Axiom Of Choice

We have the following indirect implication of form equivalence classes:

323 \(\Rightarrow\) 132 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
323 \(\Rightarrow\) 62	note-70
62 \(\Rightarrow\) 132	Sequential compactness and the axiom of choice, Brunner, N. 1983b, Notre Dame J. Formal Logic

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

Howard-Rubin Number	Statement
323:	\(KW(\infty,WO)\), The Kinna-Wagner Selection Principle for a family of well orderable sets: For every set \(M\) of well orderable sets there is a function \(f\) such that for all \(A\in M\), if \(\|A\| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 15.)
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.

Comment:

Back