We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 87-alpha \(\Rightarrow\) 43 | clear |
| 43 \(\Rightarrow\) 8 | clear |
| 8 \(\Rightarrow\) 27 | clear |
| 27 \(\Rightarrow\) 31 | clear |
| 31 \(\Rightarrow\) 34 | clear |
| 34 \(\Rightarrow\) 19 |
Sur les fonctions representables analytiquement, Lebesgue, H. 1905, J. Math. Pures Appl. |
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: | \(C(\aleph_{0},\infty)\): |
| 27: | \((\forall \alpha)( UT(\aleph_{0},\aleph_{\alpha}, \aleph_{\alpha}))\): The union of denumerably many sets each of power \(\aleph_{\alpha }\) has power \(\aleph_{\alpha}\). Moore, G. [1982], p 36. |
| 31: | \(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): The countable union theorem: The union of a denumerable set of denumerable sets is denumerable. |
| 34: | \(\aleph_{1}\) is regular. |
| 19: | A real function is analytically representable if and only if it is in Baire's classification. G.Moore [1982], equation (2.3.1). |
Comment: