Implications

We have the following indirect implication of form equivalence classes:

44 \(\Rightarrow\) 365
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
44 \(\Rightarrow\) 41 The Axiom of Choice, Jech, 1973b, page 120 theorem 8.1
41 \(\Rightarrow\) 365 Zermelo's Axiom of Choice, Moore, 1982, table 5

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

Howard-Rubin Number Statement
44:

\(DC(\aleph _{1})\):  Given a relation \(R\) such that for every  subset \(Y\) of a set \(X\) with \(|Y| < \aleph_{1}\) there is an \(x \in  X\)  with \(Y \mathrel R x\), then there is a function \(f: \aleph_{1} \rightarrow  X\) such that \((\forall\beta < \aleph_{1}) (\{f(\gamma ): \gamma < b \} \mathrel R f(\beta))\).

41:

\(W_{\aleph _{1}}\): For every cardinal \(m\), \(m \le \aleph_{1}\) or \(\aleph_{1}\le m \).

365:

For every uncountable set \(A\), if \(A\) has the same cardinality as each of its uncountable subsets then \(|A| = \aleph_1\).

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