Implications

We have the following indirect implication of form equivalence classes:

270 \(\Rightarrow\) 64
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 \(\Rightarrow\) 11 clear
11 \(\Rightarrow\) 12 clear
12 \(\Rightarrow\) 336-n clear
336-n \(\Rightarrow\) 64 Weak choice principles, De la Cruz, O. 1998a, Proc. Amer. Math. Soc.

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.

11:

A Form of Restricted Choice for Families of Finite Sets: For every infinite set \(A\), \(A\) has an infinite subset \(B\) such that for every \(n\in\omega\), \(n>0\), the set of all \(n\) element subsets of \(B\) has a choice function. De la Cruz/Di Prisco [1998b]

12:

A Form of Restricted Choice for Families of Finite Sets: For every infinite set \(A\) and every \(n\in\omega\), there is an infinite subset \(B\) of \(A\) such the set of all \(n\) element subsets of \(B\) has a choice function. De la Cruz/Di Prisco} [1998b]

336-n:

(For \(n\in\omega\), \(n\ge 2\).)  For every infinite set \(X\), there is an infinite \(Y \subseteq X\) such that the set of all \(n\)-element subsets of \(Y\) 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.)

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