Axiom Of Choice

We have the following indirect implication of form equivalence classes:

292 \(\Rightarrow\) 185 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
292 \(\Rightarrow\) 90	The axiom of choice and linearly ordered sets, Howard, P. 1977, Fund. Math.
90 \(\Rightarrow\) 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 \(\Rightarrow\) 77	Well ordered subsets of linearly ordered sets, Howard, P. 1994, Notre Dame J. Formal Logic
77 \(\Rightarrow\) 185	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
292:	\(MC(LO,\infty)\): For each linearly ordered family of non-empty sets \(X\), there is a function \(f\) such that for all \(x\in X\) \(f(x)\) is non-empty, finite subset of \(x\).
90:	\(LW\): Every linearly ordered set can be well ordered. Jech [1973b], p 133.
51:	Cofinality Principle: Every linear ordering has a cofinal sub well ordering. Sierpi\'nski [1918], p 117.
77:	A linear ordering of a set \(P\) is a well ordering if and only if \(P\) has no infinite descending sequences. Jech [1973b], p 23.
185:	Every linearly ordered Dedekind finite set is finite.

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