Axiom Of Choice

We have the following indirect implication of form equivalence classes:

384 \(\Rightarrow\) 73 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
384 \(\Rightarrow\) 14	"Maximal filters, continuity and choice principles", Herrlich, H. 1997, Quaestiones Math.
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 \(\Rightarrow\) 132	Sequential compactness and the axiom of choice, Brunner, N. 1983b, Notre Dame J. Formal Logic
132 \(\Rightarrow\) 73	clear

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

Howard-Rubin Number	Statement
384:	Closed Filter Extendability for \(T_1\) Spaces: Every closed filter in a \(T_1\) topological space can be extended to a maximal closed filter.
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.
132:	\(PC(\infty, <\aleph_0,\infty)\): Every infinite family of finite sets has an infinite subfamily with a choice function.
73:	\(\forall n\in\omega\), \(PC(\infty,n,\infty)\): For every \(n\in\omega\), if \(C\) is an infinite family of \(n\) element sets, then \(C\) has an infinite subfamily with a choice function. De la Cruz/Di Prisco [1998b]

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