We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 192 \(\Rightarrow\) 43 |
Injectivity, projectivity and the axiom of choice, Blass, A. 1979, Trans. Amer. Math. Soc. |
| 43 \(\Rightarrow\) 8 | clear |
| 8 \(\Rightarrow\) 9 | Was sind und was sollen die Zollen?, Dedekind, [1888] |
| 9 \(\Rightarrow\) 10 | Zermelo's Axiom of Choice, Moore, 1982, 322 |
| 10 \(\Rightarrow\) 288-n | clear |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 192: | \(EP\) sets: For every set \(A\) there is a projective set \(X\) and a function from \(X\) onto \(A\). |
| 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. |
| 10: | \(C(\aleph_{0},< \aleph_{0})\): Every denumerable family of non-empty finite sets has a choice function. |
| 288-n: | If \(n\in\omega-\{0,1\}\), \(C(\aleph_0,n)\): Every denumerable set of \(n\)-element sets has a choice function. |
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