Implications

We have the following indirect implication of form equivalence classes:

133 \(\Rightarrow\) 84
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 \(\Rightarrow\) 77 Well ordered subsets of linearly ordered sets, Howard, P. 1994, Notre Dame J. Formal Logic
77 \(\Rightarrow\) 185 Well ordered subsets of linearly ordered sets, Howard, P. 1994, Notre Dame J. Formal Logic
185 \(\Rightarrow\) 84 Well ordered subsets of linearly ordered sets, Howard, P. 1994, Notre Dame J. Formal Logic

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.

77:

A linear ordering of a set \(P\) is a well ordering if and only if \(P\) has no infinite descending sequences. Jech [1973b], p 23.

185:

Every linearly ordered Dedekind finite set is finite.

84:

\(E(II,III)\) (Howard/Yorke [1989]): \((\forall x)(x\) is \(T\)-finite  if and only if \(\cal P(x)\) is Dedekind finite).

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