Implications

We have the following indirect implication of form equivalence classes:

15 \(\Rightarrow\) 82
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
15 \(\Rightarrow\) 82 Models of \(ZF\) with the same sets of ordinals, Monro, G.P. 1972, Notices Amer. Math. Soc.

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

Howard-Rubin Number Statement
15:

\(KW(\infty,\infty)\) (KW), The Kinna-Wagner Selection Principle: For every  set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 81(\(n\)).  

82:

\(E(I,III)\) (Howard/Yorke [1989]): If \(X\) is infinite then \(\cal P(X)\) is Dedekind infinite. (\(X\) is finite \(\Leftrightarrow {\cal P}(X)\) is Dedekind finite.)

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