Axiom Of Choice

We have the following indirect implication of form equivalence classes:

261 \(\Rightarrow\) 316 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
261 \(\Rightarrow\) 256	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
256 \(\Rightarrow\) 259	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
259 \(\Rightarrow\) 51	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
51 \(\Rightarrow\) 316	Well ordered subsets of linearly ordered sets, Howard, P. 1994, Notre Dame J. Formal Logic

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

Howard-Rubin Number	Statement
261:	\(Z(TR,T)\): Every transitive relation \((X,R)\) in which every subset which is a tree has an upper bound, has a maximal element.
256:	\(Z(P,F)\): Every partially ordered set \((X,R)\) in which every forest \(A\) has an upper bound, has a maximal element.
259:	\(Z(TR\&C,W)\): If \((X,R)\) is a transitive and connected relation in which every well ordered subset has an upper bound, then \((X,R)\) has a maximal element.
51:	Cofinality Principle: Every linear ordering has a cofinal sub well ordering. Sierpi\'nski [1918], p 117.
316:	If a linearly ordered set \((A,\le)\) has the fixed point property then \((A,\le)\) is complete. (\((A,\le)\) has the fixed point property if every function \(f:A\to A\) satisfying \((x\le y \Rightarrow f(x)\le f(y))\) has a fixed point, and (\((A,\le)\) is complete if every subset of \(A\) has a least upper bound.)

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