Implications

We have the following indirect implication of form equivalence classes:

213 \(\Rightarrow\) 249
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
213 \(\Rightarrow\) 85 clear
85 \(\Rightarrow\) 32 clear
32 \(\Rightarrow\) 10 clear
10 \(\Rightarrow\) 249

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

Howard-Rubin Number Statement
213:

\(C(\infty,\aleph_{1})\): If \((\forall y\in X)(|y| = \aleph_{1})\) then \(X\) has a choice function.

85:

\(C(\infty,\aleph_{0})\):  Every family of denumerable sets has  a choice function.  Jech [1973b] p 115 prob 7.13.

32:

\(C(\aleph_0,\le\aleph_0)\): Every denumerable set of non-empty countable sets  has a choice function.

10:

\(C(\aleph_{0},< \aleph_{0})\):  Every denumerable family of non-empty finite sets has a choice function.

249:

If \(T\) is an infinite tree in which every element has exactly 2 immediate successors then \(T\) has an infinite branch.

Comment:

Back