Axiom Of Choice

We have the following indirect implication of form equivalence classes:

392 \(\Rightarrow\) 403 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
392 \(\Rightarrow\) 393	clear
393 \(\Rightarrow\) 403	clear

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

Howard-Rubin Number	Statement
392:	\(C(LO,LO)\): Every linearly ordered set of linearly orderable sets has a choice function.
393:	\(C(LO,WO)\): Every linearly ordered set of non-empty well orderable sets has a choice function.
403:	\(KW(LO,WO)\), The Kinna-Wagner Selection Principle for a linearly ordered set of well orderable sets: For every linearly ordered set of well orderable sets \(M\) there is a function \(f\) such that for all \(A\in M\), if \(\|A\|>1\) then \(\emptyset\neq f(A)\subsetneq A\).

Comment:

Back