Axiom Of Choice

We have the following indirect implication of form equivalence classes:

407 \(\Rightarrow\) 271-n 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 \(\Rightarrow\) 270	Restricted versions of the compactness theorem, Kolany, A. 1991, Rep. Math. Logic
270 \(\Rightarrow\) 271-n	Restricted versions of the compactness theorem, Kolany, A. 1991, Rep. Math. Logic

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.
270:	\(CT_{\hbox{fin}}\): The compactness theorem for propositional logic restricted to sets of formulas in which each variable occurs only in a finite number of formulas.
271-n:	If \(n\in\omega-\{0,1\}\), \(CT_{n}\): The compactness theorem for propositional logic restricted to sets of formulas in which each variable occurs in at most \(n\) formulas.

Comment:

Back