Implications

We have the following indirect implication of form equivalence classes:

36 \(\Rightarrow\) 276
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
36 \(\Rightarrow\) 62 On Loeb and weakly Loeb Hausdorff spaces, Tachtsis, E. 2000, Math. Japon.
62 \(\Rightarrow\) 61 clear
61 \(\Rightarrow\) 88 clear
88 \(\Rightarrow\) 276

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

Howard-Rubin Number Statement
36:

Compact T\(_2\) spaces are Loeb. (A space is Loeb if the set of non-empty closed sets has a choice function.)

62:

\(C(\infty,< \aleph_{0})\):  Every set of non-empty finite  sets  has  a choice function.

61:

\((\forall n\in\omega, n\ge 2\))\((C(\infty,n))\): For each \(n\in\omega\), \(n\ge 2\), every set of \(n\) element  sets has a choice function.

88:

  \(C(\infty ,2)\):  Every family of pairs has a choice function.

276:

\(E(V'',III)\): For every set \(A\), \({\cal P}(A)\) is Dedekind finite if and only if \(A = \emptyset\)  or \(2|{\cal P}(A)| > |{\cal P}(A)|\). \ac{Howard/Spi\u siak} \cite{1994}.

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