Implications

We have the following indirect implication of form equivalence classes:

133 \(\Rightarrow\) 92
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 \(\Rightarrow\) 51 Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic
51 \(\Rightarrow\) 337 Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.
337 \(\Rightarrow\) 92 clear

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.

51:

Cofinality Principle: Every linear ordering has a cofinal sub well ordering.  Sierpi\'nski [1918], p 117.

337:

\(C(WO\), uniformly linearly ordered):  If \(X\) is a well ordered collection of non-empty sets and there is a function \(f\) defined on \(X\) such that for every \(x\in X\), \(f(x)\) is a linear ordering of \(x\), then there is a choice function for \(X\).

92:

\(C(WO,{\Bbb R})\):  Every well ordered family of non-empty subsets of \({\Bbb R}\) has a choice function.

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