Axiom Of Choice

We have the following indirect implication of form equivalence classes:

316 \(\Rightarrow\) 13 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
316 \(\Rightarrow\) 77	"Representing multi-algebras by algebras, the axiom of choice and the axiom of dependent choice", Howard, P. 1981, Algebra Universalis
77 \(\Rightarrow\) 185	Well ordered subsets of linearly ordered sets, Howard, P. 1994, Notre Dame J. Formal Logic
185 \(\Rightarrow\) 13	clear

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

Howard-Rubin Number	Statement
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.)
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.
13:	Every Dedekind finite subset of \({\Bbb R}\) is finite.

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