Axiom Of Choice

We have the following indirect implication of form equivalence classes:

407 \(\Rightarrow\) 308-p 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 \(\Rightarrow\) 49	A survey of recent results in set theory, Mathias, A.R.D. 1979, Period. Math. Hungar.
49 \(\Rightarrow\) 30	clear
30 \(\Rightarrow\) 62	clear
62 \(\Rightarrow\) 308-p	Maximal p-subgroups and the axiom of choice, Howard-Yorke-1987 [1987, Notre Dame J. Formal 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.
49:	Order Extension Principle: Every partial ordering can be extended to a linear ordering. Tarski [1924], p 78.
30:	Ordering Principle: Every set can be linearly ordered.
62:	\(C(\infty,< \aleph_{0})\): Every set of non-empty finite sets has a choice function.
308-p:	If \(p\) is a prime and if \(\{G_y: y\in Y\}\) is a set of finite groups, then the weak direct product \(\prod_{y\in Y}G_y\) has a maximal \(p\)-subgroup.

Comment:

Back