Implications

We have the following indirect implication of form equivalence classes:

17 \(\Rightarrow\) 374-n
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
17 \(\Rightarrow\) 132 Amorphe Potenzen kompakter Raume, Brunner, N. 1984b, Arch. Math. Logik Grundlagenforschung
132 \(\Rightarrow\) 10 Amorphe Potenzen kompakter Raume, Brunner, N. 1984b, Arch. Math. Logik Grundlagenforschung
10 \(\Rightarrow\) 423 clear
423 \(\Rightarrow\) 374-n clear

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

Howard-Rubin Number Statement
17:

Ramsey's Theorem I: If \(A\) is an infinite set and the family of all 2 element subsets of \(A\) is partitioned into 2 sets \(X\) and \(Y\), then there is an infinite subset \(B\subseteq A\) such that all 2 element subsets of \(B\) belong to \(X\) or all 2 element subsets of \(B\) belong to \(Y\). (Also, see Form 325.), Jech [1973b], p 164 prob 11.20.

132:

\(PC(\infty, <\aleph_0,\infty)\):  Every infinite family of finite  sets has an infinite subfamily with a choice function.

10:

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

423:

\(\forall n\in \omega-\{o,1\}\), \(C(\aleph_0, n)\) : For every \(n\in  \omega - \{0,1\}\), every denumerable set of \(n\) element sets has a choice function.

374-n:

\(UT(\aleph_0,n,\aleph_0)\) for \(n\in\omega -\{0,1\}\): The union of a denumerable set of \(n\)-element sets is denumerable.

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