Implications

We have the following indirect implication of form equivalence classes:

7 \(\Rightarrow\) 80
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
7 \(\Rightarrow\) 9 On the existence of large sets of Dedekind cardinals, Tarski, A. 1965, Notices Amer. Math. Soc.
The Axiom of Choice, Jech, 1973b, page 161 problem 11.6
9 \(\Rightarrow\) 10 Zermelo's Axiom of Choice, Moore, 1982, 322
10 \(\Rightarrow\) 80 clear

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

Howard-Rubin Number Statement
7:

There is no infinite decreasing sequence of cardinals.

9:

Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.

10:

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

80:

\(C(\aleph_{0},2)\):  Every denumerable set of  pairs has  a  choice function.

Comment:

Back