \(\Rightarrow\) 132 |
Amorphe Potenzen kompakter Raume, Brunner, N. 1984b, Arch. Math. Logik Grundlagenforschung |
| 132 \(\Rightarrow\) 73 | clear |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 43: | \(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\) is a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\) then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\). See Tarski [1948], p 96, Levy [1964], p. 136. |
| 8: | \(C(\aleph_{0},\infty)\): |
| 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. |
| 132: | \(PC(\infty, <\aleph_0,\infty)\): Every infinite family of finite sets has an infinite subfamily with a choice function. |
| 73: | \(\forall n\in\omega\), \(PC(\infty,n,\infty)\): For every \(n\in\omega\), if \(C\) is an infinite family of \(n\) element sets, then \(C\) has an infinite subfamily with a choice function. De la Cruz/Di Prisco [1998b] |
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