Implications

We have the following indirect implication of form equivalence classes:

91 \(\Rightarrow\) 108
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
91 \(\Rightarrow\) 37 Equivalents of the Axiom of Choice II, Rubin, 1985, theorem 5.7
37 \(\Rightarrow\) 38 L’axiome de M. Zermelo et son rˆole dans la th´eorie des ensembles et l’analyse, Sierpi'nski, W. 1918, Bull. Int. Acad. Sci. Cracovie Cl. Math. Nat.
38 \(\Rightarrow\) 108 clear

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
91:

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

37:

Lebesgue measure is countably additive.

38:

\({\Bbb R}\) is not the union of a countable family of countable sets.

108:

There is an ordinal \(\alpha\) such that \(2^{\aleph _{\alpha}}\) is not the union of a denumerable set of denumerable sets.

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