Axiom Of Choice

We have the following indirect implication of form equivalence classes:

260 \(\Rightarrow\) 18 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
260 \(\Rightarrow\) 40	Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic
40 \(\Rightarrow\) 39	clear
39 \(\Rightarrow\) 8	clear
8 \(\Rightarrow\) 9	Was sind und was sollen die Zollen?, Dedekind, [1888]
9 \(\Rightarrow\) 17	The independence of Ramsey's theorem, Kleinberg, E.M. 1969, J. Symbolic Logic
17 \(\Rightarrow\) 18	Ramsey's theorem in the hierarchy of choice principles, Blass, A. 1977a, J. Symbolic Logic The Axiom of Choice, Jech, 1973b, page 164 problem 11.20

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

Howard-Rubin Number	Statement
260:	\(Z(TR\&C,P)\): If \((X,R)\) is a transitive and connected relation in which every partially ordered subset has an upper bound, then \((X,R)\) has a maximal element.
40:	\(C(WO,\infty)\): Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325.
39:	\(C(\aleph_{1},\infty)\): Every set \(A\) of non-empty sets such that \(\vert A\vert = \aleph_{1}\) has a choice function. Moore, G. [1982], p. 202.
8:	\(C(\aleph_{0},\infty)\):
9:	Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.
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.
18:	\(PUT(\aleph_{0},2,\aleph_{0})\): The union of a denumerable family of pairwise disjoint pairs has a denumerable subset.

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