Axiom Of Choice

We have the following indirect implication of form equivalence classes:

49 \(\Rightarrow\) 342-n given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
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 \(\Rightarrow\) 342-n	clear

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

Howard-Rubin Number	Statement
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.
342-n:	(For \(n\in\omega\), \(n\ge 2\).) \(PC(\infty,n,\infty)\): Every infinite family of \(n\)-element sets has an infinite subfamily with a choice function. (See Form 166.)

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