Implications

We have the following indirect implication of form equivalence classes:

391 \(\Rightarrow\) 61
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
391 \(\Rightarrow\) 399 clear
399 \(\Rightarrow\) 323 clear
323 \(\Rightarrow\) 62 note-70
62 \(\Rightarrow\) 61 clear

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

Howard-Rubin Number Statement
391:

\(C(\infty,LO)\): Every set of non-empty linearly orderable sets has a choice function.

399:

\(KW(\infty,LO)\), The Kinna-Wagner Selection Principle for a set of linearly orderable sets: For every set of linearly orderable sets \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\).

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.

61:

\((\forall n\in\omega, n\ge 2\))\((C(\infty,n))\): For each \(n\in\omega\), \(n\ge 2\), every set of \(n\) element  sets has a choice function.

Comment:

Back