We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 87-alpha \(\Rightarrow\) 43 | clear |
| 43 \(\Rightarrow\) 8 | clear |
| 8 \(\Rightarrow\) 9 | Was sind und was sollen die Zollen?, Dedekind, [1888] |
| 9 \(\Rightarrow\) 64 |
The independence of various definitions of finiteness, Levy, A. 1958, Fund. Math. clear |
| 64 \(\Rightarrow\) 127 |
Amorphe Potenzen kompakter Raume, Brunner, N. 1984b, Arch. Math. Logik Grundlagenforschung |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 87-alpha: | \(DC(\aleph_{\alpha})\): Given a relation \(R\) such that for every subset \(Y\) of a set \(X\) with \(|Y|<\aleph_{\alpha}\), there is an \(x\in X\) with \(Y\mathrel R x\) then there is a function \(f:\aleph_{\alpha}\to X\) such that (\(\forall\beta < \aleph_{\alpha}\)) \(\{f(\gamma): \gamma < \beta\}\mathrel R f(\beta)\). |
| 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. |
| 64: | \(E(I,Ia)\) There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.) |
| 127: | An amorphous power of a compact \(T_2\) space, which as a set is well orderable, is well orderable. |
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