Axiom Of Choice

We have the following indirect implication of form equivalence classes:

398 \(\Rightarrow\) 401 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
398 \(\Rightarrow\) 400	clear
400 \(\Rightarrow\) 401	clear

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

Howard-Rubin Number	Statement
398:	\(KW(LO,\infty)\), The Kinna-Wagner Selection Principle for a linearly ordered family of sets: For every linearly ordered set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(\|A\|>1\) then \(\emptyset\neq f(A)\subsetneq A\).
400:	\(KW(LO,LO)\), The Kinna-Wagner Selection Principle for a linearly ordered set of linearly orderable sets: For every linearly ordered set of linearly orderable sets \(M\) there is a function \(f\) such that for all \(A\in M\), if \(\|A\|>1\) then \(\emptyset\neq f(A)\subsetneq A\).
401:	\(KW(LO,<\aleph_0)\), The Kinna-Wagner Selection Principle for a linearly ordered set of finite sets: For every linearly ordered set of finite 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