Implications

We have the following indirect implication of form equivalence classes:

44 \(\Rightarrow\) 362
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
44 \(\Rightarrow\) 39 The Axiom of Choice, Jech, 1973b, page 120 theorem 8.1
39 \(\Rightarrow\) 8 clear
8 \(\Rightarrow\) 361 Zermelo's Axiom of Choice, Moore, 1982, page 325
361 \(\Rightarrow\) 362 Zermelo's Axiom of Choice, Moore, 1982, page 325

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 \((\forall\beta < \aleph_{1}) (\{f(\gamma ): \gamma < b \} \mathrel R f(\beta))\).

39:

\(C(\aleph_{1},\infty)\): Every set \(A\) of non-empty sets such that \(\vert A\vert = \aleph_{1}\) has a choice function. Moore, G. [1982], p. 202.

8:

\(C(\aleph_{0},\infty)\):

361:

In \(\Bbb R\), the union of a denumerable number of analytic sets is analytic. G. Moore [1982], pp 181 and 325.

362:

In \(\Bbb R\), every Borel set is analytic. G. Moore [1982], pp 181 and 325.

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