Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
164 \(\Rightarrow\) 91 Dedekind-Endlichkeit und Wohlordenbarkeit, Brunner, N. 1982a, Monatsh. Math.
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
164:

Every non-well-orderable set has an infinite subset with a Dedekind finite power set.

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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