Implications

We have the following indirect implication of form equivalence classes:

164 \(\Rightarrow\) 108
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 \(\Rightarrow\) 79 clear
79 \(\Rightarrow\) 94 clear
94 \(\Rightarrow\) 5 clear
5 \(\Rightarrow\) 38 Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.
38 \(\Rightarrow\) 108 clear

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.

79:

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

94:

\(C(\aleph_{0},\infty,{\Bbb R})\): Every denumerable family of non-empty sets of reals  has a choice function. Jech [1973b], p 148 prob 10.1.

5:

\(C(\aleph_0,\aleph_0,\Bbb R)\): Every denumerable set of non-empty denumerable subsets of \({\Bbb R}\) has a choice function.

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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