Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
303 \(\Rightarrow\) 50	Some propositions equivalent to the Sikorski extension theorem for Boolean algebras, Bell, J.L. 1988, Fund. Math.
50 \(\Rightarrow\) 14	A survey of recent results in set theory, Mathias, A.R.D. 1979, Period. Math. Hungar.
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
303:	If \(B\) is a Boolean algebra, \(S\subseteq B\) and \(S\) is closed under \(\land\), then there is a \(\subseteq\)-maximal proper ideal \(I\) of \(B\) such that \(I\cap S= \{0\}\).
50:	Sikorski's Extension Theorem: Every homomorphism of a subalgebra \(B\) of a Boolean algebra \(A\) into a complete Boolean algebra \(B'\) can be extended to a homomorphism of \(A\) into \(B'\). Sikorski [1964], p. 141.
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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