Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
89 \(\Rightarrow\) 90 The Axiom of Choice, Jech, 1973b, page 133
90 \(\Rightarrow\) 91 The Axiom of Choice, Jech, 1973b, page 133
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
89:

Antichain Principle:  Every partially ordered set has a maximal antichain. Jech [1973b], p 133.

90:

\(LW\):  Every linearly ordered set can be well ordered. Jech [1973b], p 133.

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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