Axiom Of Choice

We have the following indirect implication of form equivalence classes:

317 \(\Rightarrow\) 72 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
317 \(\Rightarrow\) 14	Limitations on the Fraenkel-Mostowski method of independence proofs, Howard, P. 1973, J. Symbolic Logic
14 \(\Rightarrow\) 72	Prime ideal theorems for Boolean algebras and the axiom of choice, Tarski, A. 1954b, Bull. Amer. Math. Soc.

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

Howard-Rubin Number	Statement
317:	Weak Sikorski Theorem: If \(B\) is a complete, well orderable Boolean algebra and \(f\) is a homomorphism of the Boolean algebra \(A'\) into \(B\) where \(A'\) is a subalgebra of the Boolean algebra \(A\), then \(f\) can be extended to a homomorphism of \(A\) into \(B\).
14:	BPI: Every Boolean algebra has a prime ideal.
72:	Artin-Schreier Theorem: Every field in which \(-1\) is not the sum of squares can be ordered. (The ordering, \(\le \), must satisfy (a) \(a\le b\rightarrow a + c\le b + c\) for all \(c\) and (b) \(c\ge 0\) and \(a\le b\rightarrow a\cdot c\le b\cdot c\).)

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