Axiom Of Choice

We have the following indirect implication of form equivalence classes:

224 \(\Rightarrow\) 224 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
224 \(\Rightarrow\) 224

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number	Statement
224:	There is a partition of the real line into \(\aleph_1\) Borel sets \(\{B_\alpha: \alpha<\aleph_1\}\) such that for some \(\beta <\aleph_1\), \(\forall\alpha <\aleph_1\), \(B_{\alpha}\in G_{\beta}\). (\(G_\beta\) for \(\beta < \aleph_1\) is defined by induction, \(G_0=\{A: A\) is an open subset of \({\Bbb R}\}\) and for \(\beta > 0\), \(G_\beta =\left\{\bigcup^\infty_{i=0}A_{i}: (\forall i\in\omega) (\exists\xi <\beta)(A_i\in G_\xi)\,\right\}\) if \(\beta\) is even and \(G_\beta = \left\{\bigcap^\infty_{i=0}A_{i}: (\forall i\in\omega) (\exists \xi < \beta)(A_{i}\in G_\xi)\,\right\}\) if \(\beta\) is odd.)
224:	There is a partition of the real line into \(\aleph_1\) Borel sets \(\{B_\alpha: \alpha<\aleph_1\}\) such that for some \(\beta <\aleph_1\), \(\forall\alpha <\aleph_1\), \(B_{\alpha}\in G_{\beta}\). (\(G_\beta\) for \(\beta < \aleph_1\) is defined by induction, \(G_0=\{A: A\) is an open subset of \({\Bbb R}\}\) and for \(\beta > 0\), \(G_\beta =\left\{\bigcup^\infty_{i=0}A_{i}: (\forall i\in\omega) (\exists\xi <\beta)(A_i\in G_\xi)\,\right\}\) if \(\beta\) is even and \(G_\beta = \left\{\bigcap^\infty_{i=0}A_{i}: (\forall i\in\omega) (\exists \xi < \beta)(A_{i}\in G_\xi)\,\right\}\) if \(\beta\) is odd.)

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