Implications

We have the following indirect implication of form equivalence classes:

89 \(\Rightarrow\) 251
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 \(\Rightarrow\) 79 clear
79 \(\Rightarrow\) 251 consequence of the axiom of choice, Ash, C. J. 1975, J. Austral. Math. Soc. Ser. A.

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.

79:

\({\Bbb R}\) can be well ordered.  Hilbert [1900], p 263.

251:

The additive groups \(({\Bbb R},+)\) and \(({\Bbb C},+)\) are  isomorphic.

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