Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
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
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.)

Comment:

Back