Implications

We have the following indirect implication of form equivalence classes:

77 \(\Rightarrow\) 84
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
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
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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