We have the following indirect implication of form equivalence classes:
Implication | Reference |
---|---|
345 \(\Rightarrow\) 14 |
Topologie, Analyse Nonstandard et Axiome du Choix, Morillon, M. 1988, Universit\'e Blaise-Pascal |
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 |
---|---|
345: | Rasiowa-Sikorski Axiom: If \((B,\land,\lor)\) is a Boolean algebra, \(a\) is a non-zero element of \(B\), and \(\{X_n: n\in\omega\}\) is a denumerable set of subsets of \(B\) then there is a maximal filter \(F\) of \(B\) such that \(a\in F\) and for each \(n\in\omega\), if \(X_n\subseteq F\) and \(\bigwedge X_n\) exists then \(\bigwedge X_n \in F\). |
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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