Implications

We have the following indirect implication of form equivalence classes:

270 \(\Rightarrow\) 127
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
270 \(\Rightarrow\) 62 Restricted versions of the compactness theorem, Kolany, A. 1991, Rep. Math. Logic
62 \(\Rightarrow\) 61 clear
61 \(\Rightarrow\) 45-n clear
45-n \(\Rightarrow\) 64 Classes of Dedekind finite cardinals, Truss, J. K. 1974a, Fund. Math.
64 \(\Rightarrow\) 127 Amorphe Potenzen kompakter Raume, Brunner, N. 1984b, Arch. Math. Logik Grundlagenforschung

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

Howard-Rubin Number Statement
270:

\(CT_{\hbox{fin}}\): The compactness theorem for propositional logic restricted to sets of formulas in which each variable occurs only in a finite number of formulas.

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.

45-n:

If \(n\in\omega-\{0,1\}\), \(C(\infty,n)\): Every set of \(n\)-element sets has a choice function.

64:

\(E(I,Ia)\) There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.)

127:

An amorphous power of a compact \(T_2\) space, which as a set is well orderable, is well orderable.

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