Implications

We have the following indirect implication of form equivalence classes:

133 \(\Rightarrow\) 6
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
133 \(\Rightarrow\) 90 Dedekind-Endlichkeit und Wohlordenbarkeit, Brunner, N. 1982a, Monatsh. Math.
90 \(\Rightarrow\) 91 The Axiom of Choice, Jech, 1973b, page 133
91 \(\Rightarrow\) 79 clear
79 \(\Rightarrow\) 94 clear
94 \(\Rightarrow\) 6 Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.

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

Howard-Rubin Number Statement
133:  

Every set is either well orderable or has an infinite amorphous subset.

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.

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.

6:

\(UT(\aleph_0,\aleph_0,\aleph_0,\Bbb R)\): The union of a denumerable  family  of denumerable subsets of \({\Bbb R}\) is denumerable.

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