Axiom Of Choice

We have the following indirect implication of form equivalence classes:

15 \(\Rightarrow\) 288-n given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
15 \(\Rightarrow\) 30	The Axiom of Choice, Jech, 1973b, page 53 problem 4.12
30 \(\Rightarrow\) 10	clear
10 \(\Rightarrow\) 288-n	clear

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

Howard-Rubin Number	Statement
15:	\(KW(\infty,\infty)\) (KW), The Kinna-Wagner Selection Principle: For every set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(\|A\|>1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 81(\(n\)).
30:	Ordering Principle: Every set can be linearly ordered.
10:	\(C(\aleph_{0},< \aleph_{0})\): Every denumerable family of non-empty finite sets has a choice function.
288-n:	If \(n\in\omega-\{0,1\}\), \(C(\aleph_0,n)\): Every denumerable set of \(n\)-element sets has a choice function.

Comment:

Back