Implications

We have the following indirect implication of form equivalence classes:

407 \(\Rightarrow\) 241
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
407 \(\Rightarrow\) 14 Effective equivalents of the Rasiowa-Sikorski lemma, Bacsich, P. D. 1972b, J. London Math. Soc. Ser. 2.
14 \(\Rightarrow\) 233 Algebraic closures without choice, Banaschewski, B. 1992, Z. Math. Logik Grundlagen Math.
233 \(\Rightarrow\) 242 clear
242 \(\Rightarrow\) 241 clear

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

Howard-Rubin Number Statement
407:

Let \(B\) be a Boolean algebra, \(b\) a non-zero element of \(B\) and \(\{A_i: i\in\omega\}\) a sequence of subsets of \(B\) such that for each \(i\in\omega\), \(A_i\) has a supremum \(a_i\). Then there exists an ultrafilter \(D\) in \(B\) such that \(b\in D\) and, for each \(i\in\omega\), if \(a_i\in D\), then \(D\cap\ A_i\neq\emptyset\).

14:

BPI: Every Boolean algebra has a prime ideal.

233:

Artin-Schreier theorem: If a field has an algebraic closure it is unique up to isomorphism.

242:

There is, up to an isomorphism, at most one algebraic closure of \({\Bbb Q}\).

241:

Every algebraic closure of \(\Bbb Q\) has a real closed subfield.

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