Implications

We have the following indirect implication of form equivalence classes:

264 \(\Rightarrow\) 364
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
264 \(\Rightarrow\) 202 Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic
202 \(\Rightarrow\) 91 note-75
91 \(\Rightarrow\) 363 Equivalents of the Axiom of Choice II, Rubin, 1985, theorem 5.7
363 \(\Rightarrow\) 364 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
264:

\(H(C,P)\): Every connected relation \((X,R)\) contains a \(\subseteq\)-maximal partially ordered set.

202:

\(C(LO,\infty)\): Every linearly ordered family of non-empty sets has  a choice function.

91:

\(PW\):  The power set of a well ordered set can be well ordered.

363:

There are exactly \(2^{\aleph_0}\) Borel sets in \(\Bbb R\). G. Moore [1982], p 325.

364:

In \(\Bbb R\), there is a measurable set that is not Borel.  G. Moore [1982], p 325.

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