Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
384 \(\Rightarrow\) 14	"Maximal filters, continuity and choice principles", Herrlich, H. 1997, Quaestiones Math.
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
384:	Closed Filter Extendability for \(T_1\) Spaces: Every closed filter in a \(T_1\) topological space can be extended to a maximal closed filter.
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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